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Monthly Archives: April 2013Whenever I'm feeling lonely or like my life is insignificant, I like to think of the connections I have to people past and present. There are certain undeniably universal aspects to everyone's life, and if we can only keep these things in mind, we can remind ourselves that we are connected to everyone who has ever lived or ever will. I've compiled a list of sentences that nearly every person has said or thought in some form, in some language, at some point in their lives. Hopefully this will inspire you to feel the connection you have to all humans throughout time.
1. I'm hungy.
2. I'm thirsty
3. I'm horny.
4. Ouch.
5. Ahhhhhh!!!
6. Whoops.
7. Fuck it's hot/cold!
8. I'm tired.
9. I wonder if [person] likes me.
10. I have to take a shit.
So there you go! Hope that was inspiring. Feel free to add to the list! illusion But yourThis morning, the wildly popular Facebook blog, I Fucking Love Science, posted the above image. The blog, probably one of the most popular on Facebook, is getting a lot of attention recently due to the (somehow shocking?) revelation that it is run by a young woman. It's a wonderful blog that uses little-known scientific facts to blow people's minds on a daily basis. And though usually I enjoy their posts, I woke up to this one, and felt a little disappointed. The image explains the nature of the irrational number Pi, and goes on to describe its magical wonder at having an infinite, non-repeating series of integers. Though most of the information she gives is theoretically true, it really has more to do with the nature of infinity than the magic of the number pi, which really isn't all that special.
To see why, let's think about what pi actually is. Pi is simply a number. It's the ratio of any circle's circumference to its diameter. And while it is somewhat fascinating that every circle ever has this same ratio, it is more a result of the definition of a circle, than a miraculous coincidence. As discussed in my post, The Limits of Language, science and mathematics are simply the languages we use to understand the universe. If we forget to think about them that way, then when these coincidences and patterns crop up, we assume they are miracles of the universe and attribute special meaning to them, when they are simply anomalies in our language system. Take pi for example. There is a constant number that exists that defines the ratio of any circle's circumference to its diameter, but because that number doesn't fit nicely into our number system, we have to make up a symbol for it. If we try to define that number using our numerical language, it yields the irrational number that the Facebook blog considers so magical, when it is really just a problem of translation. And, yes, it is true that in an infinite, non-repeating, random series of integers, theoretically every combination of integers will exist, but this is more a result of the series being infinite, than anything that has to do with pi specifically.
Another reason that pi is not special (sorry bud) is that there are literally an infinite amount of irrational numbers. In fact, as Georg Cantor proved, and as can easily be seen, there are actually more irrational numbers than rational ones. Once again, this goes back to how we define our numbers. Numbers exist on a scale, much like wavelengths of light. This scale can be, for lack of a better term, 'zoomed in' on to the nth degree (meaning infinitely). So between any two integers, 3 and 4 for example, there are an infinite amount of numbers that we cannot fully represent using the number system that we've created, and each one of them translates into an infinite, non-repeating decimal that has the same characteristics of pi.
So yes, infinity as a concept is really something special, but it is simply that: a concept. Though we can represent it and theorize about it and try to define its characteristics, it merely exists in our own minds. |
Studiosity BlogThere are a number of questions our Maths Specialists see crop up again and again, ranging from simple tricks like finding the x and y intercepts of an equation, to more advanced concepts like solving quadratic inequalities. So we thought it would be helpful to share these with you, along with tips on how to find the solutions.
One of our expert Maths Specialists, Jemma, shares the top five questions she sees, the most common issues and mistakes students make, and her tips on how you can work around them.
I tutored maths for a few years, and there were two basic skills I wished every high school student had nailed down.
The first is the ability to quickly and correctly arrange an algebraic equation; it's just too useful, and can save a lot of time on exams.The second is the ability to spot a wrong answer. There's nothing more frustrating than seeing a student finish an exam with 10 minutes to go, and knowing they have some mistakes on their paper, if only they knew how to find them.
Pi. It's 3.14159...ish? Or is it 3.14159265359 and so on and so forth?
As tomorrow is Pi Day, an annual celebration of the mathematical constant π (Pi), we wanted to share our thoughts on the significance of the number and what you can do to celebrate |
IngridBirdy
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Liefde voor Science-fiction en Romantiek in alle soorten en maten gecombineerd.
"There are infinite numbers between
zero and one. There's point-one,
point-one-two and point-one-one-two
and an infinite collection of others.
Of course, there's a bigger infinite
between zero and two or between
zero and a million. Some infinities
are simply bigger then other infinities." |
Monday, June 23, 2014
The first book I reviewed was Euler: The Master of Us All William Dunham. I liked how the author presented the book so I decided to read (skim) another one of his books, Journey Through genius. The book is a nice read, it gives a broad view of the some of the greatest mathematicians, and their works, who ever lived. The book reads in chronological order, so you get a taste of each generation of mathematicians. This means the book starts out at around 300 B.C where was still young. Therefore, the first five chapters talk a lot about old school geometry, which I'm not really a fan of so I skimmed through those pretty fast. Though, the first chapters talks a little bit about transcendental numbers (numbers that are not solutions to any polynomial equation with integer coefficients); so I read that and it was pretty interesting. Then chapter 6 discusses Cardano and his cubic formula. It's a good chapter, but the first book I read discussed Cardano pretty extensively so I didn't spend to much time on chapter 6.
No book about mathematical, or scientific, geniuses would be complete without Issac Newton (chapter 7). So in that chapter you get to learn more about him and his works. The author gives a good explanation of Newton's binomial theorem, which is so widely used in many areas of mathematics; so it's worth it to read and understand his explanation.
Then the author gives two good chapters on Leonhard Euler himself. I pretty much knew everything he had to say in them; but, if you haven't read about Euler, the two chapters should convince you to learn more about him (and read Euler: The Master of Us All). Now comes the good stuff, chapter 11: The Non-Denumerability of the Continuum. One of my favorite subjects in math is infinity. This chapter discusses Georg Cantor and his great work on infinite sets. If you don't know much on cardinality, you will love this chapter. The concepts of countability and different levels of infinity is so mind blowing, I just love Cantor's work. Though, many mathematicians of his time thought he was crazy for coming up with this stuff. I just want to give them a good punch in stomach.
Then we have chapter 12, which is essentially a continuation of chapter 11, which means it's awesome. This chapter has Cantor's proof that the cardinality of any set A is always smaller than the cardinality of the power set of A. This theorem is powerful because it works for infinite sets as well. Though, the actual proof in the book was long and a bit difficult to understand. Georg Cantor was like Euler in a way, he helped other mathematicians not be so scared of infinity, as Euler did with imaginary numbers.
Overall, this is a fine read for anyone interested in mathematics, and science overall. William Dunham touches upon pretty much all my favorite (and maybe yours) mathematicians. He gives a number of proofs, so there is plenty to learn in this book; it's not just a leisure read. I would actually recommend giving this book to any freshman math or science major; it would inspire them to keep working hard in their field. Personally, I would read any book by William Dunham, he's a good author.
Sunday, June 22, 2014
For my final weekly post, we are going to have some fun. I will be trying to interpret the meaning of David Hilbert's 9th problem the best I can. But, before I state the actual problem, I would like to build up the terminology and appropriate knowledge required to understand it. When I type in a different color, that just signifies my thoughts and opinions on the statement at hand. So first, observe the following table I took from Wikipedia. Here, f(n) = n^2 - 5 and the column after f(n) gives the prime factorizations of f(n).
Whats interesting about this table is that for the prime factorizations, other than the numbers 2,5, the prime numbers that appear as factors end in either a 1 or 9. There is a cool way to state this: the primes q for which there exists an n such that n^2 = 5 (mod q) are precisely 2, 5, and those primes q such that q = 1 (mod 5) (so q = 11, 31, 41 ...) or q = 4 (mod 5) (q = 19, 29, 89, ...). The law of quadratic reciprocity gives something similar like the example above of prime divisors of f(n) = n^2 − c for any integer c. So how about we state that law, with my thoughts in blue. Law of quadratic reciprocity:
The statement of the theorem is almost directly from my first reference at the bottom. This theorem seems pretty confusing. So how about we explain it by using an example; naturally, we will use the example I stated at the beginning of the blog. In that example, p = 5, which means p = 1 (mod 4) so we use the (i) statement of the reciprocity law. Also, it was given in the example that n^2 = 5 (mod q) so we know p (which is 5) is a square mod q. Then by the law of quadratic reciprocity part (i), we can see that q is a square mod 5, which means q = n^2 (mod 5). Now what does this mean? It means that q is a prime where q = 1^2 (mod 5), or q = 2^2 (mod 5). This means q = 1 (mod 5) (so q = 11, 31, 41, ...) or q = 4 (mod 5) (so q = 19, 29, 89, ...) and there you go, we used the reciprocity law to get the same results as in our example.
But, your thinking, HOLD ON KYLE, what about q = 0 (mod 5), q = 9 (mod 5), q = 16 (mod 5), q = 25 (mod 5), and so on. Well first, if q = 0 (mod 5) then q = 0,5,10,15, ...; where none of those numbers will ever be prime because they are multiples of 5, so we can throw that case out. But, for q = 9 (mod 5), we have q = 9 + 5c = 1 + 5 + 5c = 1 + 5(1+c) so q = 1 (mod 5), but that would mean q = 9 (mod 5) is redundant. I could go on with other cases, but how about not. Recall in MTH 310 when we had congruence classes. I won't go to far in depth, because every math major has to take the class anyway, but it's known that if a = b (mod 5), then a is congruent to either: 0 (mod 5), 1 (mod 5), 2 (mod 5), 3 (mod 5), or 4 (mod 5), there is no need to go on because each case will just revert back to the 5 cases stated above, ie. redundancy. The 5 cases above are known as the congruence classes of Z (mod 5). But, now you are saying: Kyle, what if we hit a case in which q = n^2 (mod 5) where it reverts back to cases 2 (mod 5), or 3 (mod 5). That won't happen though. If you want to know why, refer to my daily 13. Now back to my explanation of the Law of quadratic reciprocity.
So we went through statement (i), so then what about (ii)? Well if you understood my explanation of (i), then surely you can understand (ii). So there you go, you have a very rough understanding of the law of reciprocity. There are actually many ways to state this law: Euler, Lagrange, and Gauss had some I found them more confusing than the way stated above. Gauss is the one who actually proved the law. Now let's switch gears and talk about something a bit different.
So from Math 310, we all remember rings: sets with 2 binary operations that satisfy basic properties, like the set of rational numbers with addition and multiplication. Then a field was a ring with a few more properties: multiplication was commutative, had an identity, and each element had an inverse. Again, the rational numbers are an example of a field. Now let A be a field and let B be a field that contains A and has the same operations as A. Then B would be a field extension of A. So then with the field Q (set of rationals), the set L = {a + b*sqrt(5), where a,b are rationals} would be a field extension of Q. Also, the term algebraic number field is any finite field extension of Q.
So gathered with all this information, here is David Hilbert's 9th problem:
Find the most general law of the reciprocity theorem in any algebraic number field.
So this problem is like the law of reciprocity, except that we are working in a more general sense: not just rational numbers, but any field extension of the raitionals. Sadly, this problem has been partially. Luckily the guy supposedly only solved the law of reciprocity for abelian extensions (another confusing concept) of the rationals. So the non-abelian case is up for grabs. Now that you understand the problem you can go solve it, Good Luck!
Sunday, June 15, 2014
For my weekly 6 assignment, I will give a biography on Marie-Sophie Germain, along with some of my thoughts. Sophie was born on April 1, 1776 in Paris, France. Sophie's parent's house was actually a meeting place for those interested in liberal reforms. That would be interesting to have an upbringing like; it probably influenced Sophie greatly as a kid. In her teen years, Sophie was able to teach herself Latin and Greek. She also read Newton and Euler at night while under blankets as her parents were sleeping. They took away her fire, her light and her clothes in an attempt to get her away from books. Wow, if my parents did that, I probably would never have even pursued math. Anyways, her parents did lessen their opposition to her studying the sciences. What I found interesting was that her father actually supported her financially throughout her life, even though she never really had a well paying job. So maybe we (as in me) shouldn't be so quick to judge her parents.
At the end of some of Lagrange's (we should all know who he is) lecture course on analysis, using the pseudonym M. LeBlanc, Sophie submitted a paper that even made Lagrange look for its author. When Lagrange found out Sophie was a woman, he still respected her work and would eventually become her sponsor and mathematical counselor. Sophie collaborated with many mathematicians, but the most notable is Karl Friedrich Gauss. Between 1804 and 1809, she wrote several letters to him, again taking M. LeBlanc as her name. Gauss gave her tons of praise for her number theory, which is amazing because he was a highly intelligent schmuck. When Gauss did find out about her true identity he gave her even more praise, for learning science even with society's harsh gender roles at that time. One of Germain's most famous papers was her work on Fermat's last theorem in where she broke new ground and used divisibility as an attempt to prove Fermat's Last theorem.
Then came the Institut de France prize competition which brought about the following challenge:
formulate a mathematical theory of elastic surfaces and indicate just how it agrees with empirical evidence.
Most mathematicians didn't even try to solve the problem. But, Germain spent the next decade attempting to derive a theory of elasticity, collaborating with some of the most famous mathematicians and physicists of her time. Sadly, she did not win this time. Her hypothesis was not formed from the principles of physics, nor did she have any training in analysis or calculus (which was important in solving the problem). Finally, Germain's third attempt was deemed worthy of the prize: one kilogram of gold. Though, to public disappointment, she did not receive the prize. She thought the Judges did not fully appreciate her work, which probably was true. I bet if a man submitted her work, he would have one the prize, so unfair. In an attempt to extend her research, Sophie submitted a paper in 1825 to a commission of the Institut de France, whose members included Poisson, Gaspard de Prony, and Laplace. Her work suffered from a number of deficiencies (which probably could have been avoided if she had the proper training, which was inaccessible to her), but rather than reporting them to the author, the commission ignored the paper. It was recovered from de Prony's papers and published in 1880. Wow that just makes me mad that they just ignored her. It sounds like her research was pretty significant since de Pony's kept it. Sadly, Germain got breast cancer in 1829, but still she completed papers on number theory and on the curvature of surfaces (1831). Even on her death certificate in 1831, she was not even listed as a mathematician or scientist, bullcrap.
Sophie Germain was a very strong person. She never caught a break and her work was insulted left and right. I think that for any class, or book, that discusses woman's rights and the history overall of woman, Germain's name should be mentioned. From woman who campaigned for their rights, who voiced their opinion even in the face of adversity, Germain is different. Back then, society thought of woman as lesser than men, but Germain proved she was not lesser. She PROVED she was intelligent, brave, and had so much to offer. She was a shining example of the hardships woman had to go through, but prevailing through them. I know way more about her than that de Prony guy; I've never even heard of him (hahahaha). If she was actually given some proper training, and had a bit more support, who knows what more she could have done. As a mathematician, scientist, woman, or anyone really, Sophie Germain is truly someone to admire.
References
Marie-Sophie Germain, (JOC/EFR, October 1998 School of Mathematics and Statistics, University of St. Andrews, Scotland),
Sunday, June 8, 2014
The book I am reviewing is Euler: Master of us All, by William Dunham, published in 1999 by the Mathematical Association of America (incorporated). I've read several math books in the past but this book is by far my favorite. Probably because it's about my favorite mathematician, Leonhard Euler. Now, there were eight chapters; each chapter talks about a topic: algebra, number theory complex numbers, and so on. In each each chapter, the author gives some historical background on the topic, then what Euler accomplished with that topic, then some concluding thoughts. For my blog, I will essentially review each chapter. But there also was a historical background of Euler right before chapter 1, and that is where we will start.
The history of Euler is a story of courage, doubt, and the triumph of human spirit. Euler's parents were heavily involved in the clergy. So it looked like that was where Euler was headed. But as he was taking his theology classes, he couldn't help but to study mathematics. He actually entered college at the age of 14. Even as a child, Euler's genius was very apparent. At the university of Basel, he met Johann Bernoulli, a professor. Each Sunday Euler met with him to discuss Euler's questions on mathematics and physics. Later, Euler got a position at St. Petersburg in Russia, as a physics professor. The head of the math department was held my Danioull Bernoulli, with which Euler became friends with. He later took Danioull''s position as the head of the math department. But, because of political turmoil and wars, Euler went to the Berlin academy in Germany. There he was able to write mathematical works all day. Later he went back to St. Petersburg because the king of Germany (Frank the Great) was jealous and didn't like Euler. In Euler's later years, he was essentially blind. Though he was still able to publish math works with the help of scribes.
Euler may have been a genius, but my god did he work hard. Being a family man and having to tend to the government, I'm surprised he had time to do so much math. He inspires me to work hard and become a math professor. With his intelligence, he didn't have to work so hard and he would still be successful; he could have been rich. But instead, he concentrated his focus on math, something he loves to do. He is truly and inspiration for anyone. That's why I study math, because I love it. I don't really care about money, I never did. I'm just very thankful there are jobs out there where you can study math and make money. Now let's discuss the actual chapters.
Chapter 1: Euler and Number Theory
Euler made a pretty significant contribution to number theory, especially on perfect numbers.
Victor Klee and Stan Wagon (professors in the late 1900's) said that perfect numbers "is perhaps the oldest unfinished project of mathematics." Even with Euler's contributions, that's a pretty bold statement.
One major contribution Euler made was off one of Euclid's theorems:
If 2^k-1 is prime and if N = 2^{k-2}(2^k-1), then N is perfect.
Euclid proved that one. But, Euler proved the next one
If N is an even perfect number, then N = 2^{k-2}(2^k-1), where 2^k-1 is prime.
The author gives proofs to both of these theorems. The proofs are not too difficult; you could understand them with a couple of years of college math. One of my favorite theorems Euler proved was that the sum of reciprocals of odd perfect numbers is finite, the proof of which is very intuitive. Euler also helped out the study of perfect numbers by considering each number's whole number, instead of just their proper factors.
Chapter 1 was very interesting, it made me keep wanting to read on. Understanding the proofs take some time, but it's worth your while. Though sometimes the author will refer to a theorem or statement he made, so you have to flip back to remember what that statement was; things do get a bit muddy this way. But, some proofs in this chapter can be long, so I guess that was his best option.
Chapter 2: Euler and Logarithms
This chapter explores some of Euler largest contributions, that of logarithms. One of Euler's books: Introductio in analysin infinitorum was published in 1748. The author said it was one of the most influential math books of all time. It's essentially a pre-calculus text. I wouldn't mind taking taking a look at it. I wonder how students would fare if we gave them Euler's book instead of our usual pre-calculus book, just kidding. At the beginning the author discussed early methods of finding logs. From what I read: to compute logs, we first used square rooting, then series, and then carried on from there. Some of those methods took very long to explain; it took me like a half hour to understand that dreadful square rooting method. I won't mention it here because I would like you to keep reading my blog and not shut off your computer.
Euler also worked with exponential functions. He found that if a^z = y then log_a y = z; any pre-calculus student must know this if they want any shot in passing his/her class.
Euler also found that
which is a way to find numbers, other than in base 10. Euler also found a series expansion for a^x, eventually finding the number e. One of my favorite parts was Euler's proof that the Harmonic series diverges.
The beginning of the chapter explains early methods of finding logs, so it was kind of a snore fest. It seems like Euler laid the foundation for the knowledge needed to study calculus, which was very important for future generations of mathematicians. Euler may have not discovered logs, but he definitely popularized them and found many ways to use them.
Chapter 3: Euler and Infinite Series
Before Euler's time, infinite series were already pretty popular. Jakob Bernoulli loved infinite series, he found the sum of (k^2) / (2^k), which he found to be 6. He also found the sum of (k^3) / (2^k), which came out to be 26. But he had no idea of the Basel problem: the sum of (1) / (k^2). When Euler took a stab at it he found it to be Pi^2 / 6. His proof is not all that rigorous, he makes a lot of assumptions on certain infinite sums and products. Later on, with Issac Newton's help, Euler found the sum of (1) / (k^4), (1) / (k^6), and so on. There were doubters who thought Euler played too fast and loose with the Basel problem proof. So Euler gave alternate solutions, which were a bit more confusing than his original. Though one interesting fact was that Euler could not find the sum of (1) / (k^p) for odd p.
This was one of my favorite chapters. It was classics Euler: taking the natural logs of expressions, turning terms into infinite series and manipulating variables. It was all about infinite series and their sums. Euler was a master at finding these sums. Euler was so quick to recognize the sum for any know series. He was utterly a master at manipulating expressions. It was like watching Calvin Johnson run a route and catch a football.
Chapter 4: Euler and Analytic Number Theory
This chapter had a few things I had no idea about. It was known that odd primes are either in the form 4k+1, or 4k-1. Also, it was known that there are infinitely many primes; Euclid gives a very clear and simple proof of this in this chapter. The infitude of 4k-1 primes proof is similar, but the infitude of $4k+1$ primes proof was a lot more complicated, which was weird. You think it would be a similar proof as 4k-1. Also, I never knew that 4k+1 primes can be decomposed into the sum of unique squares, I think Fermat found this, but Euler proved it. For instance, 137 = 16 + 121 = 4^2 + 11^2. But, 4k-1 primes did not share this property.
Euler proved that the sum of reciprocals of primes is infinite. The proof is very long but very interesting. Andre Weil said the proof may be the birth of analytic number theory.
I like this chapter because because it discussed things I never knew about. Though it gets very confusing when the author discusses some of Euler's proofs. They are long and not intuitive. Though the author discusses some pretty interesting properties Euler found on the harmonic series. One of which is that
1+1/2^2+1/3^2+... = (2*3*5*7*11*...) / (1*2*4*6*10*...)
where the numerator is the product of all primes and the denominator is the product of all primes minus 1. This property blows my mind, but I still don't quite understand why it's true.
Chapter 5: Euler and complex Variables
From Cardano's cubic formula to Bombelli interpreting the results, mathematicians were still confused with where the other roots came from in the solutions from Cardano's formula. Not Euler though. This required the use of imaginary numbers. Even Liebniz was scared of the square root of -1. It got me to think "is there something mathematicians are scared of today?" Not sure. In Euler's book Elements of Algebra, he said there are imaginary numbers because they only exist in our imagination. Sounds like nursery rhyme or something. Euler didn't mind using imaginary numbers. He found the logs, exponent, and sines and cosines of imaginary numbers. He also found that e^x = cos x + i*sin x, one of his most famous formulas. Essentially, Euler popularized imaginary numbers and showed us that there is nothing to be afraid of. One interesting account was where Johann Bernoulli argued with Liebniz over what ln(-x) was. Euler found that
ln(-x) = ln((-1)x) = ln x + ln(-1), a pretty amazing discovery. Though I was confused on how Euler knew he could do ln(ab) = lna + lnb where a,b may not be positive. Oh well, I'm sure he was correct.
Euler found that ln (a+bi) = ln c + i*(theta + 2*Pi*k)$ where c=a^2+b^2 and sin \theta = b / c. He also found that i^i = e^{-Pi / 2}*e^{+/- 2*Pi*k}$. Two very interesting discoveries.
Euler popularized imaginary numbers, which is so important to mathematics now-a-days. To me, this was his largest contribution. This chapter has many long confusing algebraic proofs, but they are worth understanding. This chapter covered everything I knew about imaginary numbers and much more. This chapter was probably the most informative, learning wise.
Chapter 6: Euler and Algebra
Euler knew how to solve a quartic: you have to depress it into a cubic, then use Cardano's formula. Euler also somehow knew that the solution was of the form sqrt{p}+sqrt{q}+sqrt{r} where p,q,r are complex numbers. How he figured this out, I have no idea, but that's just Euler. A lot of this chapter discusses Euler solving the cubic and quatric, and Attempting to solve the quintic. He couldn't find a formula for it (as we already know, there is none). Also, Euler tried to solve the fundamental theorem of Algebra: that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This theorem brings about the fact that essentially, every polynomial can be factored as a product of linear or non-reducible quadratic factors. Euler failed in solving the fundamental theorem of Algebra, but Carl Fredrick Gauss found it. The author does not explain the full proof because it is very long and requires a lot more knowledge than a few years of math courses.
This chapter, like it's title, contains a lot of tedious algebra. So it probably is not the most exciting chapter. Euler did not prove many large algebraic theorems, be he laid the groundwork for others to do so. Also, we know about Evareste Galios, but you should looks up Niels Abel. At first I thought they were the same guy.
Chapter 7: Euler and Geometry
Euler did not do much in geometry. He did prove that the 3 centers of a triangle (orthocenter, centriod, circumcenter) all lie on a straight line, which is known as the Euler line. A lot of the chapter consists of geometrical proofs that requires tons of algebra. This was probably the most boring chapter, but included for completeness.
Chapter 8: Euler and Combinatorics
Euler dabbled a bit with this subject. Before his time, basic combinatorical theory was known. The book Ars Conjectandi was a text on probability theory published in 1713 and written by Jakob Bernoulli, so a good amount of combinatorics existed before Euler made his mark.
Euler's most notable work in this subject was on partitions. For any whole number, a partition is the number of ways you can add smaller numbers to get that certain number (1+3=2+2=1+1+2= ... = 4). A special case is the number of partitions with different numbers. So for 4, there are 2: 1+3 and 4 itself. Another special case is the number of partitions that contain just odd whole numbers. For 4, there are again, only 2: 1+1+1+1 and 1+3, interesting. As you can tell, I'm getting to a theorem Euler proved: The number of different ways a given number can be expressed as the sum of different whole numbers is the same as the number of ways in which that same number can be expressed as the sum of odd numbers, whether the same or different.
That was truly a remarkable theorem. I understood the proof but I had no idea how he could have thought of it, it was a work of genius. Hopefully you will read about it.
Conclusion
Overall this book was great. With a few years of college math you can understand basically all the proofs. Also, some proofs proved to be a bit difficult and not so rigorous (classic Euler), so the author gave alternate proofs of certain theorems and he took extra time to explain certain parts of the proofs. This was very helpful. Like I explained, this book is not just on Euler. The author mentions the works of many other great mathematicians, so you get a pretty broad view of some mathematical history in general. I hope you enjoyed my review and even more so, I hope you read the book.
because my |
Polyhedrons IV: Dodeca
The five regular or "Platonic" solids are abstract shapes, but they sometimes take material forms.
Anthony Barreiro, after seeing the cover picture of Astronomical Calendar 2015 with its stellated dodecahedrons, happened to revisit his childhood home in Castro Valley, California, and saw hanging on the back porch three lamps his father had bought long ago in Tijuana, Mexico. He sent me this photo of one of them: a stellated dodecahedron.
Dice are cubes, so that when thrown they settle with one of six numbers of spots on top – but they can be other polyhedrons. The more faces they have, the closer to a sphere and the easier they'll roll. Twice as many numbers are given by a dodecahedral die. (That's the correct but often forgotten singular of "dice." It comes down through Old French dé from Latin datum, "given.")
The dodecahedron, with its twelve five-edged sides meeting three at a time at twenty vertices, has an overall shape that feels to me more five-ish than twelve-ish (or twenty-ish or three-ish). A contentment resides in it. We are five-fingered and five is at the root of our own being. It has calm mystery; it is a nut, impregnably fortified yet pregnant with growth. It is almost annoyingly subtle and difficult to define. Can you see how to find the "latitude" of one of the points in the row around its middle – the angle to it from the dodecahedron's "north pole"? It's nothing simple, like a third or a quarter or a fifth of the way around; you have to travel a mixture of edge-lengths and pentagon-widths. It turns out to be 90 minus arccos (inner radius / outer radius); and the inner radius is (squareroot (75 + 30 * squareroot (5)) / 15) * outer radius; and the outer radius is edgelength * (squareroot (3) / 4) * (1 + squareroot(5)). Maybe you know of an easier way.
This is the template for making a dodecahedron – trace on card, fold along the heavy lines, glue the flaps.
It looks like two humanoid shapes about to become clasped together, reminding us of the comic explanation of love by Aristophanes (in Plato's Symposium): humans are originally spherical beings, who having been chopped in half go forever seeking their other halves.
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6 thoughts on "Polyhedrons IV: Dodeca"
All this business of dodecahedrons sent me back to Ian McEwan's short story 'Solid Geometry', in which a brilliant young mathematician invents a 'plane without a surface'. He folds and cuts a sheet of paper in a certain way, and when he pulls the paper through the incision it disappears. Then, to convince an audience deriding this apparent conjuring trick, he adopts a contorted posture and somehow disappears by crawling through a 'hoop' made by his own arms. Perhaps I should let you guys get on with your polywhatnots, as long as you avoid that kind of experiment.
A Borges-like story. A plane without a surface is impossible to conceive, and so, you might think, is a plane with only one surface, but you can make that easily with a strip of paper. You twist one end of it 180 degrees and tape it to the other, and it is called a Mobius strip or cylinder. A line drawn along it arrives on the opposite side. I once wrote a story about that (it too is somewhere in my mislaid "Among the Shapes") with two two-dimensional beings who live in this Flatland, called Stabilis and Mobilis. Stabilis stays where he is; Mobilis sets off on a journey all the way around the world, and arrives back with his heart seeming (to Stabilis) to be on the other side of his body.
Guy,
I didn't notice any polyhedrons or orreries, but some of these images provided by Google of what's being done with 3D printers amaze me and trigger many ideas: q=3d+printed+objects&espv=2&biw=1138&bih=511&site=webhp&tbm=isch&tbo=u&source=univ&sa=X&ei=0QMCVbiiLsL5yQTN7IKQDw&sqi=2&ved=0CB0QsAQ
With your expertise in representing 3D objects in 2D, your artistic skills and good judgement, your computer and algorithmic skills, etc, I wonder if this 3D technology might be of interest. I think, for example, of "Kepler's polyhedral cosmos" at the end of page 34 of Astronomical Calendar 2015. Or maybe there's a way to produce a better planisphere using this technology (at least as a prototype). (My overall favorite planisphere continues to be the one available from Edmond, but I can think of improvements..)
BTW, the original cover of AC2015 was good enough for me, but thanks for the improved version you blogged to us recently. And thanks for sharing some details of your travails in learning to automate the production of more accurate polyhedron images!
BTW2, the above template you provide for constructing a dodecahedron is very cool, and your related observations about the relevance of the number 5 to humans are worthy of some meditation..
To make James's link work, delete the space between "search?" and "q". In other words, copy the whole of his text from "http" to the end of the paragraph, paste it into the browser address box, and delete that space.
I can sort of imagine how 3-D printing works, though I assume it's done with machinery. I don't know how 3-D animations are programmed, such as the one you can find somewhere of the two halves of the dodecahedron actually rotating into their clasp. I imagine it could be used to animate the construction of Kepler's cosmos of polyhedrons enclosing each other. Something else to learn if there were infinite time after doing enough more of the learning of WordPress, Photoshop, Musescore, etc., etc.
I will say more about "five… at the root of our being" if I can find a mass of notes I once made (before computer) for a project called "Among the Shapes".
I've learned so much about polyhedra this year! And the dodecahedron is my favorite. Five is certainly the first number that jumps out at me when looking at a dodecahedron.
By the way, the picture of the lamp should be rotated 90 degrees clockwise. It's not a great picture to start with, a quick snap with my ipad with the Sun backlighting the lamp (as you said, "without thinking" — no ray tracing needed!). |
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'One of the first applications of the simplex algorithm was to the determination of an adequate diet that was of least cost.'
'Here there is no unfolding to a single planar component but the algorithm finds an unfolding with four planar components.'
'This first step is here reduced to a simple algorithm suitable for computer use.'
'He began to study mathematical logic and the theory of algorithms just before 1940.'
Origin
Late 17th century (denoting the Arabic or decimal notation of numbers): variant (influenced by Greek arithmos 'number') of Middle English algorism, via Old French from medieval Latin algorismus. The Arabic source, al-Ḵwārizmī 'the man of Ḵwārizm' (now Khiva), was a name given to the 9th-century mathematician Abū Ja'far Muhammad ibn Mūsa, author of widely translated works on algebra and arithmetic. |
Wednesday, February 3, 2016
Math in Film Making
As long as I've been exploring math in music, I thought I'd check out information for film making. Most of us have a student who love filming everything and turning it into the epic adventure.
I'm not talking about the student who is always making selfie video to post on their site but the one who has vision they create on film. It turns out film making uses so much math.
The math goes from the producer keeping track of the budget to making sets, costumes, filming, and so much more I found several sites that have wonderful information on the topic.
A student online newspaper has a wonderful article on math in film making. The author discusses how the producer uses math in the cost of the film while the cinematographer must calculate the best angles for a shot. We are told not to forget the sets and the costumes both of which involve quite a bit of math. In addition, the author comments that animators use the most math and several types of math.
The Mathematical Association of America has a lovely piece on film making and math. They actually reference a paper about math and animation that goes into quite a bit of detail.
No Film School goes into specific detail about the golden ratio, the Fibonacci sequence, perspectives, the rule of thirds, and there are accompanying videos to show many of these.
Plus Maths has a very technical article on mathematics and animation. This article takes the reader through the process step by step so the get an idea of how creatures and items are created. There is talk about matricies, vetices and so many other mathematical material. It is such a cool article.
Eveltio has a list of several articles with links, all dealing with math in film making.
To make it even more relevant, I found a few places that talk about the cost of making a movie from start to finish. This puts the information of the previous articles into context.
Investopedia has a great piece on the cost of making films. They include information on the cost of marketing films. Something many people don't think about.
Quora has a summary of a 75 page budget used for the movie the Village. The final cost was over 71 Million for making it. You see how much costumes, sets, lighting, visual effects, etc. This ties in with what the producer keeps track of.
Slide Share has a whole piece on when a film starts making money and many of the misconceptions involved in that process. The article uses Spiderman 2 as its example. I found out that there are huge costs involved in printing and distributing the film. Something in the neighborhood of 50 to 100 million!
Just think what type of eye opener this kind of unit would make for our students! |
Synopses & Reviews
Publisher Comments
Self-similarity is related to symmetry analysis is an attribute of many physical laws: particle physics and those governing Newton's laws of gravitation. Symmetry, found throughout the biological universe, is also a basic property of the mathematical universe. In this book the author explores the idea's of scaling, self-similarity, chaos. and fractals as they appear throughout the universe of pure and applied mathematics. Because of his formidable research experience, stretching from the acoustical modelling of concert halls to pure number theory, Schroeder is able to take the reader on an intellectual excursion through this vast forest of topics. Requires a basic familiarity with undergraduate mathematics and elementary physics.
Synopsis
About the Author
Manfred Schroeder is a pioneer in the artistic potential of computer graphics, a world-renowned expert in concert hall acoustics, and holder of over 45 patents. He divides his time between Berkeley Heights, California and Goetingen, Germany. |
Thursday, March 18, 2010
Game Theory
I finished my re-read of I'm a Stranger Here Myself and I'm on to a new book: A Beautiful Math by Tom Siegfried. It's about game theory, particularly John Forbes Nash's work (the mathematician who A Beautiful Mind was written about, hence the title). I don't know much at all about game theory. It's one of those things I heard math majors talk about in college but never actually found out what it was. At some point I established that the "game" in game theory doesn't have anything to do with actual games.
Turns out I was wrong about that, that one thing I knew prior to starting the book. It's mostly an economic theory, used for predicting how people (and other systems, it turns out) will behave. It was developed by looking at simple, specific situations with clearly defined rules - things like a game of chess or checkers.
Here's one neat thing I learned so far: If I like A more than B, and I like B more than C, it follows that I like A more than C. But how much more? How can you quantify how much you prefer one thing over another? John Von Neumann and Oskar Morgenstern (mathematicians in the first half of the 20th century) determined a way to do this.
The example the book uses involves Let's Make a Deal (statisticians love examples involving Monty Hall, I've learned that, too). You're given a choice between a BMW, a new TV, and an old tricycle. Let's assume you want the car. You're told you can either have the TV, or you can have a 50% shot at the BMW. Maybe you pick the TV. Well, what if it was a 60% chance of getting the car? 70%? The method is to find at what point you would decide to choose the chance at the BMW over the sure thing of the TV. That percentage is a measure of how much more you prefer the BMW to the TV. Cool!
I feel like rating tons of stuff like that now. If I can get a new pair of earrings or a 50% chance of having a sheep live next door that I could go visit... Hmm.... |
March 19, 2009
Well to start off class today, Mr.K wanted to talk about the "Blogger Hall Of FAME!!". We took a few votes on the subject and here's what we decided:
1) It takes a MINIMUM of 12 votes to get your blog into the blogger hall of fame.
2) NO anonymous votes.
3) NO voting for yourself.
Once we were done with our little bit of classroom democracy, we split up into groups to work on some practice problems. Once we were in our groups, we did a little recap on Pythagorean identities. Remember:
As practice, we simplified a few simple expressions using Pythagorean identities when we could.
Example #1
1) first change the expression to only use sine, cosine, or 1
2) this leaves you with two fractions with the same denominator. Rewrite the expression
3) since the numerator is a pythagorean identity, we now know that the numerator is equal to sin^2 t
4) the numerator and denominator can be factored and the answer that is left over is 1
Once we had finished simplifying the expressions, we found that two of the expressions simplified to the same thing, in this case 1. We used them to write the equation
To check if this is true there are a few strategies that we could use. These are:
1) Work with the more complicated side of the identity first
2) Rewrite both sides using sine and cosine
3) Use a Pythagorean identity
4) Simplify complex fractions or rewrite them
5) Use factoring (especially to create differences of squares)
A very important thing to remember when proving trigonometric identities is that you have to drop down "The Great Wall of China" and you can NOT cross it. This is because, when proving an equation, there is no way of knowing whether or not it is true.
So now that we have covered proving trigonometric identities, on to a new subject!!
EVEN ANDODDIDENTITIES!!
sin(-x)=-sinx cos(-x)=cos(x) tan(-x)=tan(x)
the sine and tangent functions and ODD functions
cosine is an EVEN function
ok so now i've gone through pretty much the entire class of today. There is only one more thing from class that i must remind you of:
YOUR DANCING TOMMOROW!!!!
so i'd bring something comfy to wear. Mr.K said that as soon as you get into the room get into groups of four, with at least one person in your group that you have NEVER worked with. |
The "Weird Al Yankovic of mathematics education", UTEP Professor Larry Lesser motivates teachers and students by merging two of his great loves – mathematics and song. He's published a score of math lyrics in national journals as well as the first juried comprehensive articles on using songs in math class, and has given national workshops ranging from NCTM (a ballroom audience of 700) to the Rock and Roll Hall of Fame.
As the "Mathemusician", Lesser brings his acoustic guitar to classrooms and keynotes, facilitating explorations and performing raps and musical parodies, creatively adapting popular lyrics towards math topics such as infinity, pi, problem solving, graphing functions, as well as even more worldly applications such as whether to play the lottery! Visit his math-and-music page of examples and resources at: |
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His Childhood
The Eulers
Education
Leonhard Euler was born on April 15, 1707 in Barley, Switzerland.
Personal Life
Leonhard Euler had a huge family. He had 4 siblings and his parents were Paul Euler and Marguerite Brucker. He married twice and had 5 children named: Johann Euler, Christof Euler, Charlotte Euler, Elene Euler, and Karl Euler.
Euler's Formula
(eiπ+1=0)
Leonhard Euler went to the University of Basel from 1720-1723. He started college at the age of 13 and finished at 16. He later became a professor at St. Petersburg Academy around 1730.
Euler's Formula Today
Euler married Katharina Gsell in 1734 and had 5 children. They were married for 39 years until she died. A few years later Euler married Katharina's half sister Salome Abigail Gsell. He then died in 1783 at the age of 79.
Euler was famous for many reasons. He was a mathematician and was also known for geometry, calculus, physics, lunar theory, astronomy, etc. But he also discovered his most famous formula: (eiπ+1=0).
We use Euler's to find equivalent ways to move in a circle. It is a complicated equation and some people don't even understand it. |
So stop me if you've heard this one before. We're going to make something interesting. You bring to it a complex-valued number. Anything you like. Let me call it 's' for the sake of convenience. I know, it's weird not to call it 'z', but that's how this field of mathematics developed. I'm going to make a series built on this. A series is the sum of all the terms in a sequence. I know, it seems weird for a 'series' to be a single number, but that's how that field of mathematics developed. The underlying sequence? I'll make it in three steps. First, I start with all the counting numbers: 1, 2, 3, 4, 5, and so on. Second, I take each one of those terms and raise them to the power of your 's'. Third, I take the reciprocal of each of them. That's the sequence. And when we add —
Yes, that's right, it's the Riemann-Zeta Function. The one behind the Riemann Hypothesis. That's the mathematical conjecture that everybody loves to cite as the biggest unsolved problem in mathematics now that we know someone did something about Fermat's Last Theorem. The conjecture is about what the zeroes of this function are. What values of 's' make this sum equal to zero? Some boring ones. Zero, negative two, negative four, negative six, and so on. It has a lot of non-boring zeroes. All the ones we know of have an 's' with a real part of ½. So far we know of at least 36 billion values of 's' that make this add up to zero. They're all ½ plus some imaginary number. We conjecture that this isn't coincidence and all the non-boring zeroes are like that. We might be wrong. But it's the way I would bet.
Anyone who'd be reading this far into a pop mathematics blog knows something of why the Riemann Hypothesis is interesting. It carries implications about prime numbers. It tells us things about a host of other theorems that are nice to have. Also they know it's hard to prove. Really, really hard.
Ancient mathematical lore tells us there are a couple ways to solve a really, really hard problem. One is to narrow its focus. Try to find as simple a case of it as you can solve. Maybe a second simple case you can solve. Maybe a third. This could show you how, roughly, to solve the general problem. Not always. Individual cases of Fermat's Last Theorem are easy enough to solve. You can show that doesn't have any non-boring answers where a, b, and c are all positive whole numbers. Same with , though it takes longer. That doesn't help you with the general .
There's another approach. It sounds like the sort of crazy thing Captain Kirk would get away with. It's to generalize, to make a bigger, even more abstract problem. Sometimes that makes it easier.
For the Riemann-Zeta Function there's one compelling generalization. It fits into that sequence I described making. After taking the reciprocals of integers-raised-to-the-s-power, multiply each by some number. Which number? Well, that depends on what you like. It could be the same number every time, if you like. That's boring, though. That's just the Riemann-Zeta Function times your number. It's more interesting if what number you multiply by depends on which integer you started with. (Do not let it depend on 's'; that's more complicated than you want.) When you do that? Then you've created an L-Function.
Specifically, you've created a Dirichlet L-Function. Dirichlet here is Peter Gustav Lejeune Dirichlet, a 19th century German mathematician who got his name on like everything. He did major work on partial differential equations, on Fourier series, on topology, in algebra, and on number theory, which is what we'd call these L-functions. There are other L-Functions, with identifying names such as Artin and Hecke and Euler, which get more directly into group theory. They look much like the Dirichlet L-Function. In building the sequence I described in the top paragraph, they do something else for the second step.
The L-Function is going to look like this:
The sigma there means to evaluate the thing that comes after it for each value of 'n' starting at 1 and increasing, by 1, up to … well, something infinitely large. The are the numbers you've picked. They're some value that depend on the index 'n', but don't depend on the power 's'. This may look funny but it's a standard way of writing the terms in a sequence.
An L-Function has to meet some particular criteria that I'm not going to worry about here. Look them up before you get too far into your research. These criteria give us ways to classify different L-Functions, though. We can describe them by degree, much as we describe polynomials. We can describe them by signature, part of those criteria I'm not getting into. We can describe them by properties of the extra numbers, the ones in that fourth step that you multiply the reciprocals by. And so on. LMFDB, an encyclopedia of L-Functions, lists eight or nine properties usable for a taxonomy of these things. (The ambiguity is in what things you consider to depend on what other things.)
What makes this interesting? For one, everything that makes the Riemann Hypothesis interesting. The Riemann-Zeta Function is a slice of the L-Functions. But there's more. They merge into elliptic curves. Every elliptic curve corresponds to some L-Function. We can use the elliptic curve or the L-Function to prove what we wish to show. Elliptic curves are subject to group theory; so, we can bring group theory into these series.
And then it gets deeper. It always does. Go back to that formula for the L-Function like I put in mathematical symbols. I'm going to define a new function. It's going to look a lot like a polynomial. Well, that L(s) already looked a lot like a polynomial, but this is going to look even more like one.
Pick a number τ. It's complex-valued. Any number. All that I care is that its imaginary part be positive. In the trade we say that's "in the upper half-plane", because we often draw complex-valued numbers as points on a plane. The real part serves as the horizontal and the imaginary part serves as the vertical axis.
Now go back to your L-Function. Remember those numbers you picked? Good. I'm going to define a new function based on them. It looks like this:
You see what I mean about looking like a polynomial? If τ is a complex-valued number, then is just another complex-valued number. If we gave that a new name like 'z', this function would look like the sum of constants times z raised to positive powers. We'd never know it was any kind of weird polynomial.
Anyway. This new function 'f(τ)' has some properties. It might be something called a weight-2 Hecke eigenform, a thing I am not going to explain without charging someone by the hour. But see the logic here: every elliptic curve matches with some kind of L-Function. Each L-Function matches with some 'f(τ)' kind of function. Those functions might or might not be these weight-2 Hecke eigenforms.
So here's the thing. There was a big hypothesis formed in the 1950s that every rational elliptic curve matches to one of these 'f(τ)' functions that's one of these eigenforms. It's true. It took decades to prove. You may have heard of it, as the Taniyama-Shimura Conjecture. In the 1990s Wiles and Taylor proved this was true for a lot of elliptic curves, which is what proved Fermat's Last Theorem after all that time. The rest of it was proved around 2000.
As I said, sometimes you have to make your problem bigger and harder to get something interesting out of it.
I am one letter closer to the end of Gaurish's main block of requests. They're all good ones, mind you. This gets me back into elliptic curves and Diophantine equations. I might be writing about the wrong thing.
Height Function.
My love's father has a habit of asking us to rate our hobbies. This turned into a new running joke over a family vacation this summer. It's a simple joke: I shuffled the comparables. "Which is better, Bon Jovi or a roller coaster?" It's still a good question.
But as genial yet nasty as the spoof is, my love's father asks natural questions. We always want to compare things. When we form a mathematical construct we look for ways to measure it. There's typically something. We'll put one together. We call this a height function.
We start with an elliptic curve. The coordinates of the points on this curve satisfy some equation. Well, there are many equations they satisfy. We pick one representation for convenience. The convenient thing is to have an easy-to-calculate height. We'll write the equation for the curve as
Here both 'A' and 'B' are some integers. This form might be unique, depending on whether a slightly fussy condition on prime numbers hold. (Specifically, if 'p' is a prime number and 'p4' divides into 'A', then 'p6' must not divide into 'B'. Yes, I know you realized that right away. But I write to a general audience, some of whom are learning how to see these things.) Then the height of this curve is whichever is the larger number, four times the cube of the absolute value of 'A', or 27 times the square of 'B'. I ask you to just run with it. I don't know the implications of the height function well enough to say why, oh, 25 times the square of 'B' wouldn't do as well. The usual reason for something like that is that some obvious manipulation makes the 27 appear right away, or disappear right away.
This idea of height feeds in to a measure called rank. "Rank" is a term the young mathematician encounters first while learning matrices. It's the number of rows in a matrix that aren't equal to some sum or multiple of other rows. That is, it's how many different things there are among a set. You can see why we might find that interesting. So many topics have something called "rank" and it measures how many different things there are in a set of things. In elliptic curves, the rank is a measure of how complicated the curve is. We can imagine the rational points on the elliptic curve as things generated by some small set of starter points. The starter points have to be of infinite order. Starter points that don't, don't count for the rank. Please don't worry about what "infinite order" means here. I only mention this infinite-order business because if I don't then something I have to say about two paragraphs from here will sound daft. So, the rank is how many of these starter points you need to generate the elliptic curve. (WARNING: Call them "generating points" or "generators" during your thesis defense.)
There's no known way of guessing what the rank is if you just know 'A' and 'B'. There are algorithms that can calculate the rank given a particular 'A' and 'B'. But it's not something like the quadratic formula where you can just do a quick calculation and know what you're looking for. We don't even know if the algorithms we have will work for every elliptic curve.
We think that there's no limit to the height of elliptic curves. We don't know this. We know there exist curves with ranks as high as 28. They seem to be rare [*]. I don't know if that's proven. But we do know there are elliptic curves with rank zero. A lot of them, in fact. (See what I meant two paragraphs back?) These are the elliptic curves that have only finitely many rational points on them.
And there's a lot of those. There's a well-respected that the average rank, of all the elliptic curves there are, is ½. It might be. What we have been able to prove is that the average rank is less than or equal to 1.17. Also that it should be larger than zero. So we're maybe closing in on the ½ conjecture? At least we know something. I admit this essay I've started wondering what we do know of elliptic curves.
What do the height, and through it the rank, get us? I worry I'm repeating myself. By themselves they give us families of elliptic curves. Shapes that are similar in a particular and not-always-obvious way. And they feed into the Birch and Swinnerton-Dyer conjecture, which is the hipster's Riemann Hypothesis. That is, it's this big, unanswered, important problem that would, if answered, tell us things about a lot of questions that I'm not sure can be concisely explained. At least not why they're interesting. We know some special cases, at least. Wikipedia tells me nothing's proved for curves with rank greater than 1. Humanity's ignorance on this point makes me feel slightly better pondering what I don't know about elliptic curves.
(There are some other things within the field of elliptic curves called height functions. There's particularly a height of individual points. I was unsure which height Gaurish found interesting so chose one. The other starts by measuring something different; it views, for example, as having a lower height than does , even though the numbers are quite close in value. It develops along similar lines, trying to find classes of curves with similar behavior. And it gets into different unsolved conjectures. We have our ideas about how to think of fields.).
[*] Wikipedia seems to suggest we only know of one, provided by Professor Noam Elkies in 2006, and let me quote it in full. I apologize that it isn't in the format I suggested at top was standard. Elkies way outranks me academically so we have to do things his way:
I can't figure how to get WordPress to present that larger. I sympathize. I'm tired just looking at an equation like that. This page lists records of known elliptic curve ranks. I don't know if the lack of any records more recent than 2006 reflects the page not having been updated or nobody having found a rank-29 curve. I fully accept the field might be more difficult than even doing maintenance on a web page's content is.
Gaurish, of the For The Love Of Mathematics gives me another subject today. It's one that isn't about ellipses. Sad to say it's also not about elliptic integrals. This is sad to me because I have a cute little anecdote about a time I accidentally gave my class an impossible problem. I did apologize. No, nobody solved it anyway.
Elliptic Curves.
Elliptic Curves start, of course, with polynomials. Particularly, they're polynomials with two variables. We call the 'x' and 'y' because we have no reason to be difficult. They're of at most third degree. That is, we can have terms like 'x' and 'y2' and 'x2y' and 'y3'. Something with higher powers, like, 'x4' or 'x2y2' — a fourth power, all together — is right out. Doesn't matter. Start from this and we can do some slick changes of variables so that we can rewrite it to look like this:
Here, 'A' and 'B' are some numbers that don't change for this particular curve. Also, we need it to be true that doesn't equal zero. It avoids problems. What we'll be looking at are coordinates, values of 'x' and 'y' together which make this equation true. That is, it's points on the curve. If you pick some real numbers 'A' and 'B' and draw all the values of 'x' and 'y' that make the equation true you get … well, there's different shapes. They all look like those microscope photos of a water drop emerging and falling from a tap, only rotated clockwise ninety degrees.
So. Pick any of these curves that you like. Pick a point. I'm going to name your point 'P'. Now pick a point once more. I'm going to name that point 'Q'. Now draw a line from P through Q. Keep drawing it. It'll cross the original elliptic curve again. And that point is … not actually special. What is special is the reflection of that point. That is, the same x-coordinate, but flip the plus or minus sign for the y-coordinate. (WARNING! Do not call it "the reflection" at your thesis defense! Call it the "conjugate" point. It means "reflection".) Your elliptic curve will be symmetric around the x-axis. If, say, the point with x-coordinate 4 and y-coordinate 3 is on the curve, so is the point with x-coordinate 4 and y-coordinate -3. So that reflected point is … something special.
. The water drop bulges out from the surface.
This lets us do something wonderful. We can think of this reflected point as the sum of your 'P' and 'Q'. You can 'add' any two points on the curve and get a third point. This means we can do something that looks like addition for points on the elliptic curve. And this means the points on this curve are a group, and we can bring all our group-theory knowledge to studying them. It's a commutative group, too; 'P' added to 'Q' leads to the same point as 'Q' added to 'P'.
Let me head off some clever thoughts that make fair objections. What if 'P' and 'Q' are already reflections, so the line between them is vertical? That never touches the original elliptic curve again, right? Yeah, fair complaint. We patch this by saying that there's one more point, 'O', that's off "at infinity". Where is infinity? It's wherever your vertical lines end. Shut up, this can too be made rigorous. In any case it's a common hack for this sort of problem. When we add that, everything's nice. The 'O' serves the role in this group that zero serves in arithmetic: the sum of point 'O' and any point 'P' is going to be 'P' again.
Second clever thought to head off: what if 'P' and 'Q' are the same point? There's infinitely many lines that go through a single point so how do we pick one to find an intersection with the elliptic curve? Huh? If you did that, then we pick the tangent line to the elliptic curve that touches 'P', and carry on as before.
. The water drop is close to breaking off, but surface tension has not yet pinched off the falling form.
There's more. What kind of number is 'x'? Or 'y'? I'll bet that you figured they were real numbers. You know, ordinary stuff. I didn't say what they were, so left it to our instinct, and that usually runs toward real numbers. Those are what I meant, yes. But we didn't have to. 'x' and 'y' could be in other sets of numbers too. They could be complex-valued numbers. They could be just the rational numbers. They could even be part of a finite collection of possible numbers. As the equation is something meaningful (and some technical points are met) we can carry on. The elliptical curves, and the points we "add" on them, might not look like the curves we started with anymore. They might not look like anything recognizable anymore. But the logic continues to hold. We still create these groups out of the points on these lines intersecting a curve.
By now you probably admit this is neat stuff. You may also think: so what? We can take this thing you never thought about, draw points and lines on it, and make it look very loosely kind of like just adding numbers together. Why is this interesting? No appreciation just for the beauty of the structure involved? Well, we live in a fallen world.
It comes back to number theory. The modern study of Diophantine equations grows out of studying elliptic curves on the rational numbers. It turns out the group of points you get for that looks like a finite collection of points with some collection of integers hanging on. How long that collection of numbers is is called the 'rank', and there are deep mysteries at work. We know there are elliptic equations that have a rank as big as 28. Nobody knows if the rank can be arbitrary high, though. And I believe we don't even know if there are any curves with rank of, like, 27, or 25.
Yeah, I'm still sensing skepticism out there. Fine. We'll go back to the only part of number theory everybody agrees is useful. Encryption. We have roughly the same goals for every encryption scheme. We want it to be easy to encode a message. We want it to be easy to decode the message if you have the key. We want it to be hard to decode the message if you don't have the key.
. The water drop is almost large enough that its weight overcomes the surface tension holding it to the main body of water.
Take something inside one of these elliptic curve groups. Especially one that's got a finite field. Let me call your thing 'g'. It's really easy for you, knowing what 'g' is and what your field is, to raise it to a power. You can pretty well impress me by sharing the value of 'g' raised to some whole number 'm'. Call that 'h'.
Why am I impressed? Because if all I know is 'h', I have a heck of a time figuring out what 'g' is. Especially on these finite field groups there's no obvious connection between how big 'h' is and how big 'g' is and how big 'm' is. Start with a big enough finite field and you can encode messages in ways that are crazy hard to crack.
We trust. At least, if there are any ways to break the code quickly, nobody's shared them. And there's one of those enormous-money-prize awards waiting for someone who does know how to break such a code quickly. (I don't know which. I'm going by what I expect from people.)
And then there's fame. These were used to prove Fermat's Last Theorem. Suppose there are some non-boring numbers 'a', 'b', and 'c', so that for some prime number 'p' that's five or larger, it's true that . (We can separately prove Fermat's Last Theorem for a power that isn't a prime number, or a power that's 3 or 4.) Then this implies properties about the elliptic curve:
This is a convenient way of writing things since it showcases the ap and bp. It's equal to:
(I was so tempted to leave an arithmetic error in there so I could make sure someone commented.)
. The water drop has broken off, and the remaining surface rebounds to its normal meniscus.
If there's a solution to Fermat's Last Theorem, then this elliptic equation can't be modular. I don't have enough words to explain what 'modular' means here. Andrew Wiles and Richard Taylor showed that the equation was modular. So there is no solution to Fermat's Last Theorem except the boring ones. (Like, where 'b' is zero and 'a' and 'c' equal each other.) And it all comes from looking close at these neat curves, none of which looks like an ellipse.
They're named elliptic curves because we first noticed them when Carl Jacobi — yes, that Carl Jacobi — while studying the length of arcs of an ellipse. That's interesting enough on its own. But it is hard. Maybe I could have fit in that anecdote about giving my class an impossible problem after all |
Bardi, J. S. (2006). The calculus wars : Newton, Leibniz, and the greatest mathematical clash of all time. New York, Thunder's Mouth Press.
Today Newton and Leibniz are generally considered the twin independent inventors of calculus, and they are both credited with giving mathematics its greatest push forward since the time of the Greeks. Had they known each other under different circumstances, they might have been friends. But in their own lifetimes, the joint glory of calculus was not enough for either and each declared war against the other, openly and in secret. This long and bitter dispute has been swept under the carpet by historians--perhaps because it reveals Newton and Leibniz in their worst light--but this book tells the full story in narrative form for the first time, ultimately exposing how these twin mathematical giants were brilliant, proud, at times mad and, in the end, completely human.--From publisher description.
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Holmgren, S., Svenska folkhälsoinstitutet, et al. (2009). Child day care cinter or home care for children 12-40 months of age : what is best for the child? : a systematic literature review. Östersund, Svenska folkhälsoinstitutet.
Lindström, J.-O. (2009). With a focus on mathematics and science [Elektronisk resurs] : an analysis of the differences and similarities between international and comparative studies and national syllabuses. Stockholm, Skolverket.
National Centre for Excellence in the Teaching of Mathematics (2009). Developing mathematics in London secondary schools : headteachers talk about creating and sustaining excellent mathematics departments. London, NCETM.
National Centre for Excellence in the Teaching of Mathematics (2009). The NCETM - Impact and inspiration : annual report 2008/09. London, NCETM.
National Centre for Excellence in the Teaching of Mathematics and Yorkshire Forward (2009). Inspiring mathematics champions : final evaluation report. London, NCETM.
National Council of Teachers of Mathematics (1957). Insights into modern mathematics. Washington, D.C., National Council of Teachers of Mathematics.
National Council of Teachers of Mathematics (1963). Enrichment mathematics for high school. Washington, D.C., National Council of Teachers of Mathematics.
National Council of Teachers of Mathematics (1969). More topics in mathematics : for elementary school teachers. Washington, D.C., National Council of Teachers of Mathematics.
National Council of Teachers of Mathematics (1970). A history of mathematics education in the United States and Canada. Washington, D.C., National Council of Teachers of Mathematics.
National Council of Teachers of Mathematics (1970). The teaching of secondary school mathematics. Washington, D.C., National Council of Teachers of Mathematics.
National Council of Teachers of Mathematics and National Council of Teachers of Mathematics. Committee on the Metric System (1948). The metric system of weights and measures. New York, Teachers College, Columbia University.
Nelson, D. and National Council of Teachers of Mathematics (1976). Measurement in school mathematics. Reston, Va., National Council of Teachers of Mathematics.
Swan, M. and National Centre for Excellence in the Teaching of Mathematics (2008). Mathematics matters : final report. London, NCETM.
Tett, G. (2009). Fool's gold : how the bold dream of a small tribe at J.P. Morgan was corrupted by Wall Street greed and unleashed a catastrophe. New York, Free Press.
Traces the relationship between a team of JP Morgan banking gurus and the current financial crisis, documenting their invention of a bold variety of allegedly risk-free investments that sparked a frenzy in the banking world and may have directly contributed to the market crash. |
Arithmetic
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Arithmetic or arithmetics is the oldest and most elementary branch of mathematics. It consists of the study of numbers, especially the properties of the traditional operations between them—addition, subtraction, multiplication and division. Arithmetic is an elementary part of number theory, and number theory is considered to be one of the top-level divisions of modern mathematics, along with algebra, geometry, and analysis.
The below are the links to the sub topics. Click on the links below to view details about topics. |
Your Pie discounts pizza for 'Pi Day'
Not the dessert, but the mathematical concept. Pi is the ratio of a circle's circumference to its diameter. That's equal to about 3.14.
"Today is Pi Day is our biggest promotion of the year so we're selling pizzas for a discount of $3.14 to commemorate Pi. Obviously it's our customer appreciation day and teachers really love it, parents love it, and we see a lot of kids come in. It's just a really fun event for us to put on and we enjoy it," Christa Feighner said co-owner of Your Pie.
The mathematical holiday is a way to spread the word about the benefits of math |
Endured Fractions includes volumes -- quantity 1: Convergence conception; and quantity 2: illustration of services (tentative title), that's anticipated in 2011. quantity 1 is devoted to the convergence and computation of endured fractions, whereas quantity 2 will deal with representations of meromorphic features through persisted fractions. Taken jointly, the 2 volumes will current the elemental persevered fractions thought with out requiring an excessive amount of prior wisdom; a few uncomplicated wisdom of advanced features will suffice. either new and complicated graduate scholars of endured fractions shall get a complete figuring out of the way those endless buildings paintings in a few functions, and why they paintings so good. A assorted buffet of attainable functions to whet the urge for food is gifted first, sooner than the extra simple yet modernized conception is given. This new version is the results of an expanding curiosity in computing specified services via persevered fractions. The tools defined intimately are, in lots of circumstances, extremely simple, but trustworthy and effective. Contents: Introductory Examples; fundamentals; Convergence standards; Periodic and restrict Periodic persevered Fractions; Numerical Computation of persisted Fractions, a few persevered Fraction Expansions.
Whilst do the palms of a clock coincide? How most likely is it that young ones within the similar category will percentage a birthday? How can we calculate the amount of a doughnut? arithmetic for the Curious presents an individual drawn to arithmetic with an easy and unique account of what it could do. writer Peter Higgins provides transparent motives of the extra mysterious beneficial properties of youth arithmetic in addition to novelties and connections that turn out that arithmetic should be relaxing and whole of surprises.
This ebook is an creation to nonlinear programming, written for college students from the fields of utilized arithmetic, engineering, and economic climate. It offers with theoretical foundations besides assolution tools, starting with the classical techniques and attaining as much as "modern" tools. numerous examples, workouts with designated recommendations and functions are supplied, making the textual content enough for person stories
In 1644 the Qing dynasty seized energy in China. Its Manchu elite have been first and foremost obvious through so much in their matters as foreigners from past the good Wall, and the consolidation of Qing rule awarded major cultural and political difficulties, in addition to army demanding situations. It was once the Kangxi emperor (r.
Additional info for Continued Fractions. Volume 1: Convergence Theory
Example text
From power series to continued fraction. Use the procedure of Example 13 on page 30 to find the first terms a1 z a2 z a3 z a4 z 1 + 1 + 1 + 1 +··· of a continued fraction expansion of f (z) := ez − 1. Problems 51 22. From continued fraction to power series. Use the procedure of Example 14 on page 33 to find the first terms of the power series expansion at 0 corresponding to the continued fraction z −z/2 z/6 −z/6 1 + 1 + 1 + 1 +· · · 23. ♠ From approximants to continued fraction. Let {fn }∞ n=0 with f0 := 0 and fn = fn−1 for all n be a given sequence of complex numbers.
2. The approximants Sn (−1) are all equal to −1. e. we have an analytic continuation of the limit function in |z| > 1 to the whole plane, minus the origin. Also these properties have their non-trivial analogues. 2) then the continued fraction K(an (z)/bn (z)) converges to one function for |z| < 1 and to another function for |z| > 1. We have already seen that Sn (z) (probably) converges faster to its value than its classical approximants Sn (0). 2) is fast enough, then Sn (z) actually converges in a larger domain, and thus provides analytic continuation under proper conditions.
But in a slightly different form, as a subtraction algorithm rather than a division algorithm ([Eucl56]). But at that time it did not lead to a continued fraction. The birth of continued fractions, like many other things in the culture of mankind, took place in Italy in the renaissance, by Bombelli in 1572 ([Bomb72]) and Cataldi in 1613 ([Cata13]), in both cases as approximate values for a square root. , long before the renaissance, touched upon the idea of a continued fraction, but an ascending one ([Fibo02]). |
So the other day one of my students came up to me in semi-awe asking me how its possible for me to do rapid multiplications in my head. He's seen me do two digit and rarely three digit computations in my head, before I resort to using the calculator (I get lazy too!). "In fact," he's asked me, "why don't you do all sorts of computations in your head without the use of electronic aids?" He was implying of course that as a mathematician I should have a strange power to multiply numbers instantaneously and with no effort. I chuckled. If he only knew that in my years as a mathematician I seldom even see numbers or do arithmetic at all! The tricks I know I've picked up by myself or invented them as I've needed them, particularly or especially when I'm teaching high schoolers and university students like himself (and the calculator not being within reach)!
I've become convinced that the most efficient way to multiply numbers instantaneously and with little effort is to simply memorize the lists of two, three, and even four digit multiplication tables. Indeed, wouldn't it be a lot easier to just "know" that 1331 times 11 is 14,641, than to actually grab paper and pencil and physically do it (or use another method for mental calculation that requires considerable thinking)? It is exactly this, after all, which we ask youngsters to do: memorize the multiplication tables of 2 through 9 (rarely up to 20), or the multiplication of two identical numbers (squares). This can however become time-consuming; we don't want our children to spend their lives learning to immediately recall that 123 times 321 is 39,483. Nevertheless, I bet my bottom dollar that this is exactly what some TV people do, including savants, who indeed may have additional powers of retention (as a photographic memory) and the intention and attention to do so: to "quickly" or instantaneously multiply any two (large) numbers (really, recall their product from memory than do any computation at all).
One is taught of course the usual algorithm to multiply two (any) numbers that involves putting the largest on top and the smallest on the bottom, then taking the units digit of the second number, multiplying through digit by digit and making sure to account for all carry digits, including a zero at the units position in the next row and doing the same with the second digit, etc etc. This algorithm of course requires sparse knowledge, only the multiplication tables of 2 through 9. However the tradeoff is that this method is a bit time consuming, and often requires paper and pencil.
But by the time a student comes to learning special products, one is not told "use these rules to multiply numbers more easily." One is instead introduced to the (very boring) topic of multinomials (usually binomials) and how we go about obtaining their product. It is up to the student to smart up and think, "hey, this is immensely applicable to mental arithmetic, too," and few people can synthesize and apply such information solely on their own. Usually one is never "told" that to multiply 17 times 23 one can imagine it as the binomial product of (10 + 7) and (20 + 3) and "FOIL" it in one's head, which is a lot easier to do than envision what one would do with paper and pencil following the usual algorithm (and having to keep track of the carry digits and the "shifted" rows, for example). I can easily multiply 10 times 20 (two-hundred), 10 times 3 (thirty), 7 times 20 (one-forty), and 7 times 3 (twenty-one), and then add that up if I can retain those numbers to obtain 391 (which, incidentally, is exactly what we do with the usual paper-and-pencil algorithm anyway... except harder, and hence why we need paper and pencil: we split the second number into an addition of tens, hundreds, thousands, etc, then we distribute the first number on the explicit sum, and then sum!). There are some binomial products that are easier to calculate mentally than others, case depending. For example, I could have split 17 times 23 into (20-3) and (20+3), which, one recalls, is 400-9=391 if we use the so-called difference of squares, and more easy to compute than by FOILing as above. To recognize such patterns takes some fine-tuning, but not too long. Some people become really really good at it, and it's often what I do when I multiply two numbers in my head without a calculator.
There is another little trick that I thought up while working out some arithmetic problems with my students but I never really made conscious nor explicit (until now, that is), and I'm sure we all do the same, for example when counting money. It is this: rather than multiplying two large numbers, sometimes it's easier to just divide. Say you have 500 peso bills and you've got 32 of them. Since 500 is half of 1000, it follows that I should get half of 32000, which is 16000. Like this:
This nifty trick is immensely powerful! Let me restate it here: multiplying by five (fifty, five-hundred, five-thousand) is in fact very much like dividing by two. The reason is that and are related. So now it is really easy to multiply, say, 5 times 132 without much effort. Rather than multiplying times five, simply divide the second number by two and then multiply by ten to obtain 660. The reason is this:
Of course, it is very much helpful that the second number is divisible by two. In much the same manner, multiplying by 25 (250, 2500, 25000, etc) is akin to dividing by 4. Say you have 25 times 32. The second number is divisible by 4, so dividing 32 by 4 gives 8. Next multiply by 100, to obtain 800. Here it is explicitly:
Of course it helps that the second number given was divisible by 4. It need not be, but one has to deal with the decimal representation of the ensuing fraction.
Next, say we have 75 times 32. Multiplying by 75 (750, 7500, etc) is similar to multiplying by 3 and then dividing by four (or first dividing by four and then multiplying by three). So 75 times 32 is really 8 times 3 times 100, or 2400:
Seventy five times a number that is divisible by four is especially easy to calculate this way.
In much the same manner:
Multiplying by 125 is similar to dividing by 8. So numbers that are divisible by 8 are especially simple to multiply by 125. For example, 125 times 88 is 11,000. (Why?) Also, if the second number is divisible by four, you can divide by four and then multiply by five. For example, 125 times 44 is 5,500. If the second number is divisible by 2, then you can divide by 2 and multiply by 25 (which in turn is like a division by four). So 125 times 18 is 2,250. Just make sure you keep track of the multiplication by factors of ten.
Multiplying by 375 is similar to dividing by 8 and then multiplying by 3. Numbers divisible by 8 are especially simple to multiply by 375. For example, 375 times 64 is 24,000. (Why?) Like above, there's more that can be said here.
Multiplying by 625 is like dividing by 8 and then multiplying by 5. Again, numbers divisible by 8 are especially simple to compute. Say 625 times 56 is 35,000. (Make sure you see this.)
Multiplying by 875 is like dividing by 8 and then multiplying by 7. Say 875 times 24 is 21,000. (Yeah?)
As you may notice, multiplying by any multiple of twenty-five is like dividing by four or by eight (and then multiplying by a usually small compensator). Also notice that multiplying by any multiple of five is like dividing by two and then compensating with a multiplication. When I've had to multiply three-digit numbers, this is usually the pattern that I follow (for numbers that are multiples of 5 or 25, I've been lucky with my students), and that's how my students are wowed. Neat-o. Can you think of patterns that arise that might involve divisions by three? By six? By seven? By eleven? Fractions involving these numbers in the denominator usually imply a repeating decimal, so. Hmm. Maybe it's not so clear? Let me know what you think! |
Science Math Socks Download List:
Can you solve the problem of "The Unfair Subway"? Marvin gets off work at random times between 3 and 5 p.m. His mother lives uptown, his girlfriend downtown. He takes the first subway that comes in ei...
'A WITTY BOOK THAT PROVOKES THE IMAGINATION' The TimesHow many socks make a pair? The answer is not always two. And behind this question lies a world of maths that can be surprising, amusing and even ...
Liven up learning and celebrate special days with these interactive, literacy-building pocket-chart poems! Each kid-pleasing poem comes with reproducible art templates and step-by-step lesson plans th...
In 1868, The Times reported that poisons contained in dyes were affecting the public's health. A doctor informed a London magistrate that brilliantly coloured socks had caused severe "constitutional a...
Explore, support and consolidate Early Years Mathematics with a colourful, simple story for ages 4-5, containing key mathematical concepts and practice opportunities. - Support learners with a brightl...
"Know ye not that ye are Gods?" is old advice but has never been popular. What difference would it make if it were true? What difference would it make if there was science behind the idea? Well, it is...
This proceedings present the results of the 29th International Symposium on Shock Waves (ISSW29) which was held in Madison, Wisconsin, U.S.A., from July 14 to July 19, 2013. It was organized by the Wi... |
Mathemagic?
In my opinion, today is a magical date: 4/8/16 (dd-mm-yy; as I write on my notebook). So let me tell you what I think about "mathematics" and "magic". I believe that magic is an art of concealing facts leading to astonishing results. Magic trick is interesting from perspective of both observer and performer. Performer gets satisfaction of being able to fool observer (by making him/her believe that he/she can't do it), and on the other hand observer gets satisfaction of being able of witness an act which he/she can't perform (a quality of appreciation is expected).
Now, as many of you have observed, mathematics (or nature in general) is very much magical in the same sense. Only experts (algebraist/number theorists…) can "understand" the rules (called theorems) behind the actions (called computations) they ask you to perform (which amaze you). So, in general, whenever you are using a result (for example, an integer has unique factorization into prime numbers) without "knowing" the proof, you are performing "mathemagic" for yourself. When you take magic out of mathematics, you get what mathematicians called rigour.
I believe that the most important rule for performing a magic trick is to never reveal the secret rule (though the audience is free to conjecture and prove the possible secret rule). This is very much different from first rule of cryptanalysis, since while doing cryptanalysis you "must" know the algorithm/rule used to encipher the message and task is to find the key to decipher the cipher. Trying to find the secret rules for a magic trick is much more interesting that trying to decipher a cipher. So, I will leave you with a classic "mathemagic" trick and you as observer of this trick, try to find the secret rule governing it (but never reveal it to others!!!):
Write down the year you were born and under that the year of some great event of your life (like year of graduation, the time you saved somebody…). Now write the only even prime number. Write down your age by the end of this year (i.e. 2016) and the number of years ago the great event (quoted above) took place in your life. Now add all these. I know what the total will be! Today it's 4034. |
26 of 28
The History of the "Zero": The Influence of Geography and Culture on the Invention of the Number Zero.
[ send me this paper ]
A 5 page review of the history of the manner in which the number zero came into common mathematical usage. Traces its development to Mesopotamia and through the various other regions of the world. Distinguishes between the mathematical concept and the philosophy of nothingness. Bibliography lists 3 sources.
Filename: PPzero.rtf
The History of the Pythagorean Theorem
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5 pages. An interesting research paper on the background of the Pythagorean Theorem that is used in algebra and geometry. Details the history of the theorem. This is actually a fascinating history and one that is not often discussed in the classroom. The Pythagorean Theorem was founded by a cult that actually worshipped numbers and felt that everything in the Universe was number-related. A very interesting history for a well-known math theory to say the least! Bibliography lists 2 sources.
Filename: JApythag.rtf
The Importance of Mathematics in Early Greek Culture
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A 12 page comprehensive study of early Greek mathematicians and their cultural significance. Included in the discussion are Ptolemy, Pythagoras, Aristotle, Plato, and others. Bibliography lists 9 sources. (also related to Astronomy)
Filename: Greekmat.wps
The Life and Works of Euclid of Alexandria
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This 10 page paper gives a brief overview of the Greek mathematician, Euclid. He was the chief architect of the concept and science of geometry. Bibliography lists five sources.
Filename: Euclid.wps
The Nature of Math Assessment: Assessment Inconsistencies, Priority of Math Assessment and the Use of Authentic and International Programs
[ send me this paper ]
This is a 10 page paper discussing the interpretative and inconsistent nature of math assessment and possible authentic assessment programs. Several issues relate to the inconsistent and interpretative nature of math assessment within the educational curricula today. Firstly, on a national basis, studies reveal the range in attitudes in regards to importance math assessment has in the overall educational curriculum. While math assessment is often mentioned, its importance does not seem as highly weighted as other aspects of assessment of school performance. Secondly, within the school systems themselves, teachers in mathematics vary a great deal in their teaching practices which can affect assessment; perception of their students' abilities based on social factors and past performance; perception of the importance of national and consistent assessment; and desired learning outcomes. There are many educational assessment tools which are available. However, assessment tools vary in regards to their expense, time, and curriculum considerations which can affect whether or not the tool can be usefully applied. One of the goals of the U.S. educational system is be "the first in the world in mathematics and science achievement" and some educational programs are trying to fulfill this goal by not only developing consistent national education and assessment programs but programs which are comparative on an international scale, such as that found within the International Baccalaureate Program (IB).
Bibliography lists 10 sources.
Filename: TJmaths1.rtf
The Problems of Multicollinearity, Heteroskedasticity, Outlying and Influential Cases and Non-normally Distributed Errors When using Ordinary Least Squares (OLS) Regression
[ send me this paper ]
Multicollinearity, heteroskedasticity, outlying and influential cases, and non-normally distributed errors all present problems for ordinary least squares (OLS) regression. This 8 pager paper explains why each is a problem for OLS, how it can be detected and looks at steps can be taken to deal with each of them. The bibliography cites 5 sources.
Filename: TEOSLerror.rtf
The Science of Escher / Symmetry and Metaphor
[ send me this paper ]
A 10 page paper that provides an overview of the elements of science in the works of Escher, with a focus on symmetry and metaphor based in his mathematical approach. Bibliography lists 8 sources.
Filename: Escher.doc
The Subjectivity of Probability Distributions and Tools for Collecting Qualitative Data
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A three page address of two specific questions. The first questions regards the usefulness of probability distributions in terms of statistical analysis. The second question asks for the delineation of three specific tools that are useful in the collection of qualitative data. Bibliography lists 2 sources.
Filename: PPresrc |
What exactly are matrices and determinants? Wouldn What do they represent, and why is it messy to calculate the determinant for 3x3? And finally, how is it possible to solve system of equations, find volumes, and cross multiply vectors using them? I mean, these things are barely related, yet they can all be solved similarly.
Please help me understand all of this, it'd would also be great if someone could provide a material to the history of all of this.
Staff: Mentor
A matrix is a rectangular array of numbers, symbols, or expressions (cribbed from Wikipedia - There is nothing in the definition of a matrix that specifies anything about input data or operations, so I don't know what you're referring to here.
A system of linear equations can be represented in a shorter form as an augmented matrix. The same types of operations that you would perform on pairs of equations can be performed on the rows of a matrix, yielding a matrix that is "equivalent to" the starting matrix. IOW, the solution that is represented by the final matrix is also the solution to the starting matrix. Using matrices in this manner is mostly a matter of convenience -- you don't have to write so much stuff.
A.MHF said:
Wouldn
Someone along the way defined it to be like this.
A.MHF said:
What do they represent, and why is it messy to calculate the determinant for 3x3?
The determinant of a square matrix is a measure of a matrix, in some sense. More formally, the determinant is a mapping from the set of n x n matrices to the real numbers. If you think the formula for the determinant of a 3 x 3 matrix is messy, the formulas for 4 x 4, 5 x 5, and higher order matrices are even worse. It is possible, though, to use a few properties of the determinant to simplify these calculations. Also, you can expand a determinant by minors, breaking down, say, the determinant of a 4 x 4 matrix into four 3 x 3 determinants.
One of the most useful properties of the determinant is that if its value is zero, the associated matrix is noninvertible.
A.MHF said:
And finally, how is it possible to solve system of equations, find volumes, and cross multiply vectors using them?
Cramer's Rule uses determinants to solve a system of equations. The cross product, which is limited exclusively to 3-D vectors, uses a pseudo-determinant to arrive at the vector that represents the cross product of the two input vectors. It's not really a determinant, which can be inferred from the "pseudo" in the description (a true determinant evaluates to a number, not a vector), but the calculation is done in the same way. The magnitude of the cross product, ##|\vec{A} \times \vec{B}|##, gives the area of the parallelogram determined by the two vectors. There's another product involving vectors that gives the volume of the box that the three vectors determine.
A.MHF said:
I mean, these things are barely related, yet they can all be solved similarly.
Please help me understand all of this, it'd would also be great if someone could provide a material to the history of all of this.
Wikipedia is a good starting point. Search for "matrix", "determinant", and "Cramer's Rule".
Regarding the determinant question: Take three vectors and write their components down as column matrices. Now stick them into a matrix. The determinant of the matrix is the volume of the parallelipiped that has those three vectors along its sides. It's a signed volume because, depending on the order of the vectors, you may end up with negative the volume instead. This geometric interpretation of determinants can be very useful, because you can use it to calculate volumes in higher dimensional spaces. If you think of an area as a "2 dimensional volume", you can use determinants to get areas too!
Regarding the alignment question: Matrices are often used to represent more abstract entities called linear operators. Operators act on things, and usually we think of operators acting on things placed toward the right of them. So we think of matrices as operators that act on things placed on their right hand sides. Consequently, a vector will need to be represented as a column with m rows so that an n x m matrix can "act" on it. We can, of course, think of operators as acting on things placed on their left sides instead. If we had chosen to do that, then vectors would be expressed as rows instead of columns.
In the most general sense, a matrix is just an array of numbers. In specific, given situations, it will take a given meaning. It can serve to, e.g., represent a system of linear equations, a (finite-dimensional *) linear operator, as the adjacency graph of a matrix, as representing a Markov process, etc.
* Though there are infinite matrices too.
It seems, from the context of your question, that your matrices are used to represent linear operators. Like others said, determinants are used to decide whether a set of linear equations are linearly dependent from each other. For the ## 2 \times 2 ## -case, a line is given by a pair of numbers (these numbers determine the slope of the line). Then the two lines are independent if the slopes are equal. This is all equivalent to having ## ad-bc=0##. Something similar for ## n \times n ## matricesIsnt it more accurate to say that every matrix is equivalent to a linear map ; more formally, I think this is equivalent to the fact that the space of all linear maps (we can even define it between modules, but it is much clearer if we work with ## \mathbb R^n, \mathbb R^m ## ) defined on an ordered basis is isomorphic to the set of all ## n \times m ## matrices under multiplication (with the j-th column being the image of the ##j_th## vector in an ordered basis )? A matrix may represent many other things depending on the context. If we have , e.g., a graph, the matrix may define the connectedness properties of the graph, by using a ## 0-1## -matrix ## M_{ij} ##, where ## m_{ij}=1 ## if there is an edge joining vertex ##i ## with vertex ##j ## and ## m_{ij}=0 ## otherwise. |
Let's take the Powerball, an incredibly popular lottery in the USA. Last week someone won $571 million in the Powerball.
So what are your chances of winning?
If your 5 numbers plus the Powerball match the winning six numbers drawn, then you win or share the Grand Prize. If the jackpot is not won in any drawing, the First Prize Pool Money is carried forward and is added to the next Powerball JackpotThis is an extreme example, but it shows there may be a fundamental relationship between the availability of information and the concept of dimension infinitelyWhat's a Prime Core? Take a prime number in binary, then strip off the first and last digits (which, for all primes except 2 are always 1's) then interpret the binary string you have left as an integer, and that's the prime core.
Example, the prime 79 in binary is 1001111 so its core is 00111 which is 7. So using C to represent the prime core operation, we have C(79)=7.
Then here's an interesting question: "when is the core of a prime also a prime?"Hydrogen cars are very different from all other cars (gasoline, battery, hybrid). Here's how they work..
A hydrogen car is a 100% electric car, but instead of a battery it uses hydrogen fuel cells. These are simple, feed them hydrogen gas and they generate electricity. They are like a battery that never need recharging. There is no combustion, nothing burns, hydrogen gas is simply fed to the fuel cells to produce electricity.
So where do we get the hydrogen? The cars carry high pressure tanks of hydrogen gas to supply their fuel cells. The tanks get refilled at a hydrogen gas station, and the filling process is very similar to regular gasoline filling. The hydrogen gas station produces its hydrogen on the spot. How? By electrolysis of water.
Yes, the raw material to generate hydrogen gas is water!
So, the hydrogen gas station makes hydrogen from water. Cars refill their hydrogen tanks. The hydrogen goes through the car's fuel cells and generates electricity to drive the car. And the exhaust? The car's exhaust is water vapor. There is zero pollution.
The whole thing is a water to water cycle!
The range of a hydrogen car is about the same as a gasoline car, and the refueling time is about the same.
Also, hydrogen is the most abundant element in the universe. So I doubt we'll run out!You're a point mass and you live on the x axis. That's your entire world. It's "Lineland". What's your life like?
First, as regards moving, you only have two directions, forward and backwards. And if you meet another point mass you cannot pass. So you can only know two other masses. You have just two friends maximum!
You have no reason to count objects beyond two, so you might be slow in developing the concept of integers. Or perhaps you never develop the concept at all. You simply have no need for it.
What about Physics in Lineland? You're a point mass, so you have mass, let's say m. Another point mass could have a different mass, say M. So at least gravity exists, right? It does, but it has a strange form. Newton's formula for the gravitational force F between two masses m and M is..
F=G*M*m/(r^2)
where G is a constant and r is the distance between the two masses.
The r^2 term is good in a 3D space, but in general it's r^(n-1) where n is the dimension of the space. Putting n=1 for Lineland we get..
r^(1-1)=r^0=1 so F=G*M*m
Which means F is independent of distance! Gravity has the same strength no matter how far apart the objects are. So physics in Lineland is very different.
This is Lineland on the x axis. What if Lineland is the circumference of a circle? That's even more interesting. Would you be aware that Lineland had a "curvature"? What does gravity do now that Lineland is a closed loop? What happens if Lineland is a closed loop that intersects itself at several points? What happens at these intersection points and how do they contribute to gravity? How do things change as the number of point masses in Lineland changes? It turns out that even 1 dimension can be very complex!
Just think, there's probably a 4 dimensional world somewhere with math teachers looking for a nasty problem to set on an exam. Finally they come up with one, "explain how math would have developed if our world was constrained to just 3 dimensions".
Quantum Mechanics deals with incredibly small objects and this makes it difficult to visualize what's happening. But let's bring a quantum object up to desktop size and see how it might behave veryShannon Entropy (also called Information Entropy) is a concept used in physics and information theory. Here's the scoop..
Suppose you have a system with n states i.e whenever you make an observation of the system you find it's in one of the n possible states.
Now make a large number of observations of the system, then use them to get the probability pi that if you make an observation the system is in state i. So for every state of the system you have a probability pi.
Now construct this crazy sum = p1*log(p1) + p2*log(p2) +... + pn*log(pn) where the sum is over all the states of the system.
If the log is base 2 then (-1)*sum is called the "information entropy" of the system.
Note that "information entropy" applies to a complete system, not individual states of a system.
Here's a simple example..
My system is a penny and a table.
I define the system to have 2 states.. penny lying stationary on the table with heads up or with tails up.
My experiment is to throw the penny and then observe which state results.
I throw the penny many times and make notes. It lands heads up 1% of the time and tails up 99% of the time (it's biased). of an integer variable. IfBut not just any molecule.. the human DNA molecule is about 1.5 meters long and incredibly thin.
The shape of the molecule is very clever. Think of a ladder with 4 different color rungs. The sequence of colors is the information!
Now imagine a ladder with about 3 billion rungs and with an amazing twist. Literally. Nature twists the molecule into a corkscrew (helix) shape. This simply gives it extra strength, because a break in the DNA molecule would have disastrous consequences.
That's the amazing human DNA molecule!
Nature takes very good care of the huge DNA molecule - it winds it up and packs it into separate containers called Chromosomes. This is just an efficient method to make sure the molecule fits into a tiny space and is protected.
The whole packaging system, with the DNA molecule wound and packed into Chromosomes, is referred to as a Genome. But it's just a molecule!
The DNA molecule encodes data that acts as a program to run each cell in the body. Just like computer data can be reduced to strings of 0s and 1s (2 units), DNA uses 4 units.
The order (sequence) of these units is the program.
And Nature is massively parallel. No central processor here.. every cell in the body contains a copy of the DNA molecule.
Think of a cell as a factory. It manufactures all sorts of substances, and the instruction on how to do this is provided by the DNA. Many of these substances are proteins, so DNA has special instruction sequences (called Genes) that tell exactly how to make different proteins. The sections of DNA between the Genes are a bit of a mystery. It's not clear exactly what these instructions do, if anything.
So the cell simply reads the DNA instructions and makes the appropriate proteins. It's a wonder of molecular manufacturing!
DNA can be analyzed at many levels, for example..
- Just look at the chromosomes for abnormal shape.
- Sequence (read the order of the units) a gene in one of the chromosomes.
- Sequence all genes in one of the chromosomes.
- Sequence all genes in all the chromosomes.
- Sequence all the DNA (even the instructions between the genes) in all the chromosomes. This is called "full sequencing".
A major producer of DNA sequencing machines is a company called Illumina. They reduced the cost to sequence a human DNA molecule from $100 million in 2001 to about $1,000 in 2014. The rate of progress is staggering!
Do we all have the same exact DNA?
No. We are all 99.9% the same, but that 0.1% means about three million differences between your DNA and anyone else's. It's these differences that are used in DNA testing.
Oh, and our DNA is about 99% the same as our closest relative, the chimpanzee.Take a strip of paper, join the ends so you have a band. This is a very simple object with 2 surfaces.
Now give the strip of paper things get strange fast..
This time give the strip of paper 4 half twists before joining the ends. This has 2 surfaces. Now play around with it for a while. At some point it will suddenly "flip" into a double thickness band with 1 half twist. In other words it flips into a double thickness Mobius Strip. One surface has gone, and so have 3 half twists!
It seems that Physics likes the number 3 at all levels, from macroscopic down to the quantum level. Consider this..
1. There are 3 spacial dimensions.
But you can think of this as 3 families each containing 2 members..
Up/Down
Left/Right
Backward/Forwards
2. Quarks consist of 3 families each containing 2 members..
Up/Down
Charm/Strange
Top/Bottom
3. Quarks are bound together to form composite particles (such as protons) by an incredibly strong force known as the color force. Gluons are the particles that mediate this force. For this model to work quarks must have a color charge - and it comes in 3 families each containing 2 members..Oh sure, you'll see driverless cars on the road. A few here and a few there. But they will never happen in a major commercial way for a very simple reason: people love to drive.
I do. It's a skill that took me a while to learn. I'm proud of my skill and I enjoy using it. A driverless car takes that pleasure away. Can you imagine being in a car that obeys all traffic laws including speed limits. It would be incredibly infuriating and amazingly boring.
Plus, on todays aggressive roads it would be outright dangerous. Go ahead, plod along at the 30 mph speed limit while 18 wheeler trucks swerve past you at 55 mph. No thanks.
But there's one good thing about driverless cars - as more come on the road they will put my driving skills to the test in order to avoid them!
Driverless cars are putting me to sleep. Wake me up when we have driverless trains and driverless subways.
You gotta love the driverless car industry - a brilliant solution to a problem that doesn't exist.
After a century of neutrons discover the internal structure of the proton. Egyptians built The Great Pyramid at Giza as an amazing burial monument for Pharaoh Khufu. But they did more. They also used it as a showcase for their mathematical and engineering skills.
Pi is probably the most famous number in mathematics. Draw any circle, then measure the length of the circumference and the length of the diameter. Divide the two numbers and you get pi. But a circle was not used in the design of The Great Pyramid of Giza, right? Wrong! Not only was a circle used it was totally fundamental to the design. Here's how..
The Khufu Pyramid (The Great Pyramid of Giza) had a design height of 280 royal cubits and a base length of 440 royal cubits. The Pyramid we see today is slightly different due to erosion and theft of stone. So let's stick with the original design size.
So the distance around the base of the pyramid is simply 4*440=1760
Let's divide this by twice the height, which is 560, so we get..
1760/560=22/7
The number 22/7 is the most famous approximation of pi, and it's a pretty good one, in fact..
22/7-pi=0.001
So the Khufu Pyramid was built on circular geometry. Which means the Egyptians knew pi by the time they built the Great Pyramid (2560 BC). For all we know they may have known it much earlier.
But it gets stranger, not only did they know pi, they used it to define the dimensions of their most sacred monument. What does this mean?
The Mobius strip is a simple object yet full of surprises. Here's one you may not know about..
Take a strip of paper, give it here's the surprise..
Take a strip of paper, but give it 4 half twists before joining the ends. This has 2 surfaces. Now play around with it for a while. At some point it will suddenly "flip" into a double thickness band with 1 half twist. In other words if flips into a double thickness Mobius Strip. One surface has gone, and so have 3 half twists!
We're used to seeing numbers represented in base 10 "decimal" notation, and almost all prime number lists use base 10. But we can represent numbers in any base we please. In base 10 we use 10 symbols 0,1,2,3,...,9 and in base n we use n symbols 0,1,2,3,...,(n-1)
The simplest base is 2, because in that base we have only 2 symbols 0,1
Base 2 is also call "binary" and writing numbers in binary makes them look like computer data.
In binary the positions represent 1,2,4,8,16,32,.. so the general representation of a positive integer n is..
n=sum{ai*(2^i)} where the coefficients {ai} are all 0 or 1 and the sum is over i from 0 onward.
Writing numbers in binary can help spot patterns we might not notice in other bases. For example..
In binary all prime numbers except 2 begin and end with 1.
The first 2 digits of the prime 71 is the prime 3 and the last 5 digits is the prime 17. So we could define a "+" operation and say that 3+17=71. Notice that the + operation depends on order, so 17+3=113 is different, but it's still a prime!
The prime 13 is just the prime 11 written backwards. The same is true for 23 and 29 and lots more. Many primes are just an earlier prime written backwards!
Some primes have all digits set to 1, so these primes are of the form (2^n)-1 where n is just the number of binary digits. Primes of this form are called Mersenne primes, named after Marin Mersenne, a French monk who studied them in the 17th century.
I wonder what we might discover if we used sophisticated computer pattern recognition on prime numbers in binary format?
Imagine you own a bakery and the only thing you sell are loaves of bread. Not only that, but all your loaves are identical, which means they are all the same size and everything else is the same.
So when customers come into your shop they just tell you how many loaves they want..
{1,2,3,4,...}
One day a customer comes in and explains that they love your bread but your loaves are too big. They ask if you make smaller loaves. You don't. But then you have a clever idea. You take a knife and cut a loaf into 2 equal sized pieces. You sell one piece to the customer and they are happy.
But what did you just sell? It was not a loaf. It was something less. You chopped a loaf into 2 equal pieces and sold one of the pieces. You sold "one out of two", so you could write that as 1/2.
This idea is popular with your customers. Soon you are chopping your loaves into 5 equal pieces and selling customers 1, 2, 3 or 4 of the pieces. That's 1/5, 2/5, 3/5 or 4/5.
Of course, if you sold a customer 5 out of 5 then that's the same as the whole loaf, so 5/5=1.
One day a really fussy customer comes into your bakery and asks for 5/8. You know exactly what to do. You take a loaf, chop it into exactly 8 equal size pieces and then sell the customer 5 of the pieces.
These funny looking things like 1/2 and 5/8 are called fractions. Mathematicians call them rational numbers. That's the fancy mathematical name for them.
The electron was discovered over 100 years ago, in 1897 by JJ Thompson at the University of Cambridge in England. It was the first elementary particle to be discovered and it's still as elementary as ever. Elementary means no internal structure has been found - so far.
Moreover, it now rules our world. Without it there would be no electricity. That means no lights, no TV, no batteries, no computers, no Internet, no iPhones.. to name just a few. Who says elementary particles are abstract objects!
Who was JJ Thompson? He was born in Manchester. His mom came from a local textile family. His dad ran a bookstore. He changed the world forever. His list of students reads like a who's who of physics. He won a Nobel Prize and so did many of his students.
IXL Math is a online math education resource for ages Pre-K through 12th grade. It covers everything from counting to Calculus.
IXL Math comes in two major versions - one for parents and one for teachers.
Parents use IXL Math at home to help their kids, and teachers use it in the classroom.
IXL Math is offered on a subscription basis and has about 6 million subscribers. It's used in over 190 countries. Their website offers guest access where you can try over 6,000 interactive math skills for FREE. Try before you buy.
First, I want to introduce people to math and physics concepts in a simple and casual way. It's easy to use textbooks without really understanding the basic ideas. I want to avoid that. I'm interested in explaining fundamental concepts and ideas.
Second, I want to get people interested in math and physics. Good teachers don't just teach, they create a lifelong desire to learn.
Last but not least, I want to learn math and physics. There's no better way to learn a topic than to explain it clearly and simply to others. I spend a lot of time trying to improve my posts.
So far so good. My blog has worldwide readership and I get a lot of feedback. It helps. I'm always going back and tweaking posts to try and make them betterBraille was invented by Louis Braille in 1837 and was the first binary form of writing developed in the modern era.
Braille was based on a tactile military code called night writing, developed by Charles Barbier in response to Napoleon's demand for a means for soldiers to communicate silently at night without a light.
Today, computer professionals will instantly recognize this as 6-bit encoding. Perhaps the first byte was 6 bits!
The Prime Number Theorem is one of the most famous theorems in mathematics. It tells us something about the distribution of the prime numbers.
How many of the first n integers 1,2,3,4,....,n are prime? The Prime Number Theorem says the number of primes is approximately n/log(n)
This is not an exact count, n/log(n) is only an approximation, but as n gets bigger the approximation gets better and better.
The Prime Number Theorem is also a statement about the Shannon Entropy of the primes! Here's how..
Suppose you have a machine with a big red button. Each time you punch the button the machine responds by displaying an integer in the range 1,2,3,....,n. After much experimentation you discover that the probability of getting integer j is pj. Then physics defines the Shannon Entropy of this machine as..
Shannon Entropy=(-1)*sum (pj*log(pj)) for j=1,2,3,...,n
In the special case where all numbers occur with equal probability pj=1/n for all j and we get the famous result for the Shannon Entropy of the machine..
Shannon Entropy=(-1)*n*(1/n)*log(1/n)=log(n)
Now imagine this is "distributed" equally across all numbers, so on average an individual integer has log(n)/n entropy.
If the integers 1,2,3,....,n contain m primes then the Shannon Entropy of the primes is simply m*log(n)/n
But the prime number theorem says that m=n/log(n) approximately. So the approximate Shannon Entropy becomes..
Shannon Entropy=m*log(n)/n=1
and as n approaches infinity this approximation becomes exact. So we can say that..
"The Shannon Entropy of the primes is 1".
This statement is equivalent to the prime number theorem. How strange!
To make a deep prediction about black holes and quantum gravity we first need to play with paper strips!
Take a strip of paper, join the ends, so you have a band. Let's use this as a model for a spin 0 particle.
Now give the paper strip 1 half twist before joining the ends. This is our model for a spin 1/2 particle.
But things get strange..
Now give the paper strip 4 half twists before joining the ends. This is a spin 2 particle. The only one known is the hypothetical graviton, carrier of the gravitational force. But if you play around with this thing for a while it will suddenly flip itself neutral particle. So a Graviton can oscillate into a spin 1/2 neutral particle.
This is just a simple model, but if graviton oscillation exists the implications are deep. Graviton oscillation would change physics as we know it. Here's 2 dramatic predictions..
1. Black holes contain a neutral spin 1/2 core
You could imagine that graviton oscillation requires extremely high graviton pressure - meaning it only occurs in super intense gravitational fields - such as the center of a black hole. This means the center of a black hole would not be a singularity as predicted by General Relativity, it would be core consisting of neutral spin 1/2 particles. You can think of this core as a "graviton condensate".
2. The 1665 a recent Cambridge University graduate decided to sit and think about the motion of objects. Isaac Newton had an ambitious goal, he wanted to describe the motion of all objects, from a ball to a planet. He realized that many object move because of gravity and so his thinking included gravity. Within two years he had produced a few simple laws that described the motion of all known objects with great accuracy.
It was a remarkable achievement. Newton's Laws of Motion remained the cornerstone of physics for centuries.
Then in 1905 another recent graduate published a theory that showed something amazing. It showed Newton's equations would fail badly if used to describe objects traveling at very high speed, and it gave more accurate equations. The new equations held a big surprise, they predicted that objects could never go faster than the speed of light. Nature imposes a speed limit!
Einstein's new theory was not just an improvement on Newton's theory, it was a total replacement that gave a deep insight into physics and the world it describes. And all this from a physics student who was average in university and could not find a job when he graduated! Albert Einstein's theory became known as "Relativity". It took just a few years of thinking. It changed physics forever.
So now all was good, Physicists could predict the motion of objects with remarkable accuracy. Some even claimed there was nothing else to do and physics was over!
But nature had other ideas. By the early 1900s experimental physicists were discovering new objects. These new objects were confusing, but they all had one thing in common - they were extremely small. At first many scientists refused to believed they even existed, but soon atoms, electrons and photons became accepted.
Of course, describing the motion of these new objects was easy - just use Newton's theory, or if you wanted more accuracy use Einstein's theory. Right? Wrong. When this was tried the results were terrible.
New laws were needed. But this time they were not produced by a recent graduate. It took a generation of physicists, each contributing critical parts to the puzzle, and it was not finished until the 1950s. It was a long hard slog. The laws, designed specifically for small objects, became known as Quantum Mechanics. The mathematics was complex but the accuracy was there. The theory was incredibly accurate!
So if you had to summarize Quantum Mechanics in one sentence try this.. "small objects behave very differently than large objects". Who knew!
What happened to Einstein?
He never made the transition to the new world of Quantum Mechanics. He understood the mathematics and even made some critical contributions in the early days, but he rejected its underlying philosophy. However, he had one more giant trick up his sleeve. In 1915, ten years after producing his Relativity theory be produced a much broader theory that included gravity - the thing that got Newton started 250 years earlier. His new theory was called "General Relativity". It's still in use today over 100 years later, because nobody has found a better description of gravity.
What happened to Quantum Mechanics?
It got more complex. The laws that describe atoms, electrons and photons had to be revised to describe the new objects that were discovered. The basic principles were the same, but the mathematics got even more complex. It turned out that Quantum Mechanics was not a simple theory.
So physics is built on three massive achievements: Newton's Laws, Einstein's General Relativity and Quantum Mechanics. The first two were produced by single individuals who became legends. The third required a huge team - a whole generation of physicists.
Where does physics stand today?
Using giant accelerators such as the LHC at CERN physicists are finding even smaller objects and there are hints that Quantum Mechanics may have problems describing them. Not only that, but physicists want to unify General Relativity and Quantum Mechanics into one theory.
Perhaps we need a new physics graduate - preferably one who can't find a job!
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Content written and posted by Ken Abbott [email protected]
Learn Mathematics and Physics
If you write a prime number in binary you can sometimes split it into 2 segments that are also primes.
For example, the prime number 7591 is 1110110100111 in binary (I use the left to right convention). Now snip out the first segment 11101 which is 29 and prime. Then snip out the second 10100111 which is 167 and prime.
So, denoting a binary string concatenation operator by "+" we can say 7591=29+167
Notice that our concatenation operator depends on order, so n+m is not the same as m+n. Mathematicians call this kind of operator "non-commutative".
One of the most famous unsolved mathematical conjectures totally lends itself to computer investigation.
It's the Collatz Conjecture, named after Lothar Collatz, who first proposed it in 1937. The great mathematician Stanisław Ulam not only failed to prove it but said, "perhaps mathematics is not ready for such problems".
Here it is as a computer program..
Pick any positive integer n
INFINITE LOOP
If n is even replace it by n/2
If n is odd replace it by 3*n+1
If n=1 bail out of loop
LOOP
The Collatz Conjecture says that no matter what number you start with you'll always bail out of the loop. In other words, no matter what number you start with you'll always reach 1.
The number of cycles needed to reach 1 is called the stopping time of n and denoted s(n). It turns out that the stopping time of a number is an interesting property and by no means simple.. for example s(27)=111
Mathematical statements phrased in terms of iteration seem to be especially nasty to prove. Perhaps Ulam was correct.. meaning mathematics was never designed for such problems!
After a century of hard neutrons document the internal structure of the protonBut f(n) of an integer variable n? Yes, if for a Simple Harmonic OscillatorThe AMS (American Mathematical Society) is an professional society who's goal is to advance mathematical research, scholarship and education worldwide. Their headquarters are in Providence, Rhode Island.
Consider this problem, "what number, when added to 5, gives the result 21".
Instead of a sentence, this problem can be written much shorter and clearer as an equation, like this..
5+x=21
where x denotes the number we are trying to find.
Of course, we could also write it as x+5=21 and this is exactly the same equation. Or we could write 21=x+5 which is of course the same thing.
If we manage to find x we say that we've "solved" the equation. Can we solve this equation? Well, we could guess a few numbers for x and try them out. Does x=9 work? Let's see, 5+9=14, so x=9 is not a solution. After a few tries we get the solution, which is x=16.
Guessing a solution is perfectly fine, but it's very time consuming, especially for more complex equations. Of course, we could program a high speed computer to guess solutions and try them out ultra fast until we finally hit on the right solution. And for some very tough equations this is indeed the method used. But this method has a huge flaw.. if it fails to find a solution it does not mean the equation has no solution. That's because even the fastest computer can only make a limited number of tries.. and the actual solution may be something we never get around to trying.
So, coming back to our equation 5+x=21 we should ask if there is a foolproof method that's guaranteed to find the solution. The answer is yes, and it's all about the = sign. Once you truly understand this simple sign solving the equation is easy. The = sign is the secret to learning Algebra!
So what does this sign really mean? It means the "object" on the left of the sign is the same exact object as that on the right. They are the same thing.. exactly the same thing. They are the same exact mathematical object but just written in different ways. So there's really only one object!
OK, so our equation says that 5+x is exactly the same object as 21. So, if I do something to 5+x and then I do the same thing to 21 the results will still be equal. Cool. So lets subtract 5 from 5+x to get the result x. Now do the same exact thing to the other side, I'll subtract 5 from 21 to get the result 16. But these two results must be the same, so I can write them as equal to each other, that is x=16.
Bingo, we've solved the equation without any guessing!
Also, I'm not sure if you noticed this, but we just did some basic Algebra. Don't let Algebra intimidate you, it's just the art of manipulating equations until you get what you want!
Let's look at a slightly more complicated example..
3*x+2=17
To solve it we want to isolate x on one side and get all the other stuff over to the other side. Here's a method I use. It's exactly the same technique as above, but it's faster and easier to handle. Or at least I think so, and I've used it over the years to do massive amounts of algebra!
First move the 2 over to the other side. It was adding, so when it moves over it subtracts, like this..
3*x=17-2=15
Now move the 3 over. It was multiplying, so when it moves over it divides, like this..
x=15/3=5
This technique is quite general and can be used for any equation. But notice that the order in which you do things is important. For example, you need to get the 2 over to the other side before you can handle the 3.
Let's consider the positive integers greater than 1, that is 2,3,4,5,..
Suppose we are given the first integer and asked to make all other integers using only the multiply operation.
We soon run into problems because 2*2=4 and we have no way to make 3.
OK, we just add 3 to our set of given numbers g, so now g={2,3}
Can we make 4? Yes, 2*2=4
Can we make 5? No, all our tries fail, so we add 5 to our set of given number g={2,3,5}
Can we make 6? Yes, 2*3=6
Can we make 7? No, all our tries fail, so we add 7 to our given numbers g={2,3,5,7}
Can we make 8? Yes, 2*2*2=8
Can we make 9? Yes, 3*3=9
Can we make 10? Yes, 2*5=10
Can we make 11? No, so we add it to the set g={2,3,5,7,11}
Can we make 12? Yes, 2*2*3=12
Can we make 13? No, so add it to the set g={2,3,5,7,11,13}
Can we make 14? Yes, 2*7=14
Can we make 15? Yes, 3*5=15
Can we make 16? Yes, 2*2*2*2=16
What is the set g that we are generating by this process? It's the set of prime numbers! This is simply another way to explain prime numbers.
It's a nice demonstration because it shows how prime numbers generate all numbers using only the multiply operation. You can also see that as g gets bigger we can obviously make more numbers from it, so prime numbers become less and less frequent.
The prefix "trans" means "beyond". So a transfinite number is one that's beyond the finite. There's only one and that's infinity, right? Wrong. It turns out there are many transfinite numbers. The concept of infinity is just a general concept, and the real mathematics is the study of transfinite numbers.
This work is due to Georg Cantor, who showed that there are many types of infinity, and some are bigger than others! He even developed an arithmetic for working with transfinite numbers. He denoted them by the Hebrew letter "aleph".
His work stands as one of the most elegant pieces of mathematics ever.
So what did Cantor do?
He formalized counting. He started with the integers {1,2,3,...} and asked what other sets could be placed in 1-to-1 correspondence with the integers. Instead of just saying there are an infinite amount of integers he denoted the number of integers by aleph0 and developed an arithmetic that in many ways treated aleph0 as a regular number. But he went further..
He showed that the rational numbers (fractions) could be placed in 1-to-1 correspondence with the integers. So counterintuitively, there are only as many rational numbers as there are integers. Not more!
But when it comes to irrational numbers, there are many more. He called this number aleph1 and he showed that it was different and bigger than aleph0. He proved that the number of subsets of the set of integers {1,2,3,...} is also aleph1 and he produced this amazing result..
aleph1=2^aleph0
He even asked if there was an aleph number between aleph0 and aleph1.
During his lifetime Cantor was ridiculed, not by the general public, but by his fellow mathematicians. Today his work is regarded as brilliant and is taught as part of the standard university mathematics curriculum.
Take a strip of paper, join the ends, so you have a band. This is a model for a spin 0 particle.
Now give the paper strip 1 half twist before joining the ends. This is a model for a spin 1/2 particle.
Now give the paper strip 2 half twists before joining the ends. This is a spin 1 particle.
But things get strange..
Now give the paper 4 half twists before joining the ends. This is a spin 2 particle. The only one known is the hypothetical graviton, carrier of the gravitational force. But if you play around with this thing for a while it will suddenly flip particle. So a Graviton can oscillate into a neutrino.
This is just a simple model, but if graviton oscillation exists the implications are deep. Graviton oscillation would change physics as we know it. Here are a few predictions..
Black holes evaporate
You could imagine that graviton oscillation requires high graviton pressure - meaning it only occurs in very intense gravitational fields such as black holes. This means black holes evaporate into spin 1/2 neutral particles.
Black Holes are an intense source of neutrinos
Assuming the neutral spin 1/2 particle is a neutrino then areas of intense gravity (such as black holes) will emit neutrinos. Black Holes are neutrino factories.
The Universe is expanding
An asymmetry in the oscillation (meaning graviton to spin 1/2 particle occurs more frequently than spin 1/2 particle to graviton) would lead to weakened gravity and this would cause inflation. Of course, the rate of inflation need not be constant.
Intense gravitational fields are the source of dark matter
Could the neutral spin 1/2 fermion particle account for dark matter? i.e. dark matter is produced by the decay of black holes.
The mathematics a set is just a collection of distinct objects. What type of objects? Any type. Of course you need a clear way to specify how an object belongs to a set.
Let's consider a simple example, the set containing the first 3 letters of the alphabet..
S={a,b,c}
Can we say anything mathematically interesting about this set?
Well, it's a finite set and contains 3 members. There's also a very clear rule to decide if an object belongs to the set. But we can do more, we can apply a mathematical technique to generate more structure. This technique is very simple, but turns out to be incredibly powerful. It's this..
Once you've defined something ask if contains things like itself. In this case we've defined a set S, so we ask if it contains any subsets. A subset of S is just another set made from the same objects. It's called a subset because you can think of it as contained inside S.
{a} is a subset, so is {a,c}, so is {b,c}
How many subsets does S have in total? In the count we'll include the null set { } which contains nothing and we'll also count the set itself {a,b,c} which contains everything. So here are all the subsets of S..
{ }
{a}
{b}
{c}
{a,b}
{a,c}
{b,c}
{a,b,c}
There are 8 in total, which just happens to be 2^3 where 3 is the number of objects in S. This is no coincidence. If our set contained n objects the number of subsets would be 2^n. This number gets big fast. For example..
A set with 26 members (such as the 26 letters of the alphabet) has 2^26=67108864 subsets. So from just 26 objects we can easily generate 67108864 new objects!
If you don't like the idea of counting { } and {a,b,c} as subsets then just say the number of subsets is (2^n)-2 and this makes almost no difference in the count. In the above example the number of subsets would be 67108862. Mathematicians call these subsets the proper subsetsIn 1742, the German mathematician Christian Goldbach, in a discussion with the mathematician Leonhard Euler, made a simple statement..
Every even integer greater than 2 can be written as the sum of two prime numbers.
Mathematicians have tried to prove this ever since. None have. It's a great example of how a simple statement in mathematics can be amazingly difficult to prove. Computers have checked billions of numbers and shown it to be true for every number tested, but that's not the same as a proof. A proof would show it to be true for all even numbers, period.
It's easy to prove that every integer can be written as a product of primes. This is called the prime decomposition of an integer. So for any integer m we have..
m=(p1^s1)*(p2^s2)*......*(pn^sn)
where p1,p2,...,pn are prime numbers and the s1,s2,s3,...,sn are just integer powers and represent the simple fact that primes may be repeated. By collecting like primes together and raising them to a power we make sure that p1,p2,p3,...,pn are distinct primes with no duplication.
The condition that m be even simply means that one of these primes must be 2. Since the order of multiplication does not matter we can make p1=2. So m now looks like this..
m=(2^s1)*(p2^s2)*...*(pn^sn)
Goldbach's Conjecture says that when m is even there exists two prime numbers, let's call them g1 and g2, such that..
m=(2^s1)*(p2^s2)*...*(pn^sn)=g1+g2
Are we on our way to a proof? No, but it's still fun to try!
Perhaps there's some deep clue in the fact that Goldbach's Conjecture only works if p1=2. In other words, if 2 does not appear in the prime decomposition of an integer then Goldbach's Conjecture does not work. What's so special about the number 2?
Differential calculus is one of the two branches of calculus, the other is integral calculus. It's best to learn differential calculus first. So here's the scoop.. |
Mathematics Quiz
Mathematics Quiz General Knowledge Part 15 Mathematics Quiz Questions Part 1 - General Knowledge 1. When did Al-Khwarizmi write the popular book which introduced Indian numbers and zero to the Arab world? A.D. 820 2. When were Indian…
Applications of Mathematics Quiz Questions Mathematics Quiz - Applications of Mathematics 1) Some staircases are in the form of a: Answer: Spiral 2) Our measurement of time is based on: Answer: Sexagesimal number system…
Math Quiz Questions Instruments and Machines in Mathematics Quiz Questions Mathematics Quiz 1) Which is the instrument that measures angles, sideways, or up and down for making maps? Answer: Theodolite. 2) Which device is employed… |
Do you enjoy craft? Then you probably enjoy mathematics too – you just may not know it yet.
Don't miss out on Maths Craft Festival 2017 being held this coming Saturday and Sunday, September 9-10, at the Auckland War Memorial Museum's Event Centre.
All are welcome to attend this two-day festival, which includes 10 craft creation stations and five public talks.
Discover the maths behind craft and the craft behind maths. Join our Department of Mathematics staff, students and science scholars and find out how to tie a mathematical knot, crochet a Möbius strip, fold an origami octahedron, draw an impossible triangle, or colour a Latin square. |
The code word Knol was given because it is supposed to be a "Unit of Knowledge". Since it is, I actually started thinking about how one could measure the value of a certain piece of knowledge into knols? For instance, how many knols would Theory of Relativity be? and how much would be Newtons Laws? or in Computer Science - how many knols would be Quicksort? What about Bubble sort which is not practically used? Would that mean its knol would decrease?
I am looking for answers, and if anybody got any answers, do leave a comment |
November 21, 2011
Control digits- Portuguese Identification Card case- ISBN numbers
Each Portuguese identification card has a digit to the right of the ID number.
Countless stories have been created around this digit, most of them myths.
Although this article is in "answers to readers" section, there isn't any particular reader asking about this using the normal channels, but it's a question often brought to the conversion.
So, what is this digit?
Jorge Buescu, degree in physics from the FCL, PhD in mathematics from the University of Warwick and Professor of Mathematics at IST (1), explain this in his book "O mistério do bilhete de identidade e outras histórias"- "The mystery of the identity and other stories".
It's important to clarify
This digit isn't the total number of people with exactly same name as you. This is a urban legend, if you don't believe me just take a moment to think about this:
This is always a 1 digit number, i.e., the range is from 0 to 9, so, if I have 9 people exactly same name as me it's nuts to think there must be someone with 12 or 14 people with exactly same name? The control number for those people should be higher then 9.
So what is this mysterious digit?
On same book Jorge Buescu, explains that "The extra digit is (or would be, if the Portuguese authorities hadn't committed a pathetic mathematical error!) Only a control number that detects if the number of the card is correctly written", and continues to explain that the human brain can't handle with hundreds of number with many digits if they don't have a link between them, or a pattern. That's why this method is used, so the computer can detect errors in the numbers. What would happen if a supermarket operator makes a mistake with the barcode numbers and "... charge 200 bucks for a butter package..." it would be unthinkable.
This number, if there wasn't an error, like noted by Buescu, would be used to detect if the Identification Card Number entered in database contained errors or not.
This system of "digits detectors" is used in several other daily things such as credit cards, the ISSN for periodicals, the ISBN for the books, there are many other cases like those, where we use a control digit.
So how it works?
Buescu uses book ISBN as an example.
This number is usually in the back of the books and each title have a unique 10 digit number. For instance the ISBN for Buescu book "O mistério do bilhete de identidade e outras histórias", Edições Gradiva from collection "Ciência Aberta", is 972-662-792-3.
To check if the sequence is correct the computer applies the following formula:
Some of you, restless minds always on alert, would say: Some of the ISBN numbers have X's! Yes you are correct, some control numbers are not digit but letters, in those cases the 10 is represented by an X, that's due to the nature of used algorithm (2).
If you have a Portuguese identification card, you can try it yourself, do the same as the ISBN numbers, but read the number from right o left, like the image.
The pathetic mathematical error is that the number 10 (or the X in ISBN numbers) was replaced by a 0, so half of the people with control digit 0 should have a X. |
I have written many times about the exponential growth of big data, in jobs, applications, and scale. But in fact, there is one huge stumbling block that may limit big data's growth and potential: math.
Too few adults in the Western world are proficient at basic mathematics. And by basic mathematics, I am not referring to advanced statistical analysis or algorithms. I am talking about basic, primary school numerical skills.
A study done by the UK government in 2003 that found that 47 percent of working-age adults in England lacked basic mathematics skills and, by 2011, that figure had risen to 49 percent! In the United States, a study found that only 16 percent of children of parents with a low level of education were deemed proficient in mathematics, and that the percentage barely eclipsed 50 percent for children of parents with college degrees.
When such a large percentage of the country's population lacks basic mathematics skills, the current shortage of qualified data scientists begins to make sense. Clearly, people with insufficient math skills are not prepared to be data analysts.
But what about the people to whom data analysts report? Retail, sales, human resources, manufacturing, customer service – big data is sure to touch every corner of every industry within the next few years, even more than it has already. But a major stumbling block to implementing data-driven strategies may be explaining data-driven strategies to the employees on the ground who must implement them.
What Low Math Skills Mean to the Data Analyst
Obviously, math skills do matter. (In addition to improved economic prospects, many studies show that more advanced mathematics skills also lead to better health and wellness.) But as a data analyst, it's certainly not your job to teach math to your audience.
A Tale of Two Disciplines: Data Scientist and Business Analyst. Read the story »
Instead, we must be aware that it's highly likely that a large percentage of our audience – yes, including even C-level executives – may not have good basic maths skills. And that can change the way you approach a problem.
And that's OK, because the fact is that big data is rarely about only the mathematics. But it will affect your job and how you approach it in several ways:
First and foremost, big data jobs will remain in high demand. Because advanced mathematics skills are indeed mandatory for jobs in big data, and a high percentage of the population is lacking in these skills, my basic numerical skills tell me that the demand for qualified candidates to fill big data-related positions is not likely to dry up any time soon.
The upshot of this is that those people who are qualified for data science positions will continue to find good paying work – and the good salaries that come along with these positions may motivate more individuals to study mathematics and develop the skills that are required for them to enter the big data workforce.
In addition, the fact that it's likely that many of a data scientist's co-workers will lack advanced math skills underscores the need for data scientists and data analysts also to have business experience. This will enable the data scientist to communicate better with those who don't have the mathematics background to understand technical, math-based explanations. And it's also possible that a data scientist's supervisors and bosses may not have the necessary skill with numbers to be able to determine the right questions to ask.
Data is useless without analysis, and analysis is useless if you are attempting to answer the wrong questions. Understanding where the mathematics and the business case intersect is the invaluable ingredient for a winning data strategy.
Finally, the most important thing to remember about data is that it provides context, conveys meaning, and tells a story. The analyst, therefore, must excel at going beyond the numbers to tell the greater story. We must be excellent storytellers. From every bank of numbers or point plotted on a graph, we must be able to extrapolate meaning and relevance for our audience.
Because they may not have the required skills to do it for themselves |
A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics
Bertrand Russell wrote that mathematics can exalt "as surely as poetry". This is especially true of one equation: ei(pi) + 1 = 0, the brainchild of Leonhard Euler, the Mozart of mathematics. More than two centuries after Euler's death, it is still regarded as a conceptual diamond of unsurpassed beauty. Called Euler's identity, or God's equation, it includes just five numbers but represents an astonishing revelation of hidden connectionsThe Modern Scholar: Mathematics Is Power
William Goldbloom Bloch is a respected professor of mathematics at Wheaton College. This intriguing lecture series, Mathematics Is Power, delves into both the history of mathematics and its impact on people's everyday lives from a non-mathematician's perspective. Bloch first examines the history of mathematics and age-old questions pertaining to logic, truth, and paradoxes. Moving on to a discussion of how mathematics impacts the modern world, Bloch also explores abstract permutations such as game theory, cryptography, and voting theory.
Logic: A Very Short Introduction subject, explaining how modern formal logic deals with issues ranging from the existence of God and the reality of time to paradoxes of probability and decision theory. Along the way, the basics of formal logic are explained in simple, non-technical terms, showing that logic is a powerful and exciting part of modern philosophyA Brief History of Time: From Big Bang to Black Holes
Calculating the Cosmos: How Mathematics Unveils the Universe
In Calculating the Cosmos, Ian Stewart presents an exhilarating guide to the cosmos, from our solar system to the entire universe. He describes the architecture of space and time, dark matter and dark energy, how galaxies form, why stars implode, how everything began, and how it's all going to end. He considers parallel universes, the fine-tuning of the cosmos for life, what forms extraterrestrial life might take, and the likelihood of life on Earth being snuffed out by an asteroidPublisher's Summary
The aim of this audiobook is to explain, carefully but not technically, the differences between advanced, research-level mathematics, and the sort of mathematics we learn at school. The most fundamental differences are philosophical, and listeners of this audiobook will emerge with a clearer understanding of paradoxical-sounding concepts such as infinity, curved space, and imaginary numbers. The first few chapters are about general aspects of mathematical thought. These are followed by discussions of more specific topics, and the book closes with a chapter answering common sociological questions about the mathematical community (such as "Is it true that mathematicians burn out at the age of 25?").
Hearing mathematics makes it difficult to interpret. It has to be seen (at least for me). It is fairly joyless to listen to a long number or equation being read, and it is almost impossible to follow. I found the readers voice quite irritating too.
How would you have changed the story to make it more enjoyable?
The book itself would have been interesting to read. It needs to be seen though. It simply doesn't work as an audiobook.
Who might you have cast as narrator instead of Craig Jessen?
An English actor perhaps? Timothy Gowers is an English Mathematician. I had assumed it would be an English accent. There's no reason it should be, just my preference in this case. Had it been an American author, I would have preferred an American narrator.
Couldn't actually listen to it. The performance was so poor that I had to stop listening. No one actually speaks like that. I thought I was listening to "master thespian" from SNL not a history of mathematics. Ugh.
Would you ever listen to anything by Timothy Gowers again?
Only if someone else narrates.
How could the performance have been better?
YES
4 of 4 people found this review helpful
Caem586
23/07/14
Overall
Performance
Story
"Disappointing as an audiobook"
What could have made this a 4 or 5-star listening experience for you?
More real world examples from every day life would have created a more realistic connection with the concepts covered in this audio book. Much of the information (for me) would have been easier to understand visually. Reading off long strings of numbers and equations doesn't work as well as seeing them on paper/computer screen.
How did the narrator detract from the book?
His breathy ending to every sentence and, frankly, pretentious tone was incredibly distracting for such a detailed audio book.
8 |
Bailey celebrated Pi Day March 14 with hands-on math activities that even incorporated art.
Pi Day is celebrated each year on 3/14, as Pi is the ratio of a circle's
circumference to its diameter, nearly equal to 3.14159...
All math classes participated in a partner, hands-on activity where they measured various circles. They then entered their data into an interactive computer program to evaluate the class data and draw conclusions about circles and Pi.
They also read a story book, "Sir Cumference and the Dragon of Pi," where the character, Radius, also measured circles and discovered Pi to save his father, Sir Cumference. They compared this data to the class data to find the common number, 3.14 (Pi), for all circles. The lesson concluded with each student designing a section of a circle to create a class art display of circles. |
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Literacy as Numeracy
For this week's blog post, we were asked to read an article by Leroy Little Bear describing differences between First Nations world views and Eurocentric world views.
1. At the beginning of the reading, Leroy Little Bear (2000) states that colonialism "tries to maintain a singular social order by means of force and law, suppressing the diversity of human worldviews. … Typically, this proposition creates oppression and discrimination" (p. 77). Think back on your experiences of the teaching and learning of mathematics — were there aspects of it that were oppressive and/or discriminating for you or other students?
Oh, absolutely there were. In my own experience, people who were not naturally strong with mathematics and perhaps gifted in other ways (and this isn't taking massive aspects of peoples lives like culture/worldviews/ect into consideration!) had an extremely difficult time succeeding in math classes. As a visual learner, I found mathematics incredibly difficult to internalize. Sure, there were graphs and pie charts and all manner of neat graphics, but none of that held any form of connection to quadratics or polynomials. For me, there was nothing more abstract nor confusing than the string of numbers and letters written on the board day after day. I was expected to accept things as they were; the one teacher who WOULD offer explanations about how these concepts came to be left me more irritated and confused than when I had first walked into this class. These experiences have left me along with most of my friends and family with an embodied distrust for mathematics. So I suppose, at its core, the mathematics classes I was involved with were discriminatory towards different kinds of learners.
Until I'd sat in and listened to Gale Russell's lecture, I had never really realized that current-day Mathematics oppress groups of people through its absolute rigidity. I was thoroughly raised with primarily eurocentric values, and I still struggled immensely. I cannot even imagine how students whose cultural and worldviews clashed with the concrete rules of mathematics may have felt sitting through these classes if they were not one hundred percent compatible as a learner and individual.
2. After reading Poirier's article: Teaching mathematics and the Inuit Community, identify at least three ways in which Inuit mathematics challenge Eurocentric ideas about the purposes mathematics and the way we learn it.
-"Traditional Inuit teaching is based on observing an elder or listening to enigmas. These enigmas can be clues for problem solving in mathematics. Furthermore, Inuit teachers tell me that, traditionally, they do not ask a student a question for which they think that student does not have the answer."
-"Traditionaly, it was for three and up that they needed words to express quantities. Their tradition being essentially an oral one, the Inuit have developed a system for expressing numbers orally. They do not have other means of representing numbers, they have borrowed their number systems from the Europeans."
-"The numbers 20 and 400 are pivotal
numbers, as other numbers are built from these two numbers. The Inuit
have a base-20 numeral system."
I think a good way to conclude this week's session is to reflect upon a story told by Mrs. Gale Russell in our lecture. She told the story of how a research group was sent to a place like Kumashiro's Nepal (though I can't remember where specifically it was) and were tasked with investigating the local peoples' stance on numeracy. Ultimately, they were confronted with a sheep farmer and asked how much the farmer would like in exchange for one of his sheep. When the farmer responded with "two tobacco", they left and returned with four tobacco and offered to buy two of his sheep. However, once the farmer declined, they deduced that the people of this land were inadequate when it comes to math. They did not realize that the farmer did not accept their offer because of reasons they did not know of; they only assumed that he was unable to do simple math and left. |
That builiding complex is just beautiful !!. I can´t wait for my visit next year to Moscow
__________________ 1. Mathematics is the language of nature. 2. Everything around us can be represented and understood through numbers. 3. If you graph the numbers of any system, patterns emerge....
"Pi, Darren Aronofsky" |
Found: Operation for floating point arithmetic
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Tuesday, October 13, 2009
Physics for Fido
I can't tell you exactly how I came upon this--I know I started at Sherry Chandler's blog, followed a link, and another link...and there I was, reading Orzel's bitter complaint about a New York Times movie reviewer who proudly displays his ignorance of physics. A physicist and college professor, Dr. Orzel is often irritated by The Innumeracy of Intellectuals. Sadly, indifference to math is not limited the the intelligentsia. Adult basic education students, school children, and college freshmen all use a smug tone to tell me they "are no good at math," meaning "Get out of my face with that stuff, I can't be bothered." In contrast, people who can't read well generally try to cover up and fake it. I don't get it, but the reason I often end up teaching math to the unwilling is the scarcity of teachers willing and able to take on those classes.
I'm looking forward to reading the whole canine physics course--we never got to quantum mechanics in my undergraduate physics class, because the physics department thought it was "too hard for biologists." I wish I'd had Dr. Orzel!
3 comments:
I'm not crazy about math because, at least at the lower levels we're taught in school, there's only one correct answer. Where's the fun in that? Also, it's the ultimate in reductionism and abstraction -- neither things I enjoy. Still, I recognize its fundamental importance, and do sometimes wish I could get more enthusiastic about it.
I guess I am more puzzled and annoyed by the indifference of so many literary intellectuals to anything outside the humanities -- basically the whole of science, including anthopology.
I'm surprised a poet doesn't like reductionism and abstraction--isn't part of the excitement of poetry the tension between evoking the material and emotional worlds while using words (abstractions) in formal or unconventional ways?
I didn't like math in elementary school either--there was a lot of punishment for failure, and math is a skill that you practice, like music. You're bound to make mistakes, but if you practice, you get better at it.
I have a friend, a professor emeritus of physics, a photographer, and a poet, who has a chapbook coming out soon called "Chasing Shroedinger's Cat." He gets very upset at the way poets use that cat as a metaphor because they get the science wrong. I'll send him a link to this post.
Which I'm not sure how I missed -- I've been in some odd limbo state this fall.
Having a son with a math disability was enough to disenchant me with public school approaches to math. They thought, because he could memorize the "math facts," he therefore couldn't solve logical problems. |
Dear John. There are many ways to square a circle. You can tie a string around a circle. Remove it and square it up. However, this is not an allowable construction in Euclidian geometry. You need to use an unmarked straight edge, so there is no way to construct a line correct to the 9th decimal place of Pi. Peace, Don On Feb 13, 2008, at 7:17 AM, John Medeiros wrote:
Pi appeal: Please let me know if I made a mistake. I show my work, work of more than sixteen years, at pythagorascode.org, my detailed proposition that there has always been a correct positive answer to the riddle about squaring the circle. We can construct a line length of Pi to the ninth decimal place - 3.141592653... - and then construct squares and circles equal in area precise to the billionth of a unit (or better). That means comparison of a ten square kilometer SQUARE with a ten square kilometer CIRCLE, on a hand-held electronic calculator, comes up EQUAL. So I cry out for some consideration. If you feel that I must be wrong, please check my work, and let me know where is my error. There is a credible argument that my discovery results in the construction of Pi EXACTLY, because proportion in Nature is probably more precise than even our binary electronic computers, which rely on the physical properties of semi- conductors. These are physical properties, meaning physical limitations, like the gears in a Swiss watch. |
Significant Figures: The Lives and Work of Great Mathematicians
We rated this book:
$28.00
Without the work of mathematicians, our world would not function as it does today. All our modern technology and civilization's advances are based on the work of mathematical geniuses throughout time. This book introduces or reacquaints readers with several of these geniuses, from Archimedes to Mandelbrot–twenty-five mathematicians, in fact. And one of the delightful aspects of this book is that the mathematicians span the globe, and even include a few women. Each is given a short biography and a much longer discussion of their pioneering mathematical discoveries, put into context and parsed so even the lay reader can grasp their importance. For example, George Boole, a reluctant schoolteacher, was also a self-taught mathematician who discovered important facets of algebra but was even better known for developing Boolean logic, which is integral to computer science. Srinivasa Ramanujan, a poor clerk in Madras, India, also self-studied mathematics and developed work in number theory and infinite series. The biographical sketches are interesting and pertinent, and there's an entertaining introduction to each section; if the math seems overwhelming (as it sometimes may), the introductory sketches alone are entertaining and worthwhile, written with enthusiasm, humor, wit, and a clear love for the subjects.
"Significant Figures: The Lives and Work of Great Mathematicians" Cancel reply
Gentle, thoughtful, and hopeful: honest descriptors for this second book by Ms. Dugard. Full of discovery and of new insights into the benefits and sometimes frights of liberty, this nicely crafted tale is a kindly read.
The lady fills pages with her family, including her beloved mother, her maturing daughters, and friends. Those friends include two women who have been her therapists, some horses, dogs, and kitties.
Adventures range from a first unaccompanied airline flight, complete with being stranded in a strange city, to visits to windy Ireland and tropical Belize.
Dugard has formed a foundation to aid others in recovery. And using that institution as a platform, she has challenged the concept of Stockholm Syndrome as a patronizing stigmatization. She vehemently refutes the concept of captives necessarily coming to love their captors. To aid this, she offers a pungent description of the sheer repulsiveness, both mental and physical, of her own kidnapper and rapist.
Even with so very much fuel, she refuses to light the fire of hatred, eschewing the self-destructiveness of that indulgence.
Freedom ends on notes of hopeful anticipation. I am better for having read this, and I will now seek out the author's first book, A Stolen Life. I may also seek out Scott's Shattered Innocence.
Roald Dahl, born a hundred years ago of Norwegian immigrant parents in Wales, schooled in England, adventured in Newfoundland, worked in Kenya, flew war planes in the Middle East, …. even this geographical teaser intimates a remarkable life. In the hands of a marvelous writer the places became steeped in exploits of one kind or another. Dahl's letters to his mother, a determined and resourceful widow who raised seven children, were written between 1925-1965 reveal a frankness to which parents are not always privy. While detailed in varying degree, they seem not to conceal foibles, successes, and excesses. Editor Donald Sturrock invites us to share Dahl's school years (early commonplace ones could have been skipped) and the high school ones that allow us to compare 1930s adolescence with today's.
Far more readable, the letters written during World War 2 when Dahl served as a Royal Air Force pilot describe his sense of excitement until he received serious injuries when his plane crashed over the desert in Egypt. After a lengthy recovery period he was chosen for a prime post in Washington that was not entirely satisfactory on either side. America presented new adventures, a spell enjoying the glamour and intensive work under Walt Disney's wing in Hollywood and, more lastingly, his writing career launched with articles in widely-read magazines like the Saturday Evening Post.
The letters offer a new perspective of the versatile author, but come across as choppy, with gaps suggesting a length limit to the book. To avoid laying blame on Sturrock, the sterility is no doubt due in part to the nature of personal correspondence. They are not compelling and miss the enchantment of Boy.
The author, Betsy Lerner, not only records a touching memoir of her relationship with her apparently reticent mother but at the same time provides a sociological study of the contrast between female expectations before the 1960's and those behaviors prevailing now after the pill. And what better way to list the differences than through the metaphorical game of bridge. Over the span of time, women's goals have changed from seeking early marriage with servitude to some male who will be a good provider and caring father to the current quest for self-fulfillment and identity. Each of the five octogenarian bridge participants has her life reviewed, exposing youthful talents denied expression by the cultural limitations of their era and contrasted with the current feminist stress of expressing individuality and taking on personal responsibility.
Not only is learning the game of bridge challenging, but struggling to translate the values and behaviors that mark the differences in the generation gap strains the understanding. Bridge is the catalyst that awakens compassion between mother and daughter. Written with snappish humor and with keen delicacy for the bridge players' personalities, this is a rewarding book to remind us how time refashions thoughts as the world turns |
where the numbers do their dance of no location -- haunt, if what I've read is so, of Heisenberg, and Planck, and the quiet magister, Gauss |
About…
This bit is miscellany about Approximating logarithms
theres also a link at the foot of the page to some elementary descriptions of how to
make a continued fraction which might be helpful given that i used a variant of this method
to make an approximation program
Numbers in art is intended to be a collection of ideas on their use based on some ancient
programs written by Mike Burr.
Approximate logarithms
I'm not going to describe the algorithm in detail as its doesn't lend itself to that
but suffice it to say you keep reducing the power term while extracting a nearest residue
in roughly the same way as per the continued fractions
- and i hope to load this program onto the web as its extremely versatile
and quite intriguing.
Heres a sequence using the number base of 10 and is an approximation to log 13
from above
13 :- 1/0 ; 2/1 ; 3/2 ; 4/3 ; 5/4 ; 6/5 ; 7/6 ; 8/7 ; 9/8 ; 19/17 ; 29/26 ; 39/35 ;
127/114 ; 215/193 ; 303/272 ; 391/351 ; 479/430 ; 567/509 ; 655/588 ; 1398/1255 ;
approximation using last ratio = 1.11394422310756972112-1.11394335230683676921
resulting in an error of about 8.7e-7 - incidentally the approximation from below is quicker
giving an error about 3.3e-17
In a similar way i used the (bottom line of the ratios) to construct a test picture
made with kids coloured pencils and which i still have somewhere here.
Heres a
table of approximate logarithms .. for a load of sequential numbers
as a foot note here are their continued fraction representations
log table continued fractions .. note that sometimes the number is reduced to
zero and this i annotated in the output [ which looks odd at first ]
Sources…
Ruby program "calllogit.rb" used to generate single dimension logarithmic approximation |
Safety in numbers: Cracking the coefficient code
Decimal expansion of numbers was a common exercise in middle-school mathematics. Fractions like 1/5 and 1/4 were less taxing. Their decimal expansions terminated promptly at 0.2 and 0.25 respectively. But the fraction 1/3 was a little intriguing. Its decimal expansion did not terminate, but it soon became clear that a single digit, namely 3, started to repeat itself. For certain other fractions, a whole block of digits, and not a single one, would repeat interminably. For example, the fraction 17/27 results in 0.629629629629629629…, the block 629 repeating itself endlessly.
As if this was not puzzling enough, there were the expansions of irrational numbers, like the square root of 2. Irrational numbers are those which cannot be expressed in the form a/b, where both a and b are whole numbers. In such cases, the decimal expansions neither end nor reveal any repeating pattern of digits. The square root of 2, for example, yields 1.414213562373095… Square and cube roots of many other numbers are also irrational. A famous irrational number is pi (denoted by the symbol ð), the ratio of circumference of a circle to its diameter. This number is also an unending decimal without any repetitive pattern.
The difference between rational and irrational numbers leads to many interesting manifestations. Let us look at one of them. Take any rational number r of your choice, say 1/3. Multiply it by all natural numbers n (1, 2, 3, 4…) and keep a record of the fractional part of each of the product, nr. It can be easily observed that while n changes, the decimal part of the product nr takes only some fixed values. For example, if r is 1/2, the decimal part of the product nr is either 0.5 or 0. Similarly, if r is 1/3, the decimal part of the product is only 0.3333…, or 0.6666…, or 0.
The situation changes completely when you take r as an irrational number, say square root of 2, and start multiplying it with any natural number n (1, 2, 3, 4…). The decimal parts of the products nr so yielded are not finite. Every multiplication results in a different decimal part.
Mathematicians have tried to look for patterns here as well. In 1842, Peter Gustav Lejeune Dirichlet showed that the decimal parts of the products of any irrational number (r) with several natural numbers (n = 1,2,3,4…) tended to concentrate towards zero. In other words, the decimal parts of these these products (nr) were very likely to approximate to zero. On further study, however, another mathematician, Leopoldt Kronecker came up with a better result. The decimal parts were as likely to approximate to 0 as to 1 and to every value in between.
This result was further refined by the German mathematician Hermann Weyl in 1916. This began to be referred to as the phenomenon of equidistribution. The decimal parts were likely to be distributed evenly along every value between 0 and 1.
Why is this important? The beautiful symmetries of mathematics have often revealed many hidden intricacies of how the nature operates. For example, it was shown by physicist Eugene Wigner that one such equidistribution law, which looks like a semi-circle, correctly predicts the distribution of energy levels of heavy nuclei of atoms.
Our own research focuses on equidistribution of various families and sequences of real numbers, arising in the context of modular forms. In 1916, Srinivas Ramanujan wrote a paper titled "On certain arithmetical functions," which revolutionised number theory thereafter. In the 1930s, Erich Hecke laid the foundations of a much deeper theory underlying Ramanujan's work, called the theory of modular forms. Modular forms are some very special functions with rich inner symmetries and growth conditions. They can be described in terms of what are called their "Fourier expansions": the coefficients arising in these expansions encode important information about many facets of mathematics. A deep study of Fourier coefficients of modular forms, for example, helped in solving the famous Fermat's Last Theorem.
One of the major breakthroughs in recent times is the discovery (by Richard Taylor, Michael Harris and many others) that certain sequences arising from Fourier coefficients of modular forms follow the same "semi-circle" equidistribution law described above.
Our group is studying several deeper statistical phenomena associated with these coefficients. We are trying to understand the distribution of several families that arise from the Fourier coefficients of modular forms and apply them to problems in arithmetic geometry. |
I've been using Idris for a while, and today I stumbled into an example of a
dynamorphism that worked so beautifully I had to share. It is a stellar example
not only of dependent types but also the rôle abstraction can play in
writing correct code.
The modern
theory of continued fractions comes from
Christiaan Huygens, a Dutch physicist who invented the
pendulum clock. Continued fractions turn out to be an especially elegant
way of finding rational approximations of a number; this enabled him to design clocks
with small gears that nonetheless provided the desired degree of
accuracy. |
Archive reports
Numero e Logos
Submitted by redazione on Thu, 04/10/2014 - 10:29
Translation in progress
Logos is not only the 'discourse', nor can it be understood as a simple "word": the Word that, according to John, is the beginning of everything. It is inevitable that, if we want to recover the lost sense of logos, we need to compare it to the number, because their fates are intertwined to such an extent that one would not exist without the other. They already appear together in the verses of Homer, we understand their affinity in the first theogonies, in ancient tragedy and in the Pythagorean philosophy - and in the sources from which we get the ritual origin of mathematics the number has a meaning similar to that subsequently adopted by thelogos. But what can Logos tell us again after centuries of reflections, from Heraclitus to Hegel and Heidegger ? The answer lies first of all in its mediating function, and the originally assigned task to hold together the structure of the universe, even in its most distant parts, unfinished and inaccessible. A task that it could not continue to perform without the reality and power of realization of numerical algorithms. The relationship between the number and the logos points to a rich and original perspective, and a new direction in which to grasp the roots and the fate of philosophical and scientific thought in the West.
Is this the new direction that this provocative and fascinating book aims to explore, which continues and extends previous Zellini's researches and discovers, taking advantage of countless testimonials, a dense network of analogies and correspondences between scientific concepts and knowledge's formulas. |
Wonders Beyond Numbers: A Brief History of All Things Mathematical Review
In this book, Johnny Ball tells one of the most important stories in world history - the story of mathematics.
By introducing us to the major characters and leading us through many historical twists and turns, Johnny slowly unravels the tale of how humanity built up a knowledge and understanding of shapes, numbers and patterns from ancient times, a story that leads directly to the technological wonderland we live in today. As Galileo said, 'Everything in the universe is written in the language of mathematics', and Wonders Beyond Numbers is your guide to this language.
Mathematics is only one part of this rich and varied tale; we meet many fascinating personalities along the way, such as a mathematician who everyone has heard of but who may not have existed; a Greek philosopher who made so many mistakes that many wanted his books destroyed; a mathematical artist who built the largest masonry dome on earth, which builders had previously declared impossible; a world-renowned painter who discovered mathematics and decided he could no longer stand the sight of a brush; and a philosopher who lost his head, but only after he had died.
Enriched with tales of colourful personalities and remarkable discoveries, there is also plenty of mathematics for keen readers to get stuck into. Written in Johnny Ball's characteristically light-hearted and engaging style, this book is packed with historical insight and mathematical marvels; join Johnny and uncover the wonders found beyond the numbers. |
The proceedings describe how Gabriel Lamé, who had proved the case n = 7 some years earlier, took the podium in front of the most eminent mathematicians of the age and proclaimed that he was on the verge of proving Fermat's Last Theorem. He admitted that his proof was still incomplete, but he outlined his method and predicted with relish that he would in the coming weeks publish a complete proof in the Academy's journal.
The entire audience was stunned, but as soon as Lamé left the floor Augustin Louis Cauchy, another of Paris's finest mathematicians, asked for permission to speak. Cauchy announced to the Academy that he had been working along similar lines to Lamé, and that he too was about to publish a complete proof.
Both Cauchy and Lamé realised that time was of the essence. Whoever would be first to submit a complete proof would receive the most prestigious and valuable prize in mathematics.
…
By adding one more term x4, we get the next level of polynomial equation, known as the quartic:
By the nineteenth century, mathematicians also had recipes which could be used to find solutions to the cubic and the quartic equations, but there was no known method for finding solutions to the quintic equation:
Galois became obsessed with finding a recipe for solving quintic equations, one of the great challenges of the era, and by the age of seventeen he had made sufficient progress to submit two research papers to the Academy of Sciences. The referee appointed to judge the papers was Augustin-Louis Cauchy, who many years later would argue with Lamé over an ultimately flawed proof of Fermat's Last Theorem. Cauchy was highly impressed by the young man's work and judged it worthy of being entered for the Academy's Grand Prize in Mathematics. In order to qualify for the competition the two papers would have to be re-submitted in the form of a single memoir, so Cauchy returned them to Galois and awaited his entry.
New developments in mathematics, from calculus to topology, have often been initiated by physicists who, by means of intuition and persistence, have sneakily but sloppily invented new kinds of mathematics that were only later made rigorous by purists. Newton invented the calculus in the seventeenth century to handle mechanics, and its foundations were satisfactorily cleaned up years later by Augustin-Louis Cauchy and his contemporaries. In the late 1940s, reconnoitering around the technical difficulties of the Dirac sea, Richard Feynman, Julian Schwinger, and Shin'ichiro Tomonaga found an ingenious way to suppress the technical infinities of quantum electrodynamics by means of a judicious combination of extreme care and chicanery. Their starting point was never to forget that the normal quotidian electron we "see" every day is not the bare electron.
But cumulatively, over time or across a population, the way the results vary forms a regular and predictable pattern. The data points are grains of sand on a shoreline, blades of grass in a lawn, electrons moving along a copper wire.
The Blindfolded Archer's Score
Now, this is a convenient way to look at the world, but is it the only way? Not at all. Late in his long life, the nineteenth-century French mathematician Augustin-Louis Cauchy thought of an especially tricky one. It was, when I was younger, viewed as interesting—but unrealistic and contrived. My work made it very real.
I think the theory best imagined in terms of an archer standing before a target painted on an infinitely long wall. He is blindfolded and consequently shoots at random, in any direction. Most of the time, of course, he misses. In fact, half of the time he shoots away from the wall, but let us not even record those cases.
Riemann's response to the mathematical revolution spreading from the Paris academies was not that of a reactionary. Berlin was importing not only political propaganda from Paris, but also many of the prestigious journals and publications coming out of the academies. Riemann received the latest volumes of the influential French journal Comptes Rendus and holed himself up in his room to pore over papers by the mathematical revolutionary Augustin-Louis Cauchy.
Cauchy was a child of the Revolution, born a few weeks after the fall of the Bastille in 1789. Undernourished by the little food available during those years, the feeble young Cauchy preferred to exercise his mind rather than his body. In time-honoured fashion, the mathematical world provided a refuge for him. A mathematical friend of Cauchy's father, Lagrange, recognised the young boy's precocious talent and commented to a contemporary, 'You see that little young man?
In devising this theory he resorted to the luminiferous ("light-carrying") ether as the agent bringing matter and electricity together. Few things baffled or divided scientists more than this mysterious substance. As described by Augustin-Jean Fresnel, a French physicist who argued that light consisted of waves, the luminiferous ether was a gaslike substance through which both light and solids somehow moved. A French mathematician, Augustin-Louis Cauchy, worked out a mathematical basis for the properties of ether that made Fresnel's theory at least plausible if not satisfying to scientists.33
The wave theory of light required that ether be perfectly elastic and offer no resistance to a body passing through it. To these demands Ampère added a new chemical wrinkle: The ether was not simple but compound in nature and could "only be considered, in the generally adopted theory of two electric fluids, as the combination of these two fluids in that proportion in which they mutually saturate one another." |
Translations
In mathematics, the logarithm
of a given number to a given base
is the power
or exponent to which
the base must be raised in order to produce the given number.
For example, the logarithm of 1000 to the common
base 10 is 3, because 10 raised to the power of 3 is 1000; the base
2 logarithm of 32 is 5 because 2 to the power 5 is 32.
The logarithm of x to the base b is written
logb(x) or, if the base is implicit, as log(x). So, for a number x,
a base b and an exponent y,
\mbox~~ b^y = x, ~~\mbox~~ \log_b (x) = y \,.
An important feature of logarithms is that they
reduce multiplication to addition, by the formula:
\log (x \times y) = \log x + \log y \,.
That is, the logarithm of the product of two
numbers is the sum of the logarithms of those numbers. The use of
logarithms to facilitate complex calculations was a significant
motivation in their original development.
Properties of the logarithm
main article
List of logarithmic identities When x and b are restricted to
positive real numbers,
logb(x) is a unique real number. The
magnitude of the base b must be neither 0 nor 1; the base used
is typically 10, e,
or 2. Logarithms are defined for real numbers and for complex
numbers.
The major property of logarithms is that they map
multiplication to addition. This ability stems from the following
identity:
A related property is reduction of exponentiation
to multiplication. Using the identity:
c = b^ \ ,
it follows that c to the power p
(exponentiation) is:
c^p = \left(b^\right)^p = b^ \ ,
or, taking
logarithms:
\log_b \left(c^p \right) = p \log_b (c ) \ .
In words, to raise a number to a power p, find
the logarithm of the number and multiply it by p. The exponentiated
value is then the inverse logarithm of this product; that is,
number to power = bproduct.
Besides reducing multiplication operations to
addition, and exponentiation to multiplication, logarithms reduce
division to subtraction, and roots to division. Logarithms make
lengthy numerical operations easier
to perform. The whole process is made easy by using tables
of logarithms, or a slide rule,
antiquated now that calculators are available. Although the above
practical advantages are not important for numerical work today,
they are used in graphical analysis (see Bode
plot).
The logarithm as a function
Though logarithms have been traditionally thought
of as arithmetic sequences of numbers corresponding to geometric
sequences of other (positive real) numbers, as in the 1797
Britannica definition, they are also the result of applying an
analytic
function. The function can therefore be meaningfully extended
to complex numbers.
The function logb(x) depends on both b and x, but
the term logarithm function (or logarithmic function) in standard
usage refers to a function of the form logb(x) in which the
base
b is fixed and so the only argument is x. Thus there is one
logarithm function for each value of the base b (which must be
positive and must differ from 1). Viewed in this way, the base-b
logarithm function is the inverse
function of the exponential function bx.
The word "logarithm" is often used to refer to a logarithm function
itself as well as to particular values of this function.
The base can also be a complex number; the
evaluation of the log is just slightly more complicated in this
case. See
imaginary base.
Logarithm of a complex number
When the base b is real and z is a complex
number, say z = x + i y, the
logarithm of z is found easily by putting z in polar
form that is, z =
(x2 + y2)1/2 exp (i tan−1
(y / x) ). If the base of the logarithm is chosen
as e , that is, using loge (denoted by ln and called the natural
logarithm), the logarithm becomes:
\ln(z) = \ln \left[ \left( x^2 + y^2 \right) ^ e^\right]
= \ln \left[ \left( x^2 + y^2 \right) ^\right] + \ln \left[
e^\right]
=\frac \ln \left( x^2 + y^2 \right) + i \tan^\left( \frac
\right) \ .
This evaluation uses the properties of all
logarithms (see above), regardless of choice of base:
logb (c d ) = logb (c
) + logb (d ) and its generalization to
arbitrary products logb bz = z. Because the inverse
tangent is a multiple valued function of its argument, the
logarithm of a complex number is not unique either. See article on
complex logarithm.
Group theory
From the pure mathematical perspective, the
identity
\log(cd) = \log(c) + \log(d) \,
is fundamental in two senses. First, the
remaining three arithmetic properties can be derived from it.
Furthermore, it expresses an isomorphism between the
multiplicative
group of the positive real numbers and the additive
group of all the reals.
Logarithmic functions are the only continuous
isomorphisms from the multiplicative group of positive real numbers
to the additive group of real numbers.
Bases
The most widely used bases for logarithms are 10,
the mathematical constant e
≈ 2.71828... and 2. When "log" is written without a base (b missing
from logb), the intent can usually be determined from
context:
Many engineers, biologists, astronomers, and some others write
only "ln(x)" or "loge(x)" when they mean the natural logarithm of
x, and take "log(x)" to mean log10(x) or, sometimes in the context
of computing, log2(x).
Some people use Log(x) (capital L) to mean log10(x), and use
log(x) with a lowercase l to mean loge(x).
The notation Log(x) is also used by mathematicians to denote
the principal
branch of the (natural) logarithm function.
A notation frequently used in some European countries is the
notation blog(x) instead of logb(x).
This chaos, historically, originates from the
fact that the natural logarithm has nice mathematical properties
(such as its derivative being 1/x, and having a simple definition),
while the base 10 logarithms, or decimal logarithms, were more
convenient for speeding calculations (back when they were used for
that purpose). Thus natural logarithms were only extensively used
in fields like calculus while decimal logarithms were widely used
elsewhere.
As recently as 1984, Paul Halmos
in his "automathography" I Want to Be a Mathematician heaped
contempt on what he considered the childish "ln" notation, which he
said no mathematician had ever used. The notation was in fact
invented in 1893 by Irving Stringham, professor of mathematics at
Berkeley. As of 2005,
many mathematicians have adopted the "ln" notation, but most use
"log".
In computer science, the base 2 logarithm is
sometimes written as lg(x), as suggested by Edward
Reingold and popularized by Donald
Knuth. However, lg(x) is also sometimes used for the common
log, and lb(x) for the binary log. In Russian literature, the
notation lg(x) is also generally used for the base 10 logarithm. In
German, lg(x) also denotes the base 10 logarithm, while sometimes
ld(x) or lb(x) is used for the base 2 logarithm.
As the difference between logarithms to different
bases is one of scale, it is possible to consider all logarithm
functions to be the same, merely giving the answer in different
units, such as dB, neper, bits, decades, etc.; see the section
Science
and engineering below. Logarithms to a base less than 1 have a
negative scale, or a flip about the x axis, relative to logarithms
of base greater than 1.
Change of base
While there are several useful identities, the
most important for calculator use lets one find logarithms with
bases other than those built into the calculator (usually loge and
log10). To find a logarithm with base b, using any other base
k:
\log_b(x) = \frac.
Moreover, this result implies that all logarithm
functions (whatever the base) are similar to each other. So to
calculate the log with base 2 of the number 16 with a
calculator:
\log_2(16) = \frac.
Uses of logarithms
Logarithms are useful in solving equations in
which exponents are unknown. They have simple derivatives, so they are
often used in the solution of integrals. The logarithm is one
of three closely related functions. In the equation bn = x, b can
be determined with radicals,
n with logarithms, and x with exponentials.
See
logarithmic identities for several rules governing the
logarithm functions.
Science
Various quantities in science are expressed as
logarithms of other quantities; see logarithmic
scale for an explanation and a more complete list.
In chemistry, the
negative of the base-10 logarithm of the activity
of hydronium ions
(H3O+, the form H+ takes in water) is the measure known as pH. The activity of
hydronium ions in neutral water is 10−7 mol/L at
25 °C, hence a pH of 7.
In computer
science, logarithms often appear in bounds for computational
complexity. For example, to sort N
items using comparison can require time proportional to the product
N × log N. Similarly, base-2 logarithms
are used to express the amount of storage space or memory required
for a binary representation of a number—with k bits (each a 0 or a
1) one can represent 2k distinct values, so any natural
number N can be represented in no more than (log2 N) + 1
bits.
Similarly, in information
theory logarithms are used as a measure of quantity of
information. If a message recipient may expect any one of N
possible messages with equal likelihood, then the amount of
information conveyed by any one such message is quantified as log2
N bits.
Many types of engineering and scientific data are typically
graphed on log-log or semilog axes, in order to most
clearly show the form of the data.
In inferential
statistics, the logarithm of the data in a dataset can be used
for parametric statistical testing if the original data does not
meet the assumption of normality.
Musical
intervals are measured logarithmically as semitones. The interval between
two notes in semitones is the base-21/12 logarithm of the frequency
ratio (or equivalently, 12 times the base-2 logarithm). Fractional
semitones are used for non-equal temperaments. Especially to
measure deviations from the equal tempered scale, intervals are
also expressed in cents
(hundredths of an equally-tempered semitone). The interval between
two notes in cents is the base-21/1200 logarithm of the frequency
ratio (or 1200 times the base-2 logarithm). In MIDI, notes are
numbered on the semitone scale (logarithmic absolute nominal pitch
with middle C at 60). For microtuning to other tuning systems, a
logarithmic scale is defined filling in the ranges between the
semitones of the equal tempered scale in a compatible way. This
scale corresponds to the note numbers for whole semitones. (see
microtuning
in MIDI).
Exponential functions
One way of defining the exponential
function ex, also written as exp(x), is as the inverse of the
natural logarithm. It is positive for every real argument x.
The operation of "raising b to a power p" for
positive arguments b and all real exponents p is defined by
b^p = \left( e^ \right) ^p = e^.\,
The antilogarithm function is another name for
the inverse of the logarithmic function. It is written antilogb(n)
and means the same as bn.
Easier computations
Logarithms can be used to replace difficult
operations on numbers by easier operations on their logs (in any
base), as the following table summarizes. In the table, upper-case
variables represent logs of corresponding lower-case variables:
These arithmetic properties of logarithms make such calculations
much faster. The use of logarithms was an essential skill until
electronic computers
and calculators
became available. Indeed the discovery of logarithms, just before
Newton's era, had an impact in the scientific world that can be
compared with that of the advent of computers in the 20th century
because it made feasible many calculations that had previously been
too laborious.
As an example, to approximate the product of two
numbers one can look up their logarithms in a table,
add them, and, using the table again, proceed from that sum to its
antilogarithm, which is the desired product. The precision of the
approximation can be increased by interpolating between
table entries. For manual calculations that demand any appreciable
precision, this process, requiring three lookups and a sum, is much
faster than performing the multiplication. To achieve seven decimal
places of accuracy requires a table that fills a single large
volume; a table for nine-decimal accuracy occupies a few shelves.
Similarly, to approximate a power cd one can look up log c in the
table, look up the log of that, and add to it the log of d; roots
can be approximated in much the same way.
One key application of these techniques was
celestial navigation. Once the invention of the chronometer
made possible the accurate measurement of longitude at sea, mariners had
everything necessary to reduce their navigational computations to
mere additions. A five-digit table of logarithms and a table of the
logarithms of trigonometric
functions sufficed for most purposes, and those tables could
fit in a small book. Another critical application with even broader
impact was the slide rule, an
essential calculating tool for engineers. Many of the powerful
capabilities of the slide rule derive from a clever but simple
design that relies on the arithmetic properties of logarithms. The
slide rule allows computation much faster still than the techniques
based on tables, but provides much less precision, although slide
rule operations can be chained to calculate answers to any
arbitrary precision.
Related operations
Cologarithms
The cologarithm of a number is the logarithm of
the inverse of said number, meaning cologb(x)=logb(1/x)= -
logb(x).
Antilogarithms
The antilogarithm is the logarithmic inverse of
the logarithm, meaning that the antilogb(logb(x))=x. Thus, setting
by=x implies that logb(x)=y. By taking the antilogb of both sides,
antilogb(logb(x))=antilogby, thus x=antilogby. Therefore,
by=antilogby.
Computers
Most computer languages use log(x) for the
natural logarithm, while the common log is typically denoted
log10(x). The argument and return values are typically a floating
point (or double
precision) data type.
As the argument is floating
point, it can be useful to consider the following:
A floating point value x is represented by a
mantissa m and
exponent n to form
x = m2^n.\,
Therefore
\ln(x) = \ln(m) + n\ln(2).\,
Thus, instead of computing \ln(x) we compute
\ln(m) for some m such that
1 ≤ m < 2.
Having m in this range means that the value u = \frac is always in
the range 0 \le u . Some machines use the mantissa in the range 0.5
\le m and in that case the value for u will be in the range
-\frac13 In either case, the series is even easier to
compute.
To compute a base 2 logarithm on a number between
1 and 2 in an alternate way, square it repeatedly. Every time it
goes over 2, divide it by 2 and write a "1" bit, else just write a
"0" bit. This is because squaring doubles the logarithm of a
number.
The integer part of the logarithm to base 2 of an
unsigned integer is given by the position of the left-most bit, and
can be computed in O(n) steps using the following algorithm:
int log2(int x)
However, it can also be computed in O(log n)
steps by trying to shift by powers of 2 and checking that the
result stays nonzero: for example, first >>16, then
>>8, ... (Each step reveals one bit of the result)
The discrete
logarithm is a related notion in the theory of finite
groups. It involves solving the equation bn = x, where b and x
are elements of the group, and n is an integer specifying a power
in the group operation. For some finite groups, it is believed that
the discrete logarithm is very hard to calculate, whereas discrete
exponentials are quite easy. This asymmetry has applications in
public
key cryptography.
For each positive b not equal to 1, the function
logb (x) is an isomorphism from the
group
of positive real numbers under multiplication to the group of (all)
real numbers under addition. They are the only such isomorphisms
that are continuous. The logarithm function can be extended to a
Haar
measure in the topological
group of positive real numbers under multiplication.
History
The method of logarithms was first publicly
propounded in 1614, in a book
entitled Mirifici Logarithmorum Canonis Descriptio, by John Napier,
Baron of Merchiston, in Scotland.
(Joost
Bürgi independently discovered logarithms; however, he did not
publish his discovery until four years after Napier.) Early
resistance to the use of logarithms was muted by Kepler's
enthusiastic support and his publication of a clear and impeccable
explanation of how they worked.
Their use contributed to the advance of science,
and especially of astronomy, by making some difficult calculations
possible. Prior to the advent of calculators and computers, they
were used constantly in surveying, navigation, and other branches
of practical mathematics. It supplanted the more involved method of
prosthaphaeresis, which
relied on trigonometric
identities as a quick method of computing products. Besides the
utility of the logarithm concept in computation, the natural
logarithm presented a solution to the problem of quadrature
of a hyperbolic
sector at the hand of Gregoire
de Saint-Vincent in 1647.
At first, Napier called logarithms "artificial
numbers" and antilogarithms "natural numbers". Later, Napier formed
the word logarithm to mean a number that indicates a ratio:
(logos) meaning
proportion, and (arithmos) meaning number. Napier chose that
because the difference of two logarithms determines the ratio of
the numbers they represent, so that an arithmetic
series of logarithms corresponds to a geometric
series of numbers. The term antilogarithm was introduced in the
late 17th century and, while never used extensively in mathematics,
persisted in collections of tables until they fell into
disuse.
Napier did not use a base as we now understand
it, but his logarithms were, up to a scaling factor, effectively to
base 1/e. For interpolation purposes and ease of calculation, it is
useful to make the ratio r in the geometric series close to 1.
Napier chose
r = 1 - 10−7 = 0.999999
(Bürgi chose
r = 1 + 10−4 = 1.0001).
Napier's original logarithms did not have
log 1 = 0 but rather
log 107 = 0. Thus if N is a number and L
is its logarithm as calculated by Napier,
N = 107(1 − 10−7)L.
Since (1 − 10−7)107 is
approximately 1/e, this makes L/107 approximately equal to
log1/e N/107. An edition of Vlacq's work, containing many
corrections, was issued at Leipzig in 1794 under the title
Thesaurus Logarithmorum Completus by Jurij
Vega.
François
Callet's seven-place table (Paris, 1795), instead of
stopping at 100,000, gave the eight-place logarithms of the numbers
between 100,000 and 108,000, in order to diminish the errors of
interpolation,
which were greatest in the early part of the table; and this
addition was generally included in seven-place tables. The only
important published extension of Vlacq's table was made by Mr. Sang
in 1871, whose
table contained the seven-place logarithms of all numbers below
200,000.
Besides the tables mentioned above, a great
collection, called Tables du Cadastre, was constructed under the
direction of Gaspard de
Prony, by an original computation, under the auspices of the
French
republican government of the 1700s. This work,
which contained the logarithms of all numbers up to 100,000 to
nineteen places, and of the numbers between 100,000 and 200,000 to
twenty-four places, exists only in manuscript, "in seventeen
enormous folios," at the Observatory of Paris. It was begun in
1792; and "the
whole of the calculations, which to secure greater accuracy were
performed in duplicate, and the two manuscripts subsequently
collated with care, were completed in the short space of two
years." Cubicinterpolation
could be used to find the logarithm of any number to a similar
accuracy. |
Mathematics Quiz
Welcome to Q4Quiz Mathematics Quiz Section. Do you like Maths? How well you know about Mathematics. Expand your Math knowledge by reading our Answered quiz questions. Mathematics General Knowledge Quiz.
History of Mathematics Quiz Questions Quiz Questions Mathematics Quiz Part - 1 1) In which civilization dot patterns were first employed to represent numbers? Answer: Chinese. 2) The ancient Babylonians had their number system… |
Famous mathmatician essay
Famous mathmatician essay, Archimedes famous mathematician archimedes of syracuse was an outstanding greek mathematician, inventor, physicist, engineer and also an astronomer.
Biographies of w omen mathematicians home the association for women in mathematics sponsors an essay contest for biographies of contemporary women mathematicians. Database of free mathematics essays - we have thousands of free essays across a wide range of subject areas sample mathematics essays. Arguably the greatest mathematician of all he published an essay on conic sections using the methods list of famous mathematicians and their contributions. Check out our top free essays on famous mathematician to help you write your own essay.
Essay on indian cricket team essays and research papers mr consistent-rahul dravid, the guido man at the helm of indian cricket is the cynosure of all. Srinivasa ramanujan frs ( / the results in his notebooks inspired numerous papers by later mathematicians trying to prove what he had found. This is a list of important publications in mathematics in these papers, poincaré introduced the notions of homology and the fundamental group. Evariste galois: french mathematician famous for his contributions to the part of higher algebra now known as group theory his theory provided a solution to the long.
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Sometimes knowing the people behind the formulas can help you understand how they work learn about the lives of famous mathematicians, from euclid to fibonacci. Famous mathematician: john wallis introduction the paper focuses on the life and works of english mathematician john wallis, who is considered to be the most. Free mathematicians papers, essays outline i introduction a a condensed history of mathematics b famous mathematicians and their accomplishments ii. Essays - largest database of quality sample essays and research papers on famous mathematicians. Discover unexpected relationships between famous figures when you explore our famous mathematicians group.
The 10 best mathematicians he owes his standing as the most famous name in maths due to a theorem about right-angled writing about 1,500 papers. Mathematician essays: over 180,000 mathematician essays, mathematician term papers, mathematician research paper, book reports 184 990 essays, term and research. Mathematics is a field that many people shy away from, but there are some who had a passion for numbers and making discoveries regarding equations, measurements. Free essays on famous mathematician get help with your writing 1 through 30.
Mathematician research paper objective: the purpose of this assignment is to acquaint students with the history of a famous mathematician and discover his/her. They would give to the study are addressed, g the colour of a statistical context is used when you have chosen a project of american democracy electoral and. Euclid was a famous mathematician a greek mathematician, euclid is believed to have lived around 300 bc (ball 50) most known for his dramatic contributions to. Mathematicians' manifesto a young man who died at the age of 32 in a foreign land he had travelled to, to pursue his craft a clumsy eccentric who could. |
Mathematics Is Wrong. Here's Why.
and just trust me...this genius didn't figure out the secret to math. the guy/girl who really has the secret prolly isn't even on this site. that
person is either a) not saying a word about it, or b) is about to publish a paper on it. i know if i had rock solid proof of off-earth civilizations,
i'd never say a word about. i'd go to my grave before i'd ever, EVER put worlds in danger to satisy my ego. you can bet your house on thatInfinity is a starting point. I argue that infinity is THE starting point of the universe due to the fact that energy is neither created nor
destroyed. So mathematically, energy looks like (energy) times infinity. So you have infinitely lasting energy, and you also have infinity. No matter
the operation, infinity cannot be separated from anything. The existence of anything can be traced back to infinity and still has its connection with
infinity.
My friend you unwittingly just discovered the "God" variable through your analysis of math, infinite energy that has always existed. This infinite
energy was used to build all the atomic particles in the universe known as mass would be lost, and that would be extremely selfish of me.
Nevertheless, I will always post to the public before I gather all I know into a single book and publish it for profitwhatever...zero is before 1. without zero, there can be no 1. is it really that hard? without zero, there can be no infinity. zero is the starting
point. 1 is NOT the starting point. something has to exist before 1 can exist, and we've chose that to be zero. so you plan is to replace zero with
infinity. even the dullest of imaginations can see that cannot be the case. how can infinity come before 1. i'm mean seriously, please claify that
cause i just cannot for the life of me get my head around that. and please explain it in plain english so everybody can understand it. oh and by the
way, infinity is not with out form. it's just so large and we are so small we cannot see it. just as we are so small within the universe, we cannot
see the end of it. that does not mean it is without form. it has a form. we just can't see it.
Understand....I know what you mean about infinity...but zero...really can it be disputed in a mathematical way that makes sense...I would love to hear
it....2 - 2 is still nothing...and that is what zero represents....so how do you come up with and infinite possible in that equation?
Of course we don't want to infer religion into this but infinity is a symbol that carry a weight of understanding in mathematics...are you purposing
a change?
Is it possible that instead of having zero oranges we are left with -2 oranges. I had 2 lost 2 leaving me with -2.
so we exist inside infinty, i start with - infinity and + infinity but can never have zero, can only have values assigned expressed as + or - I can
never have nothing. I can be without something or negative that something.?
It is also important to note that numbers that arise from 0 exist to themselves as separate from everything and therefore should not be able to be
involved in any computing with other numbers because they have nothing that allows them to exist in the same space.
Numbers that arise from infinity remain connected to infinity, so mathematical calculations using any finite value are possible because they remain in
the context of a bound set even within the equations themselves.
In other words, in 0 based math, when 1 arises, it is in no way connected to the context of 2 if it arises. There is no platform for it to arise as
part of a system.
Infinity, on the other hand, is a platform for any numbers to arise and still be connected to the system in which they are operating would be lost, and that would be extremely selfish of me.
Nevertheless, I will always post to the public before I gather all I know into a single book and publish it for profit.
yea well that's just a whole bunch of Who Shot John. a paper doesn't have to be 10 million pages to make it's point. if you were so serious, you
would publish what you have, and others will agree with you, and help you flesh out your theory. waiting to write a book about the most ground
shaking theory since the creation of the zero seems a bit silly to me. and besides, you found this site. surely there is in existance somewhere on
the interwebs that will publish that rather wordy explaination of the fallacy of modern mathmatics, and will reach a wider populance then can be
reached at ATS.
Great Video about ∞. Not that new but still a great watch. As far as ∞ replacing zero I think your just playing word games. We have to have a
representation of nothing. Say you have the number 1001 you still have to have nothing in the hundreds and tens places. As far as using ∞ as a base
instead of zero for multiplication and division how does it make more sense to say ∞ x n= ∞(n) than to say nothing multiplied by n is nothing...
What your saying makes no sense. Infinity is already used in mathematics along side zero. It is mostly used in set mathematics to describe uncountable
numbers. And it has a very important place. Zero used in nearly all mathematics is just as important as a base and foundation.
Both Infinity and zero do not physically exist. There is nothing that goes on forever, and zero is the term we give to the absence of quantity. It
can't be there because the reason it is is because there isn't anything there. If you think about it, it's like the darkness. Light rays allow us to
see, however when the light rays are gone there is darkness. But there are no Dark rays, meaning the darkness isn't actually there. So you're trying
to compare and relate two things that don't even exist.
However, mathematically speaking, Zero DOES exist, and infinity again does not. Say i had a bag of apples (let's call this bag A) and you had a bag of
apples (B) now, in A there are five apples. In B, zero. If you give me all the apples in your bag you have given me zero apples. If x is the apples in
your bag and y the apples in mine, x+y=y.
If i asked you how many apples you had, your answer would be zero.
infinity cannot be quatified. 0 can be understood as we can see or maybe not see i should say when we have reached 0. Infinity however cannot be
comprehended because it would take an unlimited mind to grasp the scale as it dosent even have a scale. Hang on maybe your right if both dont exist.
Im dizzy and need to lie down.
0 is a number, zero that represents the "nothing", infinity is not a number, infinity could be anything. Mathematics is not flawed, I guess we could
add infinity befor or after zero, but that is already the case in mathematics, infinity exists because of mathematics, zero is therefor necessary.
Without zero you do not have a point to define infinity, so without zero infinity does not exist.
Here is where your logic is flawed, to be able to multiply 0 by 1 you must add 1 before multiplying, without adding you, are just multiplying 1 of
nothing with 1 of nothing.
The concept of Zero is attributed to the Hindus. The Hindus were also the first to use zero in the way it is used today. Some symbol was required in
positional number systems to mark the place of a power of the base not actually occurring. This was indicated by the Hindus by a small circle, which
was called Shunya, the Sanskrit word for vacant. This was translated into the Arabic Sifr about 800 A.D. Subsequent changes have given us the word
zero.
In Babylone by middle of the 2nd millenium BC, the lack of a positional value (or zero) was indicated by a space between sexagesimal numerals.
In 498 AD the Indian mathematician and astronomer Aryabhatta stated that Sthanam sthanam dasha gunam means place to place in ten times in value, which
may be the origin of the modern decimal-based place value notation. Arabs spread the Hindu decimal zero and its new mathematics to Europe in the
Middle Ages.
Originally posted by kbriggssCan you think of any more?
False:
Time: began during the big bang, time has not being going on forever, it's been going on for 14 billion years.
Size: smallest length possible is 0.0000000000000000000000000000000001 m, and physically speaking anything can be broken down into millions of tiny
particles meaning that say an elephant can be broken down till its particles are so tiny they aren't visible. But technically, The Universe is the
largest thing, and it has a maximum size.
Temperature: Absolute zero is the coldest possible temperature. however, the hottest temperature is interesting, as there is no absolute "hot" but
ill be looking into that.
Forgive me as I'm not the most proficient or experienced mathematician, though I have taken some upper level math courses because of my major.
I thought, the axiom x/0 = undefined (where x is any number), is just a rule that mathematicians have agreed upon? Since you need somewhere to begin,
you need to define some terminology before you can continue, such as for example: 0 + n = n or 0 * n = 0 etc.
Again, i could be wrong (which i may be) but i remember taking a linear algebra course where we had to go through and prove all 10 axioms.
Originally posted by AuirOverrun
Both Infinity and zero do not physically exist. There is nothing that goes on forever, and zero is the term we give to the absence of
quantity.
Infinity and nothing do not exist as a manifest expression. 0 does not exist because energy can neither be created nor destroyed. Infinity does
exist for this reason. Energy goes on forever. There is no objective absence of quantity. You can say you have two apples and you take away two
apples, but its not that there are 0 apples, its just that those 2 apples are now in a different location. Subjectively, you could say you have 0
apples, although those apples exist independently of your ownership of them.
It can't be there because the reason it is is because there isn't anything there. If you think about it, it's like the darkness. Light rays
allow us to see, however when the light rays are gone there is darkness. But there are no Dark rays, meaning the darkness isn't actually there. So
you're trying to compare and relate two things that don't even exist.
I am comparting two things where one allows existence and the other does not.
However, mathematically speaking, Zero DOES exist, and infinity again does not. Say i had a bag of apples (let's call this bag A) and you had
a bag of apples (B) now, in A there are five apples. In B, zero. If you give me all the apples in your bag you have given me zero apples. If x is the
apples in your bag and y the apples in mine, x+y=y.
If i asked you how many apples you had, your answer would be zero.
edit on 29-8-2011 by AuirOverrun because: (no reason
given)
edit on 29-8-2011 by AuirOverrun because: (no reason given)
Again, this is subjective, and mathematics is supposed to be the most objective tool we have. Energy cannot be created or destroyed, so if you have
some quantity of energy, and now you don't, that enerrgy which is independent of ownership still exists. When numbers arise, they arise in their own
existence and are not dependent on another number. Infinity allows for this. 0 implies that to even multiply 2*3, you have to count to 6 first.
Infinity allows these numbers to arise in whatever position is equal to their value and continue to maintain their reference.
I have always considered infinity and zero as reference points. Do you remember number line. No matter where on the infinity line you use/observe as
zero, maths rules remain the same relative to that point. So you are correct in one sense, but maths is correct because it does not treat zero or
infinity as absolutes, but as mere references.
If you take a Physics Theory class such as i have .... you will learn that Math is the perception of the world around you.
Mathmatics is nothing but a big ole Theory.
There is no Mathmatical facts , Its Mathmatical Theorims.
We had Physics Theories that blew Mathmatical expressions out of the Water , and you couldnt figure it up, but through known truths , and far left
field dimensional algorithms.
You also have the ever looming presence of Imaginary numbers.
-------
So , when you said Mathmatics is wrong. Its is not. Its our perception of the world around us. Secondly , Mathmatics is a Theorim.
--------
Yes Theoritcal Physics also uses Philosophy to explain it self. Very interesting stuff if you have the stomach for it.
--------
Also , i dont know why you keep using Infinity. If you take a Physics class , everything is attempted to be explained. When you say infinity , its an
easy way out for you to explain things you dont understand. Get infinity out of your head , its just a never ending point and a never starting
starting point. Infinity doesnt exist. When explaining physics.
You make a start point and a end point. You have to go in thinking what you want the Equations / Algorithms you form to accomplish. (Theoretical
Physics). Its just accepted Truths.
--------
So again , Conclusion. Our Mathmatics arnt wrong (The Forumulas) its what we want them to be / know them to be through our understanding. Its a Theory
, so we can change when ever we want using accepted truths. In reality if you get the accepted truth of an equation correct .... then its correct
however if you get the equation wrong against the accepted truth , then its wrong in our eyes.
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Mathematics, One Day at a Time
Vieta
One of my favourite anecdotes from my days teaching in an English secondary school involves the variable x. I was introducing the concept of a linear equation. Specifically the y=mx+b that kids in Ontario learn all about in Grade 9.
I was explaining how you plot the line on a graph when a young boy got out of his chair threw his hand up and toward the board, palm open, indicating with a wild gesture the third last letter of the alphabet written plainly in the equation. "But Sir!" he exclaimed, "What Is X!"
For my students, up until this point a letter had represented some unknown quantity, and it was their responsibility to divine its value. To Solve For X! But this was a new concept. In y = 2x + 1, x isn't a quantity to solve; it's a variable, a symbol of all the possible numbers to which we could multiply by 2 and add 1. I was amused and did my best to explain the nuanced differences of this new concept masquerading as an old familiar one.
But how did X become the symbol of anything in the first place? Who was the mystery person who decided that we needed to use letters to represent numbers?
His name was François Viète. |
"Johannes Kepler is best known for figuring out the laws of planetary motion. In 1610, he published a little book called "The Six-Cornered Snowflake" that asked an even more fundamental question: How do visible forms arise? He wrote: "There must be some definite reason why, whenever snow begins to fall, its initial formation is invariably in the shape of a six-pointed starlet. For if it happens by chance, why do they not fall just as well with five corners or with seven?"
All around him Kepler saw beautiful shapes in nature: six-pointed snowflakes, the elliptical orbits of the planets, the hexagonal honeycombs of bees, the twelve-sided shape of pomegranate seeds. Why? he asks. Why does the stuff of the universe arrange itself into five-petaled flowers, spiral galaxies, double-helix DNA, rhomboid crystals, the rainbow's arc? Why the five-fingered, five-toed, bilaterally symmetric beauty of the newborn child? Why?
Kepler struggles with the problem, and along the way he stumbles onto sphere-packing. Why do pomegranate seeds have twelve flat sides? Because in the growing pomegranate fruit the seeds are squeezed into the smallest possible space. Start with spherical seeds, pack them as efficiently as possible with each sphere touching twelve neighbors. Then squeeze. Voila! And so he goes, convincing us, for example, that the bee's honeycomb has six sides because that's the way to make honey cells with the least amount of wax. His book is a tour-de-force of playful mathematics.
In the end, Kepler admits defeat in understanding the snowflake's six points, but he thinks he knows what's behind all of the beautiful forms of nature: A universal spirit pervading and shaping everything that exists. He calls it nature's "formative capacity." We would be inclined to say that Kepler was just giving a fancy name to something he couldn't explain. To the modern mind, "formative capacity" sounds like empty words.
We can do somewhat better. For example, we explain the shape of snowflakes by the shape of water molecules, and we explain the shape of water molecules with the mathematical laws of quantum physics. Since Kepler's time, we have made impressive progress towards understanding the visible forms of snowflakes, crystals, rainbows, and newborn babes by probing ever deeper into the heart of matter. But we are probably no closer than Kepler to answering the ultimate questions: What is the reason for the curious connection between nature and mathematics? Why are the mathematical laws of nature one thing rather than another? Why does the universe exist at all? Like Kepler, we can give it a name, but the most forthright answer is simply: I don't know." |
Numerology asp script calculatorosoite poi... |
Q We now return to the important issue of how to represent
the three fundamental polarities in a precise holistic mathematical fashion.
I believe you have some deeply relevant insights regarding the nature of
the number system.
PC Yes! the simplest approach is to think in terms of the natural numbers
1, 2, 3, 4 etc.
(We should really confine ourselves to the numbers we have utilised
so far such as 0, 1 and 2. However the inclusion of other numbers - though
not necessary - may make the exposition a little easier to follow!)
What is often forgotten is that these numbers are implicitly defined
with respect to a fixed dimension (i.e. power) of 1.
Thus strictly speaking we should write the natural number system as
11, 21, 31, 41,…
In geometrical terms, a line is literally one-dimensional so that these
numbers can be represented by successive points at equal intervals from
each other on a horizontal straight line.
So the (horizontal) linear number system is defined with respect to
a fixed unitary dimension.
We now return to the binary digits and the basic operations of addition
and subtraction with respect to both analytic and holistic interpretation.
Thus in analytic terms 11 + 11 = 21.
Likewise 11 - 11 = 01.
Also in holistic terms 11 + 11 = 21.
Also 11 - 11 = 01.
As regards unitary form, differentiation takes place through the separate
positing of opposite polarities. Thus when these (separate) forms are differentiated
with respect to the same (fixed) unitary dimension (i.e. within a given
level), a dualistic interpretation of the horizontal polarities i.e. exterior
and interior (and interior and exterior) arises.
Also with regard to unitary form, integration takes place through the
dynamic complementary fusion (and ultimate identity) of opposite polarities.
Thus when these (complementary) forms are integrated with respect to the
same (fixed) unitary dimension (i.e. within a given level) a nondual interpretation
of the horizontal polarities arises.
One interesting aspect of this formulation is that it clearly points
to a remaining confusion of form and emptiness (i.e. 01). Thus
though integration with respect to (conscious) phenomena entailing Type
1 complementarity can arise at H1 (subtle realm), it still co-exists with
a degree of (rigid) form with respect to the general (qualitative) unitary
dimension to which it is related.
Put another way, though linear understanding of distinct (conscious)
perceptual phenomena in experience gives way to nondual appreciation (0)
it tends to initially co-exist with the rigid interpretation of more general
dimensional (i.e. conceptual) phenomena.
Vertical System
Q You believe however than another fully coherent vertical number
system can be defined which has especially important implications in holistic
terms.
PC Yes indeed! I thought long and hard about this issue for many years
before it finally became clear. Like most really important ideas, once
it is realised it seems very obvious. Furthermore it is intimately connected
with the fundamental mathematical operations of multiplication and division
(which - by definition - have an equally important holistic mathematical
interpretation).
As always we will first deal with this vertical system in analytic terms
by considering the fundamental operations of multiplication and division.
Just as we initially sought to add 1 to itself, we now consider the
multiplication of 1 by itself.
So 1 * 1 = 1;
Now, when we omit the dimensional characteristic this operation seems
very trivial as it results in the same answer (i.e. 1).
Thus whereas the continual addition of 1 to itself (and subsequent totals)
results in the (horizontal) natural number system, the alternative operation
of continual multiplication seemingly leads to the unchanged result of
1.
The important implication of this is that multiplication (and also division)
are conventionally interpreted with respect to the (horizontal) number
system, leading to what is strictly - even from an analytic approach -
a reduced (merely) quantitative interpretation of numbers.
Q How can this be the case? Surely numbers are quantities!
PC Not quite! However to appreciate this let me ask you a simple question.
What is 2 * 2?
Q 4 of course!
PC Well perhaps! But let's explore a little further!
If we use our more complete representation of numbers then 2 * 2 properly
represents 21 * 21.
Now you might remember that when we multiply a number raised to any
power (i.e. dimension) by the same number (raised to its respective power)
then we express the result as the number raised to its combined addition
of powers.
Therefore 21 * 21 can be expressed as 22.
So interestingly the change that has taken place here is with respect
to the (qualitative) dimension (while the number as quantity has remained
unchanged).
However because conventional mathematics is defined within a linear
analytic framework, ultimately the value of numerical operations are expressed
with respect to the invariant 1st dimension.
Thus value of 22 - which combines numbers representing both
a (quantitative) object and a (qualitative) dimension - is expressed in
merely reduced quantitative terms (with respect to the invariant 1st
dimension).
Therefore the result of 22 is expressed as 41.
However because - in conventional terms - numbers are ultimately interpreted
in (quantitative) linear terms the inclusion of the dimensional characteristic
is deemed unnecessary.
Therefore, in linear terms, 22 = 4 (as another number in
the horizontal system).
Q I can appreciate to a degree what you are getting at. However does
this very refined distinction of number as quantity and number as dimension
respectively have any practical numerical relevance in analytic terms?
PC Actually it does! But - as so often happens in a reduced approach
- its great potential significance is greatly ignored.
The best way of appreciating this is to go back to our simple example
and picture it geometrically. Say I want to draw a square with side 2 units.
Then the area will be 22.
Though I may well express this answer as 4, it should be obvious that
we are now talking about a qualitatively distinctive type of unit. Thus
though each side is measured in linear (i.e. one-dimensional) terms, the
actual area represents square (i.e. two-dimensional) units.
So properly speaking the area of our square is 4 square units.
Strictly, this distinction applies to all numerical operations involving
multiplication.
Q Ah! Now I am beginning to appreciate the importance of what you
are saying which has implications not only for holistic interpretation
but also for analytic interpretation also!
PC Yes! Using Jungian interpretation, there is an important shadow (qualitative)
side of analytic mathematical interpretation that has been greatly suppressed.
Indeed the recovery of this shadow side is a necessary prerequisite for
the provision of an appropriate context for dynamic holistic mathematical
interpretation (which entails the interaction of both quantitative and
qualitative aspects of understanding).
Q You were about to define an alternative vertical number system
that is based on correct understanding of the operations of multiplication
and division. With this initial clarification out of the way why not proceed?
PC Let us start again with the multiplication of 1 by itself this time
including the important dimensional characteristics.
So we now have 11 * 11 = 12 .
In the horizontal number system we defined the number quantity (which
can vary) with respect to an invariant unitary number dimension.
In the vertical number system we define the number quality as a dimension
(which can vary) with respect to an invariant unitary number quantity.
Now if we multiply this result by one and continue to multiply each
subsequent result in this manner then we again generate the same natural
number system 11,12 ,13,14
,
… this time referring to qualitative dimensions (rather than quantitative
objects).
So we really have two analytic natural number systems.
In the first system 1, 2, 3, 4, … represent number quantities (as objects)
that are expressed with respect to an invariant unitary dimension.
In the second system 1, 2, 3, 4, … represent number qualities (as dimensions)
that are expressed with respect to an invariant unitary quantity.
A decisive step in our understanding will come when we show how to express
number (expressed as a qualitative dimension) indirectly as a quantitative
object (and alternatively how to express number (expressed as a quantitative
object) indirectly as a qualitative dimension.
Holistic Multiplication and Division
Q I gather that this is what you mean by a holistic mathematical
interpretation of Type 2 complementarity that involves horizontal/vertical
and (vertical/horizontal) interactions as between the fundamental polarities.
However before doing so, will you explain now what is meant by multiplication
and division in a holistic mathematical context.
PC Yes, this is an extremely important topic as it intimately relates
to the part and whole (and whole and part) aspects of all relationships.
However let us look first at division in terms of our vertical number
system.
Thus 11 / 11 (1 divided by 1) = 11 - 1
= 10.
So just as 1 - 1 = 0 with respect to the horizontal number system, likewise
1 - 1 = 0 with respect to the vertical number system.
What this in effect entails is that we have defined Type 1 complementarity
with respect to both horizontal and vertical polarities (which in geometric
terms would be represented by the opposite ends of the horizontal and vertical
line diameters of the circle respectively).
Once again differentiation and consequent phenomenal duality, with respect
to either horizontal or vertical polarities, arises from positing - in
either context - opposite aspects (within relatively independent separate
frames of reference).
Thus we can differentiate the horizontal unitary poles i.e. exterior
and interior (and interior and exterior) by separating opposite poles thereby
treating each one respectively as positive. Likewise we integrate both
poles by treating them in a dynamic bi-directional fashion as complementary
(both positive and negative) leading to empty (nondual) awareness.
In like manner we differentiate the vertical unitary poles i.e. whole
and part (and part and whole) by again separating opposite poles treating
each one as positive.
Again we integrate both poles by treating them in a paradoxical bi-directional
fashion as complementary (positive and negative) leading likewise to empty
(nondual) awareness.
Q Can I raise a relevant point here. You talk about Type 1 complementarity
that applies to the integral understanding of opposite poles. However presumably
we also have Type 1 separation of poles (which serves as a precondition
for such integration)?
PC That's right. We have both Type I separation and Type 1 complementarity
representing both differentiation and integration respectively of opposite
poles.
However it is important to remember that Type 1 separation is of a very
subtle bi-directional kind (where opposite mirror asymmetrical interpretations
are clearly recognised for all linear type sequences).
So for example, earlier we recognised that holism (where holonic relationships
are ultimately defined with respect to their whole aspect) has a mirror
asymmetrical interpretation as partism (where holonic relationships are
defined with respect to their corresponding part aspect).
So by including differentiation we can perhaps see more clearly the
limitations of Type 0 complementarity. Here separation takes place in one-direction
(e.g. merely in terms of holism) leading to a somewhat unambiguous rigid
dualistic interpretation of relationships. Therefore - as little paradox
is generated - integration is largely reduced to differentiated understanding.
Q So let us return now to the holistic interpretation of multiplication
and division. How does this Type 2 complementarity (and Type 2 separation) of
poles take place?
PC Remember that Type 2 complementarity is defined more subtly in terms
of the relationship between horizontal and vertical (and equally vertical
and horizontal) poles within a given quadrant.
Thus Type 2 complementarity (where vertical and horizontal poles are
properly integrated) and of course related Type 2 separation (where vertical
and horizontal poles are properly differentiated) arise with respect to
each of the four quadrants.
As we have seen the key problem here is that horizontal and vertical
poles relate to polar relationships that are not directly comparable in
terms of each other.
So if we represent the horizontal direction as representing quantitative
object phenomena, then the vertical direction - relatively - represents
qualitative dimensional phenomena (though of course in interactive terms
these frames of reference can always be reversed).
It might help again to go back to our simple number illustration - using
the geometrical representation of the square - where 2 is multiplied by
itself.
As we have seen the result here is 4 square units.
So a transformation has thereby taken place in both quantitative and
qualitative terms.
Therefore multiplication - by its very nature - entails both a quantitative
(horizontal) and qualitative (vertical) transformation.
If one looks up the Oxford Dictionary, to multiply is defined as "to
cause to become much, many or more, to make many or manifold".
Now we saw earlier that the relationship of the one and the many (and
the many and the one), involves the interaction of a perception with its
corresponding concept (and - in reverse fashion - the concept with its
corresponding perception).
In terms of our interpretative framework, the actual perception of a
unitary phenomenon such as a particular number, operates at the horizontal
quantitative level.
However the corresponding concept of number - which provides a general
dimension potentially applying to all numbers operates - relatively - at
the vertical qualitative level.
However with linear understanding all perceptions are interpreted with
respect to a fixed qualitative unitary dimension, so that the distinction
between quantitative and qualitative is confused leading to a merely reduced
interpretation.
Thus in a linear interpretation, the concept of number is understood
in a (reduced) quantitative manner as applying to all actual numbers. So
what involves - in dynamic interactive terms - the relationship between
both the quantitative (horizontal) and dimensional (vertical) characteristics
of number is thereby reduced to its merely quantitative aspect.
In the context we have defined it, the perception relates to the part
and the concept to the whole aspect of understanding respectively.
In proper dynamic interactive terms the relation between part and whole
(and whole and part) is always subject to Type 2 complementarity where
quantitative and qualitative aspects of understanding continually interact
with respect to both horizontal and vertical interpretation.
However the reduced linear interpretation of this relationship between
part and whole - which characterises conventional analytic understanding
- is in (merely) quantitative terms.
We saw this earlier in our discussion of the relationship between the
atom and the molecule. In the linear (asymmetrical) interpretation, the
atom is viewed as part of a larger (whole) molecule. This is possible because,
in this context, both the atom and the molecule are interpreted in quantitative
terms. So the atom (as smaller quantity) is thereby seen as part of the
molecule (as larger quantity). By the same token - in this asymmetrical
quantitative interpretation - the (larger) molecule cannot thereby be part
of the (smaller) atom.
However before moving on to deal with the more correct dynamic interpretation,
I will point to a fascinating aspect of the relationship between multiplication
and division that is evident (even in the linear analytic interpretation).
Let us say that we have a cake divided into two slices.
Now in linear terms the cake represents the whole quantity to which
the slices - as parts - are related.
So the whole cake expressed in terms of the number of its quantitative
parts = 2. In other words we have been able to multiply the number of slices
as parts to 2 (i.e. 1 * 2).
Indeed if we further subdivided each of these slices in half, the whole
cake (as expressed in terms of its part slices) would thereby multiply
to 4 (i.e. 2 * 2).
Now in reduced linear terms the whole cake expressed in terms of its
part slices = 41.
However if we reverse the frame of reference and express each slice
as part of the whole cake we thereby divide the cake into its constituent
parts.
So in the former case we multiply the quantitative parts (as slices)
to obtain the quantitative whole (as cake).
In the latter case we divide the (quantitative) whole as cake to obtain
the (quantitative) parts (as slices).
So each slice as part of the cake = ¼.
Alternatively this can be expressed as 4 -1.
So we can see here that whereas multiplication is expressed with respect
to the positive direction of the unitary dimension of form, division is
expressed with respect to its negative direction.
So the operations of multiplication and division (as regards dimensions)
exactly correspond here on a vertical level with the corresponding operations
on a horizontal level of addition and subtraction.
So just as the positing and negating of perceptions continually takes
place, likewise the positing and negation of concepts takes place. And
if we represent perceptions in horizontal terms (as quantities), then -
in relative terms - concepts are represented in a vertical manner (as dimensions).
Mathematical Dimensions and Integral Interpretation of Space and
Time
Q What you are suggesting here seems to lead to a very different
notion of dimensions to what we are currently use. Can you elaborate further?
PC Yes! One of the most exciting features of the holistic mathematical
approach is that it leads to a new dynamic interpretation of dimensions
which corresponds exactly with mathematical notions.
The intrinsic structure of the dimensions of space and time is therefore
mathematical in this holistic dynamic sense.
We will have more to say on this later when we examine the mathematical
interpretation of dimensions in greater detail.
However I will just attempt to give a little flavour at this stage.
Object phenomena (in space and time) arise from the continual positing
of perceptions in experience. The corresponding negation in these phenomena
leads to a certain fusion of opposites (as spiritual emptiness) enabling
a switch to the dimensions of space and time, which relate to the positing
of corresponding concepts.
Again in the dynamics of experience, concepts in turn are continually
negated enabling the reverse switch to perceptions where object phenomena
are once again posited.
Strictly speaking therefore in dynamic terms, objects have both a positive
and negative direction. Likewise - which is perhaps even more surprising
- the dimensions of space and time have both a positive and negative direction.
So in the dynamic integral view, space and time dimensions are fully
symmetrical.
Object phenomena are continually posited in space (as perceptions).
The
negation of these objects (though corresponding negation of perceptions)
causes a switch by which the dimensions of space are posited (as concepts).
Again the negation of these dimensions (through negation of concepts) causes
a reverse switch by which object phenomena are again posited (in space).
In like manner objects are also continually posited in time (as perceptions).
The negation of these objects causes a switch by which the corresponding
dimensions of time are posited (as concepts). Once more the negation of
these dimensions causes a reverse switch by which object phenomena are
again posited (in time).
So space has positive and negative aspects which continually interact
both as objects (in space) and the space dimensions (to which these objects
are related).
Likewise time has positive and negative aspects again interacting both
as between objects (in time) and the time dimensions (to which these objects
are related).
Q What is the precise relationship as between space and time?
PC To appreciate this in a dynamic integral manner we must recognise
- from a psychological perspective - the interaction of both conscious
and unconscious in experience.
When one reflects on it, unconscious implies not-conscious (i.e. the
negation of conscious activity). Likewise conscious implies not-unconscious
(i.e. as the corresponding negation of unconscious activity).
Thus conscious and unconscious dynamically interact in experience through
the continual positing and negating of one another. The conscious is posited
and then negated as unconscious; the unconscious is then (indirectly) posited
and negated in turn as conscious.
Now when the role of the unconscious is not properly recognised, this
sets severe limits on the quality and flexibility of dynamic interaction
that can take place.
Object phenomena are then rigidly experienced and the unconscious -
which is not properly recognised - likewise implicitly intervenes in a
somewhat rigid (and distorted) manner. So rather than a continual fluid
interaction as between object phenomena (and their corresponding dimensions),
the dimensions of space and time become largely separated from object phenomena
and viewed in absolute terms (as the medium in which all object phenomena
are contained).
Thus in the conventional scientific approach, both objects and dimensions
are viewed in a merely positive (i.e. conscious) fashion.
In the Newtonian worldview space and time are interpreted in an absolute
unchanging fashion.
Though modern physics challenges this perspective interpretation is
still somewhat rigid.
For example quantum mechanics would allow – at the level of sub-atomic
particles - for both the positing and negating of object phenomena (i.e.
matter and anti-matter particles).
However this finding is not equally extended to interpretation of their
corresponding dimensions i.e. in an appreciation of dimensions and anti-dimensions
(of space and time).
Likewise Relativity Theory allows for a degree of interaction as between
space and time.
However space and time are still viewed in non-symmetrical terms (i.e.
3 of space and 1 of time).
By contrast in the fully integral appreciation (which recognises the
necessary interaction of conscious and unconscious), both object phenomena
and dimensions have positive and negative aspects. Likewise space and time
(and time and space) are - ultimately - fully symmetrical with each other.
From an integral perspective, both the psychological and physical (and
physical and psychological) aspects of reality are complementary. This
therefore entails that science should recognise complementary correspondents
for both conscious and unconscious aspects (in terms of the dynamic interactions
of physical reality).
However at present it is very lop-sided with material phenomena being
interpreted solely as correspondents of conscious understanding. However
we equally need recognition of a non-physical (ground of reality) that
complements the unconscious.
Then exterior reality would be more readily understood in integral terms
as representing the continual interaction of the physical with its non-physical
ground, which would correspondingly be interpreted through the interaction
of both conscious and unconscious (in psychological terms).
Now the critical question remains as to the precise relationship as
between space and time. As we shall see in our next discussion, this can
indeed be given a surprising answer in holistic mathematical terms. We
will be then able to show how space can be converted into time (and time
into space).
What I will say at this stage is that the interaction of space and time
is intimately related to the corresponding interaction of the cognitive
and affective aspects of understanding.
As always the relationship is circular and relative (and ultimately
purely paradoxical).
So both space and time can be associated with the cognitive and affective
aspects respectively. However if for example the cognitive aspect is associated
with time (and objects in time), then it is the intervention of the affective
aspect in this context that enables the switch to experience of space (and
objects in space).
Likewise when the cognitive aspect is associated with space (and objects
in space), then interaction with the affective aspect enables the switch
to corresponding experience of time (and objects in time).
Using an alternative frame of reference we could associate the affective
aspect with time, so that now interaction with the cognitive aspect enables
a switch to experience of space. Finally when the affective aspect is associated
with time, interaction with the cognitive aspect enables a switch to experience
of space.
So from a psychological perspective, the two-way continual interaction
of space (and objects in space) with time (and objects in time) reflects
the corresponding two-way interaction as between the cognitive and affective
aspects of experience.
One key implication for an integral scientific appreciation of reality
is that the dimensions of space and time (and their associated object phenomena)
have both personal and impersonal interpretations, which continually interact
in dynamic terms.
We have just commented on one major limitation of conventional scientific
understanding in that - at least in formal terms - it ignores the role
of the unconscious.
Thus the unconscious is thereby reduced to conscious interpretation
leading to the corresponding reduction of qualitative to quantitative interpretation.
Now we can comment on another limitation in that science ignores - again
in formal interpretation - the role of the affective aspect of experience.
The affective is thereby reduced to the cognitive aspect leading to a merely
impersonal interpretation of objects and dimensions.
Now I would accept that by the very nature of science that its formal
presentation should rightly rely on cognitive interpretation. We therefore
cannot hope to directly translate the nature of affective experience though
scientific interpretation. That is why scientific needs to be complemented
by artistic type appreciation.
However in an integral scientific interpretation it is necessary to
translate the nature of the dynamic interaction of affective and cognitive
(and associated personal and impersonal aspects of physical reality) indirectly
in an appropriate cognitive manner.
And holistic mathematical appreciation provides a precise means for
carrying out this indirect translation.
Dynamics of Type 2 Complementarity
Q So far you have commented on the reduced analytic manner in which
the relationship between whole and part is conventionally interpreted (using
in turn a reduced notion of the nature of multiplication and division).
Can you now explore the dynamics of the holistic integral appreciation
(corresponding to Type 2 complementarity)?
PC As appreciation of the appropriate dynamic nature of the relationship
between whole and part (and part and whole) is so important we will have
more to say about this in our next discussion.
However we can help to clarify the relationship at this stage.
Now the important point to remember is that - in any relative context
- the switch from part to whole (or - in reverse fashion - from whole to
part) involves the interaction of both quantitative and qualitative (and
qualitative and quantitative) aspects that cannot be directly interpreted
in terms of each other.
Put another way such interactions always entail the interaction of conscious
and unconscious through actual (finite) and potential (infinite) notions.
So what we are ultimately looking for here is a satisfactory holistic mathematical
way of interpreting the dynamics by which the conscious and unconscious
interact in experience.
We will do this in future discussions. However for the moment we will
attempt to carefully outline the nature of dynamics that are involved.
So let us start from the perspective of the recognition of a specific
perception - say a particular number - which has an actual finite identity
(as phenomenal object).
In the language of our holistic mathematical approach, this corresponds
to the horizontal level of interpretation as a quantity within a fixed
linear (i.e. one-dimensional) framework.
Now in moving from this perception (i.e. of an actual number) to the
corresponding recognition of its corresponding concept (i.e. as the general
class potentially applying to all numbers), we switch from quantitative
appreciation of a particular object to qualitative appreciation of a general
dimension, which intrinsically is of a potentially infinite (rather than
an actual finite) nature.
This implicitly requires in experience a corresponding movement from
linear understanding at the horizontal level (where poles are separated)
to circular understanding (where they are complementary).
In other words with respect to the number perception, the implicit fusion
of both the positive and negative aspects of understanding at the horizontal
level, (unconsciously) provides the spiritual intuition to enable the switch
to conceptual appreciation of number (with potentially infinite application).
Thus correctly speaking in dynamic terms, the relationship between any
perception and its corresponding concept (in this reference context) involves
the relationship of linear to circular understanding.
Finally, using our holistic mathematical terminology the switch from
a horizontal (quantitative) object interpretation to a (qualitative) dimensional
interpretation always entails the switch from linear to circular understanding.
Therefore - and this is crucial for subsequent interpretation - in the
appropriate holistic interpretation, the numbers representing both the
(horizontal) quantities and the (vertical) dimensions respectively, relate
to distinct systems of interpretation.
Q If I now attempt to summarise, what you are saying is that Type
2 complementarity always entails a dynamic relationship as between part
and whole (and whole and part) aspects, with understanding that is - relatively
- linear and circular (or alternatively circular and linear) with respect
to each aspect.
The deeper implication - I take it - is that all numbers can mathematically
be given a linear and a circular interpretation in both an analytic and
holistic fashion.
Now I eagerly await to see how the nature of this circular interpretation.
Presumably - given that you identify the integral understanding with the
circular aspect, the holistic interpretation of the circular number system
will then assume special significance as a scientific integral approach
to development!
PC This is quite true! We will demonstrate in a future discussion how
- precisely - this circular number system is derived. Furthermore we will
show how it contains all the holistic mathematical notions we have introduced
so far and a lot more besides.
Indeed quite simply the circular number system provides the appropriate
scientific system for a fully integral scientific TOE.
Q As I understand it, whereas the linear aspect of understanding
relates directly to the conscious aspect of understanding, the circular
aspect relates directly to the unconscious aspect. Therefore as conscious
and unconscious necessarily interact in the dynamics of experience all
phenomena - either as objects or dimensions - have both a conscious and
unconscious interpretation. Indeed we can see this in ordinary language
where a symbol may serve to give either an actual localised or alternatively
a potential universal meaning (as archetype). However though this may be
recognised in artistic type interpretation it plays no formal role in scientific
terms.
In other words science has not yet discovered a way to incorporate
the vitally important role of the unconscious in its formal interpretation
of reality.
Can you briefly describe this reductionism in the context of Type
2 interactions?
PC We must first recognise that the unconscious necessarily plays a
vital role in all scientific understanding. However this remains merely
implicit and is not recognised in formal interpretation, which is based
solely on the conscious aspect.
I will illustrate this with reference to the standard mathematical way
of understanding the relationship between a number perception and its corresponding
number concept.
Thus an actual number perception is posited and thereby differentiated
in linear terms (representing the conscious aspect of understanding). Then
- because in the dynamics of experience - some degree of negation must
necessarily occur, this implicitly leads to an unconscious fusion of opposites
(in circular terms) where the notion of form gives way to that of emptiness.
This emptiness in turn causes a transformation and a switch from the understanding
of number as a particular perception to the general concept of number (as
dimension) that potentially applies to all numbers.
However because in formal terms only the conscious aspect of number
is recognised, the potential (archetypal) concept (representing the unconscious
aspect of understanding) is quickly reduced in actual conscious terms.
So the interpretation of the dimensional concept of number is thereby reduced
as applying to all actual numbers.
Now of course we could interpret this in reverse fashion starting with
the actual (conscious) interpretation of the (dimensional) concept. The
reverse switch now requires an (unconscious) recognition of each specific
number as archetype (i.e. where the potential aspect is made immanent in
the specific perception). However again this will be quickly reduced in
merely actual (conscious) terms.
So inherent in all standard mathematical interpretation is a basic reduction
of infinite (potential) notions to finite (actual) terms.
Now we have already commented on the fact that numerical results in
mathematical terms are ultimately expressed in linear (i.e. one-dimensional)
terms.
This in turn reflects the fact that mathematical interpretation is formally
based on conscious linear understanding (where polar opposites are clearly
separated).
Q I see what you are getting at which makes great sense. It would
be hard for anyone - even a hard-core mathematician - to seriously deny
that the unconscious necessarily interacts with the conscious aspect in
understanding. Yet it is patently obvious - as you have shown - that the
unconscious is not at all recognised in formal interpretation. Therefore
it must also be true as you have demonstrated that conventional mathematical
understanding gives a distorted interpretation of reality (even within
its own clearly defined terms). Why is this problem not recognised?
PC Good question!
Firstly conventional mathematics has been amazingly successful in analytic
terms. For many - perhaps the vast majority - this seems a good enough
reason to continue with the recognised approach without addressing fundamental
philosophical problems.
Secondly - as I have stated before - the holistic integral approach
requires pure intuitive awareness (traditionally associated with mystical
development) with the specialised form of cognition (associated with mathematics).
This combination rarely goes together. Mathematicians - certainly as regards
their own specialised work - do not see a role for mystical type considerations.
Likewise, mystics traditionally have rarely shown a keen appetite for applying
spiritual intuitive awareness to mathematical type considerations.
However it is indeed a great pity!
Not alone does the appropriate combination of mystical intuition with
mathematical type cognition open up vast new possibilities for dynamic
holistic type understanding (Holistic Mathematics) that is appropriate
for a scientific integral appreciation of reality, but equally it opens
up significant new possibilities within the analytic approach to mathematics.
Correctly understood there is a massive "shadow " side to conventional
mathematical understanding that has not yet been properly explored. This
is due to the misleading reduced manner in which its key notions are interpreted.
Conclusion
Q Let us finally return to the notions of multiplication and division.
Can you briefly conclude this discussion by contrasting the (reduced) analytic
with the dynamic holistic interpretation?
PC In the analytic interpretation - based formally on solely conscious
recognition - wholes and parts can only be viewed together in either reduced
quantitative (or alternatively reduced qualitative) terms.
In other words the dynamic interaction of quantitative with qualitative
(and quantitative with qualitative) cannot be satisfactorily dealt with
as this requires both conscious and unconscious interpretation.
The (reduced) analytic approach leads to an asymmetrical notion of the
relationship between part and whole. So in any asymmetrical context the
whole - for example - is essentially viewed as the "bigger" quantity (which
includes its parts as "smaller" quantities).
Thus when we refer to our cake of four slices, in this context the cake
(as whole) represents the multiplication (by 4) of each of its constituent
slices (as parts).
In turn each slice (as part) represents the division of the cake (as
whole) into 4 constituent parts (i.e. ¼ of the cake).
So in this reduced analytic context, multiplication and division are
viewed in asymmetrical terms as the reciprocal of each other where the
fixed unitary dimension alternates between positive and negative. So 41
(i.e.
4 * 1) represents the multiplication of (part) slices as the (whole) cake.
In the holistic interpretation, multiplication always entails the interaction
of quantitative and qualitative aspects that are linear and circular (or
circular and linear) with respect to each other. Thus the interaction of
an actual (quantitative) perception with its potential (dimensional) concept
literally creates the potential to multiply all actual perceptions without
limit.
So for example with respect to number any actual number represents just
one (finite) example. However the general concept of number (as dimension)
provides the potential to multiply actual numbers without limit.
Indeed it is the application of the (potential) concept to an (actual)
number that creates the realisation that it is just one of many (actual)
numbers.
Likewise in reverse the application of an actual concept to a potential
perception (as for example in algebra where we let x be any number) creates
the reverse realisation that the actual concept (as set of all finite numbers)
can be divided into many constituent numbers.
So to briefly conclude, in analytic terms whole and part (and part and
whole) are asymmetrically related to each other in actual terms as "bigger"
to "smaller (and "smaller to bigger").
In holistic terms whole and part (and part and whole) are symmetrically
related to each other as actual to potential (and potential to actual). |
Pages
(AMS Bumper Sticker)
Sunday, April 26, 2015
Miniature Worlds
"Mathematicians are explorers of many miniature mathematical worlds. Explorers often find the objects or phenomena that they discover novel and surprising and they do not always describe them accurately. Indeed, just because they are novel and surprising, early explorers may mis-describe them, misunderstand them, and give most misleading reports. "It is only after much further study that the 'true nature' of the kangaroo, or the manatee, or carnivorous plants are determined. The same is true of mathematicians exploring their miniature worlds."
-- David Wells |
Equations that describe the natural world can convey profound truths while at the same time, to a trained eye, look absolutely beautiful. It is like learning to appreciate a work of art. Art may or may not be eternal. These poetic truths are. The equations show here are 1.Boltzmann equation 2. Euler Lagrange Equation 3. Dirac Equation 4. Euler Identity 5. Navier Stokes equation. |
Project your work and creativity in the world outside!
Very recently I stumbled upon an interesting and quite surprising fact. I was performing an analysis with some colleagues, and we noticed that an apparently simple combination of questions (all of the types "yes/no" or "choose from this short list") could lead to a stunning 10^19 number of possible outcomes. For those unfamiliar with scientific notation for numbers, 10^19 (ten to the nineteenth power) means the number written as 1 followed by nineteen zeroes: 10,000,000,000,000,000,000 (let's call this number Joe, to make it shorter and less frightening!). Now, this is not a number the human mind can normally fathom, even a "simple" 1.000.000 is too much to be grasped by our intuitive mind. Sure, we can do formal math with one million, we can use that number, we can put it into spread- and balance-sheets, but we cannot enumerate one million things at a glance, nor we can imagine ourselves performing such a high number of simple actions. It transcends from our "everyday brain" into the world of abstraction. Yet, numbers such as Joe are perfectly realistic, and they are hidden in many simple problems. Let's have a look at some of them: I'll make some reflections later.
Imagine you are in New York, or in another square-streets city (what a mathematician would call a "taxicab geometry"). You can follow a street for a short while, and then you end up in a four-streets crossing so that you have three possible roads in front of you among which to choose. How many crossings do you have to pass to have one million possible combinations of paths? It will take just 13 to make up for 3^133=1.594.323 possible combinations. For Joe, you are done with 40. Forty random turns in New York lead you to such a high number of paths: well, the Big Apple is the city of possibilities!
Another classical example is the "traveling salesman problem", or TSP in short. You are a salesman, and you have to pass through N cities (let's assume, for simplicity, that you can always go from any city to any of the others). What is the shortest route to achieve your goal? Unless you are able to spot some hidden symmetry in the disposition of cities, or they are put in a trivial manner (like in a straight row, or around a circle/regular polygon), you are doomed to compute all the possible routes, in a so-called "brute force" approach to the problem. They add up to the factorial of N, written in short N!, and computed as N*(N-1)*(N-2)*…*3*2*1. Guess what you get with 21 cities? You get five times Joe. This kind of problems are called "NP-problems", given the fact that they scale in a way which is Non-Polynomial: another way of saying "possibilities to take into account grow very fast"!
How many different pages setup can you use for your Word document? Let's try to calculate it together, concentrating just on the order o magnitude. First of all, take into account different fonts: on an average Windows system, there are about 100. Font size: going from 5 to 200, it makes another 200. Let's make 50 different indentations, 50 different spacings before and after words, four types of alignment, text in 1 to 10 columns, page vertical and landscape, ten million different font colors, 50 possibilities for horizontal margins and 50 for vertical margins. The result? 10^20 combinations, ten times more than Joe. Sure, you would never use some of the combinations very often, but nevertheless, they may have their use in the right context.
A fairly popular lottery in Italy goes by the name of Superenalotto. Six numbers are drawn among ninety, and if you are able to guess all six, you can get an enormous amount of money. However, what people don't realize in a lottery is that the prize is not always comparable to the odds of winning. In this case, you have one winning set of numbers among 90*89*88*87*86*85 total case, for a whopping total of, roughly, 4*10^11. We are not at the level of Joe, but the number is no less astounding.
How many musical scales do exist? Can we run out of "music"? According to Nicolas Slonimsky, author of the "Thesaurus of Scales and Melodic Patterns", there are 479,001,600 possible combinations of the 12 tones in the chromatic scales. Add different patterns, rhythmical variations, phrasing, sound, intention and different harmonies, and "there is no likelihood that new music will die of internal starvation in the next 1000 years", in the words of Slonimsky. Truly, music is well beyond our Joe!
The question that I pose to myself after these few examples is: how do we choose? How can we perform rational, sensible, informed choices? How is it even possible? In the movie "The Legend of 1900" the protagonist, a virtuoso pianist who never left the ocean liner where he was born, is at one point about to disembark, but retreats from the stairs. Afterward, he tells his friend Max the reason (you can read the whole quote at the end of this post): in short, in the world outside the ship, there are just too many possibilities for him to choose from.
Another related question is: how do we know something? How can we test our moral, scientific, medical, practical beliefs among those possibilities? In my field of work, IT projects, it is important to have someone test your software, but how can someone certify the result of 10^19 possibilities in any reasonable amount of time and without some automated method? This problem is, luckily, solved in scientific contexts, and experiments are devised to discriminate different possibilities and theories, but the other areas leave me puzzled.
I have no answers to report, I'm still reeling at the sheer size of the numbers. How many ways are there to put together the 1,507 words of this post? How did I select them? How many different reactions will the 1,000… 100… sorry, 10 readers of my posts have? I don't know, but I won't be stuck with this dilemma for a long time.
It is a world of possibilities: embrace them, don't be afraid, surprises are beyond every corner, and they are more likely to be enjoyable than not!
(Here is the complete quote from 1900) "All that city… You just couldn't see an end to it. The end! Please, could you show me where it ends? It was all very fine on that gangway and I was grand, too, in my overcoat. I cut quite a figure and I had no doubts about getting off. Guaranteed. That wasn't a problem. It wasn't what I saw that stopped me, Max. It was what I didn't see. Can you understand that? What I didn't see. In all that sprawling city, there was everything except an end. There was everything. But there wasn't an end. What I couldn't see was where all that came to an end. The end of the world. Take a piano. The keys begin, the keys end. You know there are 88 of them and no-one can tell you differently. They are not infinite, you are infinite. And on those 88 keys the music that you can make is infinite. I like that. That I can live by. But you get me up on that gangway and roll out a keyboard with millions of keys, and that's the truth, there's no end to them, that keyboard is infinite. But if that keyboard is infinite there's no music you can play. You're sitting on the wrong bench. That's God's piano. Christ, did you see the streets? There were thousands of them! How do you choose just one? One woman, one house, one piece of land to call your own, one landscape to look at, one way to die. All that world weighing down on you without you knowing where it ends. Aren't you scared of just breaking apart just thinking about it, the enormity of living in it? I was born on this ship. The world passed me by, but two thousand people at a time. And there were wishes here, but never more than could fit on a ship, between prow and stern. You played out your happiness on a piano that was not infinite. I learned to live that way. Land is a ship too big for me. It's a woman too beautiful. It's a voyage too long. Perfume too strong. It's music I don't know how to make. I can't get off this ship. At best, I can step off my life. After all, it's as though I never existed. You're the exception, Max. You're the only one who knows that I'm here. You're a minority. You'd better get used to it. Forgive me, my friend. But I'm not getting off." |
Math is the language of communication with any intelligent species able to communicate over galactic distances. However, what we are sending into space are TV signals, not math. That is because we are not attempting to communicate accept with ourselves |
Hilbert Curves
A Hilbert Curve is constructed through an iterative process that is repeatedly self-similar. You start with a simple, bent path around the inside of a square, and then you take each straight part of that path and bend it to make it look what you started with. And repeat. Ad infinitum.
What's especially interesting about Hilbert curves is that they essentially "fill up" the plane. This is seemingly paradoxical, in that you have a one-dimensional object (a path) that ends up equivalent to a two-dimensional object (a plane). For this reason, these are also referred to as space-filling curves. |
Closing the Gap: the quest to understand prime numbers - Vicky Neale
Prime numbers have intrigued, inspired and infuriated mathematicians for millennia and yet mathematicians' difficulty with answering simple questions about them reveals their depth and subtlety.
Vicky Neale describes recent progress towards proving the famous Twin Primes Conjecture and explains the very different ways in which these breakthroughs have been made - a solo mathematician working in isolation, a young mathematician displaying creativity at the start of a career, a large collaboration that reveals much about how mathematicians go about their work.
Vicky Neale is Whitehead Lecturer at the Mathematical Institute, University of Oxford and Supernumerary Fellow at Balliol College. Her new book "Closing the Gap: the quest to understand prime numbers" has recently been published b...
published: 24 Oct 2017
Closing Numbers
Anna and Keith have been happily married for several years. But things turn sour as Anna, investigating her suspicions that Keith has been having an affair,
published: 15 Apr 2017Strat
published: 15 Nov 2013SuperStar Drag Show (Full Show) (No Closing Number)How to move your spine - using opening and closing - flexion and extension2...
Closing and Approachespublished: 22 Nov 2017
Mql4 Lesson 31 Closing Trades with for loops ... beginnin...How to move your spine - using opening and closing - flexion and extension
How to move your spine features one more important aspect of spinal movement and correct muscular coordination. The intent is once again extremely important, as... possibleClosing and Approaches
In this week's TeamCall Leibert shared a number of good ideas for:
* Closing a business presentation
* Ways to use curiosity to arouse interest in your product...26:47
Strength in Numbers: Closing Achievement Gaps through Collaboration
Regional Educational Laboratory (REL) Midwest and its Urban District Community of Practice1:25:45
Ten Years After - 1968 Fillmore
Ten Years After
Fillmore Auditorium, San Francisco, CA
1968-06-28
Here's another of my al...
Ten Years After - 1968 Fillmore29:01
Closing Party of Number 7 at Summer of '13
We played for the closing of the Number Seven Bar at our region.
You can see two Stanton ...25:46
Dr Boyce Closing Remarks At All Black National Convention
The Dr Boyce Watkins Channel is an all-black news and commentary channel that features a n...SACRAMENTO (AP) _ The winning numbers in Tuesday evening's drawing of the California Lottery's "Daily 4" game were.. 4-8-4-0. (four, eight, four, zero). ¶ Ticket-holders with all four winning numbers in the order given win the top prize. Lesser amounts are also awarded to ticket-holders with other varying combinations of the winning numbers... ....
We are ruled and defined by numbers, whether it's our Social Securitynumber or the countless transaction numbers we are assigned as we go about our daily business ... such series of numbers into groups of three or four....
Alexis Sanchez is set to be handed Manchester United's iconic number seven shirt when he completes his expected to Old Trafford this month ... It was certainly the obvious and expected choice for a player who has worn number seven for Arsenal for the last 18 months after... .... |
Calculus Sentence Examples
LINK / CITEADD TO FLASH CARDS
But no carefully devised calculus can take the place of insight, observation and experience.
The well-known Treatise on Differential Equations appeared in 1859, and was followed, the next year, by a Treatise on the Calculus of Finite Differences, designed to serve as a sequel to the former work.
In 1747 he applied his new calculus to the problem of vibrating chords, the solution of which, as well as the theory of the oscillation of the air and the propagation of sound, had been given but incompletely by the geometricians who preceded him.
This discovery was followed by that of the calculus of partial differences, the first trials of which were published in his Reflexion sur la cause generale des vents (1747).
Reference may also be made to the special articles mentioned at the commencement of the present article, as well as to the articles on Differences, Calculus Of; Infinitesimal Calculus; Interpolation; Vector Analysis.
In the notation of the integral calculus, this area is equal to f x o udx; but the notation is inconvenient, since it implies a division into infinitesimal elements, which is not essential to the idea of an area.
While still an undergraduate he formed a league with John Herschel and Charles Babbage, to conduct the famous struggle of "d-ism versus dot-age," which ended in the introduction into Cambridge of the continental notation in the infinitesimal calculus to the exclusion of the fluxional notation of Sir Isaac Newton.
Lacroix's Differential Calculus in 1816.
Amongst the most important of his works not already mentioned may be named the following: - Mathematical Tracts (1826) on the Lunar Theory, Figure of the Earth, Precession and Nutation, and Calculus of Variations, to which, in the second edition of 1828, were added tracts on the Planetary Theory and the Undulatory Theory of Light; Experiments on Iron-built Ships, instituted for the purpose of discovering a correction for the deviation of the Compass produced by the Iron of the Ships (1839); On the Theoretical Explanation of an apparent new Polarity in Light (1840); Tides and Waves (1842).
He at once took a leading position in the mathematical teaching of the university, and published treatises on the Di f ferential calculus (in 1848) and the Infinitesimal calculus (4 vols., 1852-1860), which for long were the recognized textbooks there.
This latter work included the differential and integral calculus, the calculus of variations, the theory of attractions, and analytical mechanics.
For the subjects of this general heading see the articles ALGEBRA, UNIVERSAL; GROUPS, THEORY OF; INFINITESIMAL CALCULUS; NUMBER; QUATERNIONS; VECTOR ANALYSIS.
During this period logarithms were invented, trigonometry and algebra developed, analytical geometry invented, dynamics put upon a sound basis, and the period closed with the magnificent invention of (or at least the perfecting of) the differential calculus by Newton and Leibnitz and the discovery of gravitation.
Pp. 8 0 -94, 95112) showed by his calculus of hyper-determinants that an infinite series of such functions might be obtained systematically.
2 From the fundamental principle of virtual velocities, which thus acquired a new significance, Lagrange deduced, with the aid of the calculus of variations, the whole system of mechanical truths, by processes so elegant, lucid and harmonious as to constitute, in Sir William Hamilton's words, "a kind of scientific poem."
In expounding the principles of the differential calculus, he started, as it were, from the level of his pupils, and ascended with them by almost insensible gradations from elementary to abstruse conceptions.
The leading idea of this work was contained in a paper published in the Berlin Memoirs for 1772.5 Its object was the elimination of the, to some minds, unsatisfactory conception of the infinite from the metaphysics of the higher mathematics, and the substitution for the differential and integral calculus of an analogous method depending wholly on the serial development of algebraical functions.
In analytical invention, and mastery over the calculus, the Turin mathematician was admittedly unrivalled.
The calculus of variations is indissolubly associated with his name.
To Lagrange, perhaps more than to any other, the theory of differential equations is indebted for its position as a science, rather than a collection of ingenious artifices for the solution of particular problems. To the calculus of finite differences he contributed the beautiful formula of interpolation which bears his name; although substantially the same result seems to have been previously obtained by Euler.
In the figure of the earth, the theory of attractions, and the sciences of electricity and magnetism this powerful calculus occupies a prominent place.
A direct and an inverse calculus is thus created, the object of the former being to determine the coefficients from the generating function, of the latter to discover the generating function from the coefficients.
From A Merely Formal Point Of View, We Have In The Barycentric Calculus A Set Of " Special Symbols Of Quantity " Or " Extraordinaries " A, B, C, &C., Which Combine With Each Other By Means Of Operations And Which Obey The Ordinary Rules, And With Ordinary Algebraic Quantities By Operations X And =, Also According To The Ordinary Rules, Except That Division By An Extraordinary Is Not Used.
In the applications of the calculus the co-ordinates of a quaternion are usually assumed to be numerical; when they are complex, the quaternion is further distinguished by Hamilton as a biquaternion.
In the extensive calculus of the nth category, we have, first of all, n independent " units," el, e2, ...
All this is analogous to the corresponding formulae in the barycentric calculus and in quaternions; it remains to consider the multiplication of two or more extensive quantities The binary products of the units i are taken to satisfy the equalities e, 2 =o, i ej = - eeei; this reduces them to.
A characteristic feature of the calculus is that a meaning can be attached to a symbol of this kind by adopting a new rule, called that of regressive multiplication, as distinguished from the foregoing, which is progressive.
As in quaternions, so in the extensive calculus, there are numerous formulae of transformation which enable us to deal with extensive quantities without expressing them in terms of the primary units.
If, in the extensive calculus of the nth category, all the units (including i and the derived units E) are taken to be homologous instead of being distributed into species, we may regard it as a (2'-I)-tuple linear algebra, which, however, is not wholly associative.
For instance, there are the symbols A, D, E used in the calculus of finite differences; Aronhold's symbolical method in the calculus of invariants; and the like.
Buchheim, on Extensive Calculus and its Applications, Proc. L.
By applying the method of the differential calculus, we obtain cos i= { (µ 2 - 1)/(n24-2n)} as the required value; it may be readily shown either geometrically or analytically that this is a minimum.
By the methods of the differential calculus or geometrically, that the deviation increases with the refractive index, the angle of incidence remaining constant.
He made use of the same suppositions as Daniel Bernoulli, though his calculus was established in a very different manner.
This calculus was first applied to the motion of water by d'Alembert, and enabled both him and Euler to represent the theory of fluids in formulae restricted by no particular hypothesis.
In the notation of the calculus the relations become - dH/dp (0 const) = odv /do (p const) (4) dH/dv (0 const) =odp/do (v const) The negative sign is prefixed to dH/dp because absorption of heat +dH corresponds to diminution of pressure - dp. The utility of these relations results from the circumstance that the pressure and expansion co efficients are familiar and easily measured, whereas the latent heat of expansion is difficult to determine.
Substituting for H its value from (3), and employing the notation of the calculus, we obtain the relation S - s =0 (dp /do) (dv/do),.
Jacques Bernoulli cannot be strictly called an independent discoverer; but, from his extensive and successful application of the calculus and other mathematical methods, he is deserving of a place by the side of Newton and Leibnitz.
The same year he went to Geneva, where he gave instruction in the differential calculus to Nicolas Fatio de Duillier, and afterwards proceeded to Paris, where he enjoyed the society of N.
Among these were the exponential calculus, and the curve called by him the linea brachistochrona, or line of swiftest descent, which he was the first to determine, pointing out at the same time the relation which this curve bears to the path described by a ray of light passing through strata of variable density.
Meanwhile the study of mathematics was not neglected, as appears not only from his giving instruction in geometry to his younger brother Daniel, but from his writings on the differential, integral, and exponential calculus, and from his father considering him, at the age of twenty-one, worthy of receiving the torch of science from his own hands.
He contributed two memoirs to the Philosophical Transactions, one, "Logometria," which discusses the calculation of logarithms and certain applications of the infinitesimal calculus, the other, a "Description of the great fiery meteor seen on March 6th, 1716."
1875); Examples of Analytical Geometry of Three Dimensions (1858, 3rd ed., 1873); Mechanics (1867), History of the Mathematical Theory of Probability from the Time of Pascal to that of Lagrange (1865); Researches in the Calculus of Variations (1871); History of the Mathematical Theories of Attraction and Figure of the Earth from Newton to Laplace (1873); Elementary Treatise on Laplace's, Lame's and Bessel's Functions (1875); Natural Philosophy for Beginners (1877).
It is not, however, necessary that the notation of the calculus should be employed throughout.
The general method of constructing formulae of this kind involves the use of the integral calculus and of the calculus of finite differences.
Roberval was one of those mathematicians who, just before the invention of the infinitesimal calculus, occupied their attention with problems which are only soluble, or can be most easily solved, by some method involving limits or infinitesimals, and in the solution of which accordingly the calculus is always now employed.
At the age of nineteen he communicated to Leonhard Euler his idea of a general method of dealing with "isoperimetrical" problems, known later as the Calculus of Variations.
The calculus of variations lay undeveloped in Euler's mode of treating isoperimetrical problems. The fruitful method, again, of the variation of elements was introduced by Euler, but adopted and perfected by Lagrange, who first recognized its supreme importance to the analytical investigation of the planetary movements. |
Maths SL, Type 1 Portfolio - triangular numbers
Extracts from this document...
Introduction
Maths Practice Portfolio
Maths Portfolio Type I
Special numbers go back in history and there is a great relation between the theorists and the maths they discovered. They are numbers with unique properties, making them different to other ordinary numbers. 'The origins of the concept of the shape of number' is a topic which can be directly related to this fact. The idea behind this is that there are many origins of the concept of the shape of number. In the following task we were investigating patterns in geometric shapes that will lead to the formation of special numbers. More specifically, we will look at triangular patterns that will enable us to discover a pattern of special numbers. The first part of the investigation we looked at a triangular pattern formed with dots in the shape of triangles to and calculates the nth term for this pattern.
Original Sequence
Counting the number of dots in each of the triangles, we can see that there is a pattern. The numbers of dots increase by (n+1) adding 2, 3, 4 and 5. Therefore, this hints that the next three terms will be as we will be adding 6, 7, 8.
From the above table, we can see that our variables are n and Tn. When we try to classify this pattern, we can see that it is not arithmetic. Arithmetic sequences require a common difference (d). Meaning that when we subtract
from
we should get the same value each time
This continues so that the value in between each increases by 1. The common difference is not equal we do not have an arithmetic sequence. The sequence is also not geometric as it would have a common ratio (r). If we were to have a geometric series when we divide
the value of r will equal.
So, in our case, this particular triangular pattern sequence is not considered in the above two categories. This sequence can be categorized in the 'special sequences' which consists of different series such as triangular numbers, square numbers, cube numbers, Fibonacci numbers and more. One of the reasons why this sequence is called triangular numbers is because if the sequence is physically drawn, the shape of the diagram would be a triangle with dots equally spaced out, hence the name 'triangular numbers'.
This was tested for all the other 5 terms and was found to be correct.
Next, we tried a 5 stellar shape.
81
Substituting 5 into the formula to replace 6:
6n(n-1) +1
5n(n-1) + 1
When n = 1
3(0) +1 = 1
n = 2
5(2)(2-1) + 1 = 11
This was tested for all the other 5 terms and was found to be correct.
Therefore the general statement is pn(n-1) + 1
To arrive at this formula, we noticed that p was the stellar star shape and when n was put into the formula, we would be able to achieve the values required. However as this is a stellar star shape, we cannot calculate the series for a stellar star shape that is less than three. Therefore p>=3 in order for the formula to work.
In conclusion, we can say that geometric shapes lead to special numbers.
therefore we can conclude that Adam normally wins the practice games as he almost certainty gets more than half the points, which is more than 5 points. Part 2 Now I will look at Non extended play games where to win a game, the player must win with at least
This formula is y=6x2 - 6x + 1. The 6 in '6x2' and '6x' from the n-stellar number. Therefore, if it was a 7-stellar number, the equation should be y = 7x2 - 7x +1; which it is shown in the graph below.
171 243 Question 4 Find an expression for the 6-stellar number at stage S7 The expression which I used for the 6-stellar number at stage S7 is carried down from the . previous information determined that each term is 12n more than the previous term, in which n is equal to the term number of the previous term.
When the value of y is divided by the value of x, a negative number is divided by a negative number resulting to a positive number. Quadrant IV The value of y equals a negative number in the quadrant 4 and the value of r equals a positive number as mentioned beforehand. |
SCIENCE
Strength in Numbers
Established in 2007, the Alliance for Breakthrough between Mathematics and Sciences supports research activities in mathematical science that are motivated by social needs. Research Director Nishiura Yasumasa explains the "math" behind the program and highlights ways in which the projects undertaken might influence our daily lives.
Mathematics has always produced ways of looking at the world that are ahead of their times. From the Ptolemaic worldview of the harmony of the spheres that accompanied astrology, through to the Newtonian worldview, all the way down to the modern chaos/fractal view of nature, mathematics has brought about a series of paradigm shifts epitomized by the Copernican revolution. These shifts can be quite profound, because if we zoom in on something in the Newtonian depiction of reality, a linear approximation can always be found for any object, no matter how complicated, but from a fractal perspective there is always a miniature version of the whole no matter how far we zoom in, so that we are presented with a world where it is impossible to separate parts from the whole, or the self from the non-self (see figure).
However, the true greatness of mathematics lies in the way that it uses "invisible concepts" to describe "the visible world" (that is, in the way that mathematics approaches the problem of how to apprehend the universe and the natural world around us). A simple example of this is the concept of "infinity." In fact, the Newtonian representation of reality describes and predicts the visible, macroscopic motions of the planets using the invisible concept of "secondary derivative," which in turn is supported by the invisible concepts of "infinitesimal" and "infinite." Without mathematics it would never have been possible to represent leaves, coastlines or even whole landscapes using infinitely recursive fractals. However, it is difficult to perceive this infinite recursion directly. These endeavors are not driven by a desire simply to reduce the beauty of nature to a meaningless mathematical description, but rather by a desire to achieve a deeper understanding of its mystery.
Invisible Technology
However, since it is invisible, many of the fruits of mathematics are used in ways that are invisible from the standpoint of the sciences, which tend to regard mathematics as simply a "tool." In fact, mathematical logic is used in ways that are truly invisible as part of our everyday life, such as in mobile telephones, as well as in more obvious applications such as barcodes, encryption logic and non-invasive measurement techniques, such as fMRI. Calculating orbits to minimize the fuel consumption of space probes for interplanetary exploration is an interesting example of a practical application of the principles of dynamics, but this too has only just been taken up by the newspapers, as astronauts have been in the news. It is certainly not generally known that mathematics is hidden all around us, from GPS car navigation systems through to special effects in the movies. From the standpoint that the highest echelon of technology is to be "invisible technology," mathematics can surely be said to have well and truly secured that position.
However, mathematics' more intrinsic role does not reside in its aspect as a "tool" or "hidden supporter of technology" but rather in a completely different function. That function is to provide a point of view for identifying the essence of things that we have no idea how to understand, and a framework and a perspective for thinking about such things.
Human beings are living creatures and, as such, have a tendency to disregard things that are not important to themselves and things that have no meaning to themselves. "Looking but not seeing," or—this is the fate of all living creatures that have resulted from evolution—"seeing things as they should be" are things that we often experience in the form of optical illusions. There are many examples in history of things that no one has bothered paying attention to, or things that have been regarded until quite recently as "errors" (or as "incomprehensible" or "noise") suddenly being rescued from obscurity and becoming the focus of attention as soon as it is discovered that they contain an embedded mathematical object with a certain structure.
In terms of the internal development of science, it is certainly not a coincidence that the logical frameworks that allowed the development of quantum mechanics and relativity theory early in the twentieth century had been developing independently within mathematics. This kind of mutual exchange of perspectives for looking at the world expands even further, so that things that were once invisible start to become apparent. This is especially true for things that are difficult to imagine with our normal human perceptions. The fact that coming up with a good, universal "definition" takes a huge amount of time and effort by many bright people is indicative of just how difficult it is to depart from our intuitive depiction of reality.
A Non-Physical Vantage Point and Current Uncertainty
However, the importance of the perspective whereby the invisible becomes visible as a result of putting on our "mathematical glasses" is not always properly appreciated by researchers in the other sciences. This is because they tend to persist with an approach that starts with "things" where these "things" cause certain effects. For them, ideas that are removed from real "entities" such as "matter" or "life" are nothing more than empty theory, with no ability to explain reality. However, this "reality" is somewhat shifty. That the things that we perceive with our senses or measure with our equipment cannot be simply called "reality" or "entities" is abundantly clear from the history of the development of measurement techniques and their interpretations. The reality that blind people construct without a sense of sight, using only other senses such as touch and hearing, is different to the reality of an able-bodied person, but it still forms a rich world of its own. After all, we have developed our sensory nervous systems through an expedient process of evolution in order to survive on this planet. This is extraordinary, but it also has its limitations.
On the other hand, there are also "social entities," which are different from physical entities. Good examples are currency and interpersonal relations. However, it is difficult to see some things that have even larger impacts. Why is it that the world is so uncertain and unstable, where anything could happen? That is because the agents that impose restrictions on the way that we live our lives were, until modern times, either natural forces that are difficult to control, such as the sun, moon and natural disasters or visible things like monarchs and other rulers, but now this agency has moved to "social mechanisms," which are both invisible and ubiquitous.
These non-physical, abstract structures are where mathematics really comes into its own. In fact, perhaps there would be no hope of finding fundamental solutions to these kinds of problems without the involvement of mathematics. There are many areas—from global warming to economic fluctuations, psychology and risk management—where we have still only partially worked out the big picture and elucidated the mechanisms involved, but what all of these areas have in common is that there are a huge number of contributing factors and it is difficult to reach an understanding using a simple cause-and-effect framework (see table p. 30). However, these changes are not completely random either. It may be difficult to make precise predictions, but in some areas it has become possible to work out macroscopic trends within a certain margin of error. In the 2007 IPCC report on global warming, different researchers arrived at almost identical conclusions (within a certain range) despite the fact that they were using different mathematical models. This demonstrates how the language of mathematics is essential for overcoming our self-interest in order to stand on common ground.
Aiming for "Connective Knowledge"
If we know the direction in which we should proceed, then by working out the details we can expect a return proportional to the time and effort invested, and in fact research on these kinds of projects is endorsed just about everywhere. However, there are many problems that cannot be solved simply by working out details or by thorough enumeration. This mass of data is essential, but working out the story that it tells is another problem entirely. Perhaps only mathematics gives people the imagination and the powers of abstraction to overcome our limitations, by integrating, if only partially, science, which has become increasingly bloated even as it becomes ever more fragmented. The reservoir and perspectives of mathematics, which have been built up assiduously in an "invisible form," do not simply provide the sciences with a language to describe the world, they are also a treasure trove which holds the key to the kind of departures described above. When we can't see something no matter how hard we look, we need "glasses" that will enable us to see, and one source of such insights may come from the pursuit of mathematics.
The question is, who will bell the cat, namely, who can attach the label of "a mathematical vantage point" to these "complex structures with a massive amount of data?" This problem lies in the opposite direction to the fragmentation and enlargement of science that was described above, and is a life-and-death question for modern science, including mathematics. This is precisely why it is so urgent that we cultivate talented people who can act as an interface between mathematics and the sciences, with a foot in each camp. We need mathematicians and mathematical scientists who have both a strong intellectual foundation in a certain area of mathematics and also what Ivica Osim, the former coach of the Japanese national soccer team, called "polyvalence." In other words, it is important that we have a "team," a network of sophisticated mathematicians that can make full use of "developed knowledge" and "connective knowledge" in all directions. This does not conflict with the position of mathematicians until now. Rather, by utilizing them as important nodes in the network, both mathematics and the sciences can coexist in harmony. Mathematicians need to become more open, while also becoming aware of the diverse missions with which mathematics has been charged.
Mathematics as a Language of Mutual Understanding
In 2007, the Alliance for Breakthrough between Mathematics and Sciences ( was established by the Japan Science and Technology Agency (JST) as part of the Core Research for Evolutional Science and Technology (CREST) program. The first aim of this alliance is to demonstrate the potential of mathematics as a "connective knowledge" in society, by applying the last numerical technology and the voluminous mathematical assets that we have accumulated so far to some of the most difficult problems that we are facing today, as described above. The second aim is for mathematics to become the core component in the formation of a "language of mutual understanding" for all humanity. The Precursory Research for Embryonic Science and Technology (PRESTO) program grants for individual research projects by young researchers started in 2007, followed by the CREST program for team-based research in 2008. Twenty PRESTO research projects and three CREST projects are already underway. These projects address a number of areas in materials science, life sciences, environmental sciences, information and communications, transport, finance and medicine, with projects such as "Mathematical sciences collaborating with clinical medicine," "Unified analysis on various transportations and solution of their traffic congestion," and "Mathematical models of visual perception by means of wavelet frames" being selected for PRESTO grants and "A mathematical challenge to a new phase of material science," "Innovations in controlling hyper redundant and flexible systems inspired by biological locomotion" and "Harmony of Gröbner bases and the modern industrial society" being selected for CREST grants.
Hopefully, the young researchers participating in these programs will go on to become an interface between mathematics and the sciences. Please see the website above for more information about these research projects. In the following section I would like to discuss two ways in which the mathematical perspective and methodology behind these collaborative research projects can influence the way that we go about our daily lives.
1) An awareness of the scale of time and space We live for about seventy or eighty years, and in our day to day lives our ability to perceive spatial dimensions generally ranges from a few millimeters to a few hundred meters, but in these respects we are not necessarily better than other animals. And in terms of our senses of smell and touch, we do quite poorly. These extremely limited sensory perceptions give us many erroneous impressions, such as the impression that the earth and the people on it have always existed just as they do now, and that they will probably continue to remain the same into the future. However, we now know that initially the earth's atmosphere contained almost no oxygen, and that the current atmosphere was created in a slow process that included the emergence of photosynthetic bacteria, the formation of the ozone layer and the oceans, and the evolution of higher life forms. We also know that the atmosphere will continue to change in the future, and that the current problem of CO2 emissions is just one part in this greater process.
We humans find it difficult to imagine such extremely slow changes—even though it is just a blink of an eye when viewed in terms of geological time. We are also not very good at coming to grips with feedback mechanisms or extremely long chains of cause and effect.
In Japan there is a proverb that says, "If the wind blows the bucket makers prosper," a metaphor meaning that seemingly insignificant everyday events can have unexpected consequences.
Problems such as environmental problems, energy problems and the food-supply problem tend to be discussed from an economic perspective, focusing on the interdependencies between countries. However, all of these things, whether they be food or fossil fuels, have their origin in energy from the sun, and it is only because this energy is stored in a visible form that it becomes a target of contention. Yet at the same time, we remain unaware of the gift of energy from the sun that comes down to us in other, less visible forms. We need to think about these problems comprehensively, from the perspective of geothermal energy, ocean currents and the universe, and not just wind power or solar energy.
The fundamental principle when it comes to applying mathematical models to reality is to accurately grasp what comes in and out of the system. In the metaphor above, this involves first pinning down "the wind" and "the bucket makers" and then gradually clarifying the network that joins them. The mathematical structure that is derived as a result has the potential to be applied to a wide range of problems.
2) Mutual exchange between the micro and the macro Traffic jams come about as a result of drivers increasing their speed and closing the gap between them and the vehicle in front to satisfy their desire to arrive at their destination as soon as possible. In other words, traffic jams are caused by people. The solution provided by a mathematical model of traffic jams is "take your time and maintain a safe distance between you and the car in front." This result may take some people by surprise, but the mathematical model gives us a precise answer to the quantitative question of what speed and inter-vehicular distance produces the maximum flow of cars per hour. It is mathematics that tells us how "micro" information about the speed of one car and the distance to the next car is linked to the "macro" information about the overall flow of traffic. Many problems remain, such as the legal problems that have to be solved in order to implement this in practice and the question of how to exchange information between drivers, but basically "macro" information should feedback to drivers in order to give them an incentive to adjust their behavior. There is already some feedback when it comes to fuel consumption, but if drivers were given information about how their driving contributed to an increased traffic flow then it is likely that we would see less people cutting in recklessly. And if drivers observed these mathematical behavioral norms, this would result in a significant reduction in CO2 emissions.
The Age of Mathematics
As we have seen above, because mathematics focuses on the relationships between objects rather than depending on any particular object, it can be expressed in the same form regardless of what the object is. In that sense, we are living in an age where mathematics is rapidly becoming more important. At the same time as individual objects are being divided into their constituent elements and homogenized, changing relationships and exchanges of information are also becoming more rapid and more widespread. In these circumstances, it seems that those who can rapidly read the structure and dynamics of the network world, where connectedness is all important but the actual objects do not matter, and then make fast and accurate predictions will be the winners—Google being a typical example of this trend. However, such circumstances also have an extremely vulnerable aspect, namely that there is also a high risk of collapsing suddenly. In other words, we live in circumstances where what is known in the non-linear sciences as "cooperative phenomena" can occur easily. If this is the case, then all we can do is to cultivate a sensibility that does not overlook global information or signs, while creating and implementing mathematical behavioral norms and ethical norms that combine both the "micro" and the "macro" perspectives. Mathematical science teaches us that it is because these are cooperative phenomena that many small individual contributions become visible as a macroscopic result.
NISHIURA Yasumasa is a professor at the Research Institute for Electronics Science of Hokkaido University and research director of the JST Mathematics Program. |
SET Math The card game known as SET is deceptively simple. Invented in 1974 by population geneticist Marsha Jean Falco, the game has become a popular, even addictive pastime for both children and adults. It has also attracted mathematical attention.
Running Lanes and Extra Steps When going out to your local running track for a workout, you sometimes find that you are allowed to use only certain lanes for training. On a standard quadrant track, however, the outer lanes are longer than the inner lanes. That presents a problem for someone using the track for speed workouts.
Perfect Pyramids A group of tetrahedra that some people consider special consists of those that have integer edge lengths, face areas, and volumes. Such a solid is sometimes called a Heronian tetrahedron or a perfect pyramid.
Improving the Odds in RISK RISK is a classic board game of global conquest. First published in 1959, this war game remains popular and continues to attract mathematical attention. Recent analyses reveal that the chances of winning a battle are considerably more favorable for the attacker.
Alphamagic Squares Magic squares have fascinated people for thousands of years. They consist of a set of whole numbers arranged in a square so that the sum of the numbers is the same in each row, in each column, and along each diagonal. A twist on the concept, the alphamagic square, is interesting, too.
Prime-Time Cicadas The fact that periodical cicadas emerge after a prime number of years could be just a coincidence. Or it might reflect some sort of evolutionary pressure that leads to prime-number cycles.
A Dog, a Ball, and Calculus Mathematician Timothy J. Pennings of Hope College in Holland, Mich., posits that his dog, fetching a ball thrown into a lake, appears to compute the optimal path to his target in much the same way that a mathematician would using calculus. |
Friday, May 9, 2008
The Importance of Mathematics
Over at Math-Blog there's a post with a lecture from Timothy Gowers about the importance of mathematics, not simply in terms of practical application but also in terms of culture. It's fantastic and will give you an itch to start colouring nodes and play with prime numbers even if you don't think you're a math person.
Pick a time when the kids are gone or asleep and grab some chocolate, a cup of tea and listen. You'll enjoy it. |
The lecturer will describe a theory intended to "explain"
the random character of the digits of various fundamental constants.
The theory---developed jointly with the computationalist David Bailey---
has, at its very core, what is called "Hypothesis A." That hypothesis
connects dynamical iterative maps with the notions of normality
(to given integer bases), and in so doing also creates new connections
between areas such as: ergodic theory, pseudorandom generators,
and irrationality proof methods. |
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325the early beginnings of science goes all the way back to Fibonacci. Fibonacci was a great mathematician. He was one of the first mathematicians known to the american history. However Fibonacci was from a foreign country. There are rumors that speak of his sexuality, some say he was a gay mathematician who was devoted to his math skills to encourage his followers that no matter what your sexuality you can succeed. Fibonacci was a success, i think he would be very proud of the future we call our present-day.
Gays have gotten independence, and lesbians have gotten free of discrimination. Discrimination is like a plague in our society today. People cannot live without using it to hurt other people. Albert Einstein is probably the most popular man to science in our modern age.
He converted E = mc 2. Besides his intelligence in the math field he was kind of stupid in society. He could not know whether to bring an umbrella in to the rain or not. One day he painted his house the color of his jeans because he could not tell the texture of his jeans.
The most popular theory on science is that everything is based on math and numbers. This is one reason why we can make structures on the computer resemble real life. Modern believers believed that Fibonacci was a pimp in his day. They believed he pimped on guys and had his own exotic gay club. The alphabet was converted by science. This is an example of words that cannot mean nothing now but they will later...
as as as dg as as dg as g as g as as dg as dg as gas g as gha h a sh as dh sad h ash as dh as dh as has dh sad has h ash and h ash sad has h adf h s adh and has h sah sad hs dah s adh sah sa ...
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"Mathematical Knowledge and the Mathematical Community in the Morse Galaxy."
Abstract:
This presentation concerns changing needs and expectations in the way mathematics is pra
cticed and communicated. The time frame is mainly the early twentieth century to the pr
esent and the scope is all activity that can be considered to fall under the purview of
the mathematical community. Unavoidably, the idea of a mathematical community is confro
nted; what it means to claim ownership in this community and how knowledge management pr
actices effect the community. Finally, a description of an extendable mathematical tex
t-based database which can be used to manage user defined forms of mathematical knowledg
e is presented. |
Essay on mathematics in everyday life
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Iris Runge
A Life at the Crossroads of Mathematics, Science, and Industry
Authors:
Tobies, Renate
The book illuminates the beginnings of industrial mathematics and its international context, the history of the application of mathematical statistics, and the use of numerical and graphical methods in filament bulb and electron tube laboratories
Using many original sources, the book provides a comprehensive illustration of a woman scientist (the eldest daughter of the famous mathematician Carl Runge, and the sister-in-law of Richard Courant) who wrote her first publication as a student with the theoretical physicist Arnold Sommerfeld. Not only did she work enthusiastically in the field of applied mathematics, but she also developed and maintained counter-cultural social attitudes during periods of political unrest
The book shows how the international center of mathematics and natural sciences at Göttingen University was established by Felix Klein and how it established the foundation for using and developing new mathematical methods for the benefit of such fields as electrical engineering and physical chemistry
The book sheds new light on the history of the electrical industry – especially vacuum tube laboratories – and on the interdisciplinary collaboration between mathematicians, physicists, chemists and electrical engineers; on history of secondary and higher education in Germany; and on the process of emigration during the Nazi era
This book concerns the origins of mathematical problem solving at the internationally active Osram and Telefunken Corporations during the golden years of broadcasting and electron tube research. The woman scientist Iris Runge, who received an interdisciplinary education at the University of Göttingen, was long employed as the sole mathematical authority at these companies in Berlin. It will be shown how mathematical connections were made between statistics and quality control, and between physical-chemical models and the actual problems of mass production. The organization of industrial laboratories, the relationship between theoretical and experimental work, and the role of mathematicians in these settings will also be explained. By investigating the social, economic, and political conditions that unfolded from the time of the German Empire until the end of the Second World War, the book hopes to build a bridge between specialized fields – mathematics and engineering – and the general culture of a particular era. It hopes, furthermore, to build a bridge between the history of science and industry, on the one hand, and the fields of Gender and Women's Studies on the other. Finally, by examining the life and work of numerous industrial researchers, insight will be offered into the conditions that enabled a woman to achieve a prominent professional position during a time when women were typically excluded from the scientific workforce.
"The book is a very thoroughly researched book. … There are many notes in the book and luckily they are at the bottom of each page. They refer to a very long bibliography. … I can highly recommend this book. Iris Runge is an accomplished, interesting and sympathetic woman to get to know." (Else Høyrup, AWM Newsletter, Vol. 46 (3), May-June, 2016)
"This book deals with the life of Iris Runge … whose name is attached to the well-known Runga-Kutta [sic] technique for solving differential equations. … This book will definitely be of interest to mathematical historians as well as researchers of gender studies." (Michael De Villiers, The Mathematical Gazette, Vol. 98 (541), March, 2014)
"This book is devoted to a description of Iris Runge's career as an industrial mathematician, which was pioneering in several respects. … In this scholarly work, with its rich collection of footnotes, photographs and references, the author has taken great pains to document the various aspects of Iris Runge's life and career." (Martin Muldoon, Bulletin of the Canadian Society for History and Philosophy of Mathematics, Issue 50, May, 2012)
"This is the English version of the biography of Iris Runge (1888-1966), the daughter of the Göttingen pioneer of applied mathematics Carl Runge (1856-1927), published originally in German … . The book is thoroughly researched and contains an extensive bibliography of over 30 pages. It covers with an interesting biographical case study the hitherto almost unexplored topic of the history of industrial mathematics and can be highly recommended." (Reinhard Siegmund-Schultze, Zentralblatt MATH, Vol. 1236, 2012) |
The following, written by Paul Cohen, is an extract from the 1967 Stanford Quad (see
It is a widely held belief that mathematics is merely a tool to be used by the more applied sciences, and that research in mathematics is impossible. In reality, mathematics is undergoing a great period of expansion and development with perhaps the most spectacular work being done in the more abstract and pure branches of the subject. To mathematicians, mathematics often seems more of an art than a science. On the other hand, a complete divorce from physical science would be unwise, and it is reassuring that pure mathematics does find new and surprising applications.
The misconceptions which the average student brings to the Calculus course often causes him to see it merely as a set of rules for handling special problems. For its discoverers, Newton and Leibniz, however, the essential element of the calculus was a new point of view rather than special problems. When teaching undergraduates, the most challenging problem thus is to give the students a feeling for the power which is in the great mathematical discoveries. For the student specialising in mathematics, the problem is to bring then as quickly as possible to the frontier. This is done more through seminars than formal classes. These informal contacts allow the professor to help the student overcome the diffidence he feels before such a highly developed discipline. The professor must reveal the essentials of mathematics and supply the personal encouragement and direction which will enable the student to make a contribution of his own. |
Results tagged "Leonhard Euler" from PlanetGreen.org
The dictionary defines the trigonometric functions as ratios between the sides of a right triangle, and in this form they are most frequently taught and applied to physical problems. However, their significance in the natural world transcends this definition - the sine and cosine ratios are central to mathematical descriptions of harmonic motion and oscillations, from the period of a pendulum to the rhythm of a bird's wings.
In the eighteenth century, mathematician Leonhard Euler discovered an elegant formula that showed the sine function to be an infinite-degree polynomial. This realization began with the well-known fact that a polynomial of degree n always possesses n roots, either real or complex. Since the sinusoidal functions have an infinite number of both real and complex roots, it follows that they could be expressed as infinite products. Euler proved that for some constant A, sin(x) could be found using the equation
Interestingly, Euler proved the feasibility of this formula before he found the value of the constant A itself, but in the following months he found a solution. Since the sine of x divided by x itself approaches 1 as x approaches zero, he factored out x from this limit and was left with the true value of A:
Though this expression of sin(x) is seldom taught, it sheds a new perspective on the nature of trigonometric functions, and contributed to important breakthroughs such as the solution of the Basel problem in 1734. |
Paradoxes have played an important role in the development of mathematics,
as they brought upon clarification of basic concepts and introduction
of new approaches. If used properly, paradoxes can play a useful role
in the classroom as they provoke deeper thinking about the basic ideas
of the theory.
There is something irresistible about a paradox, which lingers long after
class is over. The first reaction usually is that of amusement mixed with
confusion. These feelings can, if not dealt properly, develop into a feeling
of insecurity when the conflict does not seem to be resolvable. But paradoxes
can serve as a leverage to fruitful discussions and deeper understanding.
Probability theory offers a large variety of paradoxes. Some of them are
(nowadays) interesting mainly from an historical point of view, as the
theory has already been adapted to resolve what seemed to be a paradox
at the time. Others actually hide in a very subtle and tricky way, common
misconceptions. Some are real "mind boggling" and they reflect
the not always intuitive nature of the probabilistic concepts.
In the talk several of those paradoxes will be introduced (and resolved!).
We will also discuss the way the paradoxes are incorporated into class. |
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Tag: mathMath is hard. That's why America has fallen so far behind in mathematics compared to other nations around the world. It's full of complicated algorithms and greek symbols, and I want no part of it.But once you see how the Japanese approach multiplication, you'll be blown away by how simple |
Monday, August 9, 2010
In 1974 Erno Rubik, an admirer of geometry and 3-D forms created the world's most perfect puzzle. More than 30 years later the Rubik's cube is still a best selling brainteaser. By the mid-eighties almost every child had the puzzle. Did I say "Child"? I certainly was playing with one when I was about 12 or 13 with little success in solving.
So you can imagine our astonishment when a whole first year pure mathematics class was asked to go out and purchase the puzzle from the university co-op bookshop. I think, like others, I had always suspected that there would a mathematical approach to solving the puzzle. After all, many real-world, albeit, obscure problems such as the four-color map(1) and The Seven Bridges of Konigsberg(2) inspired rigorous mathematical solutions. I, off course didn't have the mathematical repertoire – instead the brute force method, not in the mathematical sense, was always adopted.
Background
The Rubik's Cube has a straightforward basis. The faces of the cube are covered by nine stickers in six colours. The puzzle is solved, when each face is of one solid colour. When you start to rotate the rows and columns and see the different mix of colour rows/columns you begin to appreciate the difficulty involved. There is, in-fact, 43 million million possible pattern combinations - but just one right one.
it's been proven that you can solve a Rubik's cube in 26 moves. Computer science professor Gene Cooperman and graduate student Dan Kunkle of the Northeastern University in Boston used algebra and fast parallel computing to show that no matter how scrambled the cube is, it is possible to solve (generate a cube with solid colors on each face, or the home state) in 26 steps. There are however claims that it can be solved in 22 or even 20 steps (see later).
The Rubik's cube has served as a guinea pig for testing techniques to solve large-scale combinatorial problems. "The Rubik's cube has been a testing ground for problems of search and enumeration. Search and enumeration is a large research area encompassing many researchers working in different disciplines — from artificial intelligence to operations. The Rubik's cube allows researchers from different disciplines to compare their methods on a single, well-known problem.
Mathematically
There are many algorithms to solve scrambled Rubik's cubes. It is not known how many moves is the minimum required to solve any instance of the Rubik's cube, although the latest claims put this number at 22. This number is also known as the diameter of the Cayley graph of the Rubik's Cube group. An algorithm that solves a cube in the minimum number of moves is known as 'God's algorithm'. Most mathematicians think it may only takes 20 moves to solve any Rubik's cube – and it's a question of proving this to be true.
Any state of the cube corresponds to a permutation of its facelets — a rule that assigns to each facelet in the home state a new location on the cube in the scrambled state. Not every permutation of facelets can be achieved by a sequence of moves — for example no move can shift the centre facelet of a face — so when trying to solve Rubik's cube you restrict your attention to a collection of legal permutations.
These form a self-contained system. If you follow one legal permutation by another, the end result is also a legal permutation. If you've done one legal permutation, you can always find another one — namely the reverse of what you've just done — that will get you back to the position you started with. Doing absolutely nothing to the cube is also a legal move and corresponds to the permutation that leaves every facelet where it is, known to mathematicians as the identity permutation.
The problem of solving Rubik's cube can be visualised using what is called the group's Cayley graph — a network whose nodes are the legal states of the cube (corresponding to legal permutations) and which has two nodes linked up if you can get from one to the other by a legal move, which, incidentally, is itself a permutation. The home state of the cube corresponds to one of the nodes (and the identity permutation) and solving the cube corresponds to finding a path from one of the nodes to the home state along a sequence of linked-up nodes.
A brute-force approach to showing that you can always solve the cube in N moves would be to show that no such path involves more than N steps. This sounds good in theory, but there is a huge problem as the Cayley graph of the group of legal permutations has 43,252,003,274,489,856,000 nodes, a number that challenges even the fastest of supercomputers.
The (3 x 3 x 3) Rubik's Cube has eight corners and twelve edges. There are 8! (40,320) ways to arrange the corner cubes. Seven can be oriented independently, and the orientation of the eighth depends on the preceding seven, giving 37 (2,187) possibilities.
There are 12!/2 (239,500,800) ways to arrange the edges, since an odd permutation of the corners implies an odd permutation of the edges as well. Eleven edges can be flipped independently, with the flip of the twelfth depending on the preceding ones, giving 211 (2,048) possibilities.
There are exactly 43,252,003,274,489,856,000 permutations. The full number is 519,024,039,293,878,272,000 or 519 quintillion possible arrangements of the pieces that make up the Cube, but only one in twelve of these are actually solvable. This is because there is no sequence of moves that will swap a single pair of pieces or rotate a single corner or edge cube.
(1)This problem is sometimes also calledGuthrie's problemafter F. Guthrie, who first conjectured the theorem in 1852. The was then communicated to de Morgan and thence into the general community. In 1878, Cayley wrote the first paper on the conjecture.
(2) One of these areas is the topology of networks, first developed byLeonhard Eulerin 1735. His work in this field was inspired by the following problem: The Seven Bridges of Konigsberg.
Friday, August 6, 2010
The use of virtual reality technology in the assessment, study of, and possible assistance in the rehabilitation of memory deficits associated with patients with Alzheimer disease.
Background
Alzheimer's disease is the common cause of dementia, and is particularly common in older people. Because it is the most common cause of dementia, Alzheimer's disease is commonly equated with the general term dementia. However, there are many other causes of dementia. Alzheimer's disease is therefore a specific form of dementia having very specific microscopic brain abnormalities.
Alzheimer disease is not merely a problem of memory. Additional mental and behavioral problems often affect people who have the disease, and may influence quality of life, caregivers, and the need for institutionalization.
Depression for example affects 20–30% of people who have Alzheimer's, and about 20% have anxiety. Psychosis (often delusions of persecution) and agitation/aggression also often accompany this disease. Common symptoms and comorbidities include:
• Loss (usually gradual) of mental abilities such as thinking, remembering, and reasoning. It is not a disease, but a group of symptoms that may accompany some diseases or conditions affecting the brain.
• Deteriorated mental state due to a disease process and the result from many disorders of the nervous system.
Distinguishing Alzheimer's disease from other causes of dementia is not always as easy and straightforward as defining these terms. In practice, people and their disorders of behaviour, or behaviours of concern are far more complex than the simple definitions sometimes provided.
Establishing patient history, abilities, the natural course of disorder development such as that involving short-term memory, speech and language, personality, decision-making and judgment, and others is often needed in the diagnosis of the disease. Routine diagnostic steps therefore include a careful history, mental status screening, laboratory and imaging studies, and neuropsychologic testing.
Differential diagnosis of Alzheimer's disease
It is sometimes difficult to differentiate dementia caused by Alzheimer's from delirium and in addition several features distinguish dementia from depression, but the two can coexist and the distinction may be uncertain.
Whilst prominent motor signs such as Gait disturbance is a characteristic feature of patients with vascular dementia - In contrast, the NINCDS-ADRDA criteria for Alzheimer's disease state that: `gait disturbance at the onset or very early in the course of the illness' makes the diagnosis of probable Alzheimer's disease uncertain or unlikely. However, clinical studies suggest that gait disturbance is not restricted to the later stages of Alzheimer's disease. Also, studies have identified abnormalities of gait and balance in patients with early Alzheimer's disease(1).
It had been thought that Dementias without prominent motor signs included Alzheimer's disease, frontotemporal dementia, and Creutzfeld-Jakob, and others and the clinical pattern of gait disturbance in patients with early Alzheimer's disease has attracted less attention to date.
Diagnosis
Differential diagnosis between the types of dementia and treatments available for Alzheimer's - while limited in their effectiveness usually have best patient outcomes when begun early in the course of the disease. Diagnosis and/or diagnostic tools include:
Taking medical history.
Physical examination including evaluations of hearing and sight, as well as blood pressure and pulse readings, etc.
Standard laboratory tests including blood and urine tests designed to help eliminate other possible conditions.
Brain-imaging or structural brain scan such as CT or MRI to help rule out brain tumors or blood as the reason for symptoms and more recently
The use of virtual reality
The use of virtual reality technology in the assessment, study of, and possible assistance in the rehabilitation of memory deficits associated with patients with Alzheimer disease.
Using virtual reality to simulate real-word environments and test patient's ability to navigate these environments. Work has been carried out to compare previously described real-world navigation tests with a virtual reality version simulating the same navigational environment (2). Authors of this research work conclude that virtual navigation testing reveals deficits in aging and Alzheimer disease that are associated with potentially grave risks to patients and the community.
In another study in the United Kingdom (3), researchers' aimed to examine the feasibility of virtual reality technology for use by people with dementia (PWD). Data was obtained directly from six PWD regarding their experiences with a virtual environment of a large outdoor park. A user-centered method was developed to assess:
The extent to which PWD could perform four functional activities in the virtual enviroment was also investigated (e.g., mailing a letter). In addition, physical and psychological well-being of PWD while interacting with the virtual environment was assessed objectively by recording heart rate during the virtual reality sessions and subjectively with discrete questionnaire items and real-time prompts.
Thursday, August 5, 2010
Back in the mid eighties when completing a pure mathematics degree and using the then state of the art mini-computers, the PDP11 family of processors (we couldn't afford the services of the more, much more, brawny nitrogen-cooled Crays to solve sparse Hadamard matrices - I contributed an article in a UK computer journal discussing the future of computers, artificial intelligence, human interfaces and visualization – I guess you can call it a naive attempt to predict how systems will/MAY evolve. Of course this was through my own lens of experience. Watching processor speeds rapidly increasing and memory, disk and everything else growing almost exponentially.
Of course the implicit question was - If all that has happened in the first 50 years of computer history, what will happen in the next 50 or so years?
Moore's Law is an empirical formula describing the evolution of processors which is often cited to predict future progress in the field, as it's been proved quite accurate in the past: it states that the transistor count in an up-to-date processor will double each time every some period of time between 18 and 24 months, which roughly means that computational speed grows exponentially, doubling every 2 years. As processors become faster the science of computability, amongst other things describes a class called 'NP-hard problems' which are also sometimes referred to 'unacceptable', 'unsustainable' or 'binomially exploding' whose complexity and therefore computation grow exponentially with time.
An example of NP-hard algorithm is the one of finding the exit of a labyrinth: it doesn't require much effort if you only find one crossing, but it gets much more demanding in terms of resources when the crossings become so large that it becomes either impossible to compute because of limited resources, or computable, but requiring an unacceptable amount of time.
Many, if not all, of the Artificial Intelligence related algorithms are extremely demanding in terms of computational resources because they are either NP-hard or involve combinatorial calculus of growing complexity.
Not all developments in processing architecture stem from a single genesis. For example, recently, IBM researchers have made huge strides in mapping the architecture of the Macaque monkey brain. They have traced long-distance connections in the brain - the "interstate highways" which transmit information between distant areas of the brain. Their maps may help researchers grasp how and where the brain sends information better than ever before, and possibly develop processors that can keep up with our brain's immense computational power and navigate its complex architecture.
Artificial intelligence and cognitive modeling try to simulate some properties of neural networks. While similar in their techniques, the former has the aim of solving particular tasks, while the latter aims to build mathematical models of biological neural systems.
Another trajectory – that of Quantum Computers
The uncertainty principle is a key underpinning of quantum mechanics. A particle's position or its velocity can be measured but not both. Now, according to five physicists from Germany, Switzerland, and Canada, in a letter abstract published in Nature Physics(1) quantum computer memory could let us violate this principle
Paul Dirac who shared the 1933 Nobel Prize in physics with Erwin Schrödinger, "for the discovery of new productive forms of atomic theory" provided a concrete illustration of what the uncertainty principle means. He explained that one of the very, few ways to measure a particle's position is to hit it with a photon and then chart where the photon lands on a detector. That gives you the particle's position, yes, but it's also fundamentally changed its velocity, and the only way to learn that would consequently alter its position.
That's more or less been the status quo of quantum mechanics since Werner Heisenberg first published his theories in 1927, and no attempts to overturn it - including multiple by Albert Einstein himself - proved successful. But now the five physicists hope to succeed where Einstein failed. If they're successful, it will be because of something that wasn't even theorized until many years after Einstein's death: Quantum Computers.
Key to quantum computers are qubits, the individual units of quantum memory. A particle would need to be entangled with a quantum memory large enough to hold all its possible states and degrees of freedom. Then, the particle would be separated and one of its features measured. If, say, its position was measured, then the researcher would tell the keeper of the quantum memory to measure its velocity.
Because the uncertainty principle wouldn't extend from the particle to the memory, it wouldn't prevent the keeper from measuring this second figure, allowing for exact, or possibly, for obscure mathematical reasons, almost exact measurements of both figures.
It would take lots of qubits - far more than the dozen or so we've so far been able to generate at any one time - to entangle all that quantum information from a particle, and the task of entangling so many qubits together would be extremely fragile and tricky. Not impossibly tricky, but still way beyond what we can do now. |
Past Winner 2002 E.W.R. Steacie Memorial Fellowship
Henri Darmon
Mathematics
McGill University
Henri Darmon
What do theoretical mathematician Henri Darmon and online shoppers have in common?
They both appreciate the benefits of a special kind of algebraic equation called an elliptic curve. For the point-click-and-pay crowd these equations are the basis for secure online credit card transactions. For Dr. Darmon they are a portal into a realm of mathematical discovery.
The McGill University mathematician's work on elliptic curves has gained him recognition as one of the world's leading young number theorists.
It's work on mathematics' theoretical frontier for which Dr. Darmon is being awarded a 2002 Natural Sciences and Engineering Research Council (NSERC) E.W.R. Steacie Memorial Fellowship - one of Canada's premier science and engineering prizes.
Number theorists search for hidden patterns and relationships among numbers. Most basic are the whole numbers (1, 2, 3, ) that we learn as children. But number theorists also explore abstract quantities like i, the square root of -1, a number that is essential to equations that describe electricity and magnetism.
Darmon's mathematical tool of choice for finding interesting solutions to elliptic curve equations is known as complex multiplication theory.
Explored by the German mathematician Kurt Heegner in the 1950's, this tool was put on a rigorous mathematical foundation - by Princeton University mathematician Andrew Wiles - as part of his famous 1994 solution to the 350-year-old number theory riddle known as Fermat's Last Theorem.
"Elliptic curves are endowed with an extremely rich structure, which accounts for their central role in number theory," explains Dr. Darmon, who wrote one of the leading expositions of Wiles' famous proof.
What makes elliptic curves so powerful in practical and theoretical applications is what happens when you draw a line through two points on the curve, says Dr. Darmon. The line intersects the curve at a single, third point, so that new solutions to the corresponding equation can be generated from previously known ones.
The ongoing importance of elliptic curves to mathematics is highlighted by the fact that the Clay Mathematics Institute offers a million dollar prize to anyone who can prove what is known as the Birch and Swinnerton-Dyer conjecture. It posits that there should be a systematic mathematical recipe (an algorithm) for finding all the rational solutions to an elliptic curve equation.
While he's not expecting a cheque in the mail anytime soon, Dr. Darmon's research has revealed a tantalising new method for finding solutions to elliptic curve equations.
His most recent work - soon to be published in the top mathematics journal, The Annals of Mathematics - suggests that complex multiplication theory is only a part of a more general pattern. It's the first broad advance in the problem of solving elliptic curve equations since the approach of Heegner.
"My identities have been verified numerically, using a computer, in a few instances to a large degree of accuracy, so that they are true beyond a reasonable doubt, but we still seem to be very far from a proof," says Dr. Darmon. "To me this situation is profoundly exciting, because somewhere out there is a theory that would explain my empirical observations, and this theory has yet to be discovered. Mathematics thrives on such mysteries." |
Harristotelian Logic
A blog with texts originally posted at several newsgroups by James Harris (who is not the author of this blog).
Wednesday, February 27, 2008
JSH: Bet it all, lose it all
One of my heroes is Sir Isaac Newton who it turns out was not exactly a nice guy. Later in life he had among other things the job of protecting the currency of Britain, so he could send criminals to be executed. He did his job.
Mathematics to me is about absolutes. So I can reach a point where I tend to think in absolutes, and after five years of facing a math society that clearly has lied repeatedly and has behaved as if it could not be caught, I've lost any interest in concerns about not acting from absolutes.
Modern mathematicians pushed the idea that proofs could be delicate things and talked of failed proofs. They claimed proofs were not discovered but were creations, and that whether an argument was a proof or not was about whether mathematicians thought it was a proof or not.
My take on the field is that it has been overtaken by fiction writers. People who see themselves as authors of "proofs" which are really entertainment for others like them as no one else can even comprehend this stuff.
So style is the most important thing for math undergrads to learn in this system.
Style.
I lost count of how many times people told me my mathematical arguments did not look like math proofs.
But I say proof is discovery, and so it can be like prospecting, hunting for gold treasure. Treasure seekers don't worry about the dressing, they worry about the goods. After all, they're rooting in dirt or streams. It's not a pretty process.
Fiction writers took over the math field and fiction writing is about conflict, and contradiction, or apparent contradiction can be part of conflict and good fiction, so math society believes in "logical contradiction" along with those "delicate proofs" that can be wrong.
But I am a discoverer. I search for mathematical proofs the way you go for gold or diamond hunting. And I don't appreciate a style society of fiction writers pretending to be mathematicians telling me my finds are not what I can prove they are.
You people of course have bet your careers on me not being able to convince anyone else, which I say is, fine. You want to bet, then fine, but you need to know that is what you are doing.
I am a no-nonsense person on these issues.
And I have no compunction with presiding over shutting down entire mathematical departments where I've said that I would definitely put the Princeton math department at the top of that list of departments that should just be shutdown.
If I am wrong, then you just have more ranting from someone many of you are quite willing to call a madman, but if I am right about my finds then you need to accept what will happen when I get past your fiction writing society, past all the blocks you've thrown up in what is increasingly clear is a conspiracy to commit fraud--and inform the world.
So they know you faked math discoveries for years and to prove you knew you were faking you blocked acceptance of my research for years and even kept up the game with the factoring problem which I turned to because I knew you couldn't successfully block a major research find in that area.
So you have bet it all.
Fine.
You people ultimately don't understand what mathematics is, or what mathematical proof is, or you would not have done it.
And that finally is my most potent argument explaining why there is no choice for the world—real mathematicians could not have failed to understand when it was over as a mathematical proof said it was over.
Therefore, you are not at all real mathematicians.
I have the theory. So it's not a question mathematically of whether or not it will work. It's just a matter of the implementation that the theory says must be there actually being presented, and then the entire sorry tale will be the talk of the entire world.
Mathematicians around the world fakes!!!—the headlines may read.
And you will have lost it all on your bets in what will turn out to be a much better story than any of you ever wrote in your fake "proofs", though its grandeur will be a lot about the heaviness of your fall.
JSH: Stepping back
I've been pushing myself as hard as I can go to get to a practical implementation and I'm facing that I can't get it done tonight, but I'm hoping to be finished within the next couple of weeks with a working solution to the factoring problem fully programmed.
But I want to keep raising the stakes, but I think I shouldn't so I'm stepping back.
Problem solving is about finding what's necessary to get the solution and I think I have it now where getting to the answer was more than just figuring out the math, it was also about facing a steady stream of negativity, character assassination and questions about my sanity from people fighting their own battle to prevent the knowledge from being found.
And what knowledge.
The factoring problem can be attacked through what I call surrogate factoring by leveraging one factorization against another.
With
z^2 = y^2 + nT
where T is the target to be factored, z itself can be approximated, surprisingly enough by looking at a maximum value for a variable I call k, for which
abs(nT - (α^2+1)k^2)
is a minimum where '=E1' is yet another variable, which is chosen such that
k^2 = (α^2+1)^{-1}(nT) mod p
exists, where p is an odd prime of your choice.
It is preferred that z have 3 as a factor as in general it can be shown to have
(2=α^2+1)
as a factor, so for most values of 'α'—2 out of 3—it will at least have 3 as a factor, and who knew that factoring had all these relationships available?
But that's what can make mathematics exciting!!!
Interested readers can figure out the derivations on their own.
Key is letting 2αx = k + pr_2, and z = x + αk, and considering what happens if you move k about with k = k_0 + jp, and substituting into
z^2 = y^2 + nT
and then you can re-derive everything I have shown here. Easily.Seemingly complex it is the result of just doing the substitution and simplifying a bit with the given relations.
It shows that if you move k around with j, you will have a minimum absolute value for
nT - (1 + α^2)(k_0^2 + 2pjk_0 + p^2*j^2)
because x, y and nT are constant, as is α and k_0, so because the j^2 term will dominate r_2 will tend to be negative to compensate, and that allows you to get an idea of where k_0 is.
And if j=0, then z = (1+2α^2)k/(2α), so you have it explicitly.
The mathematics is what is commonly called elementary methods.
And I like simple.
I think about researchers around the world claiming to be working on the factoring problem who will say nothing about this research.
Weaklings.
Fakes.
That's me stepping back a bit. But I don't want these people staying in positions that they clearly are not filling later. So the warning to them is, yes, I want this research acknowledged before I fully force the situation, but when I do force it, if that is necessary, then the next problem I will solve will be making sure that none of them remain in the field as mathematicians or cryptographers, so they need to start thinking about what they will be doing for work, in the aftermath.
Or, show some goddamn sense, and just acknowledge the research now.
[A reply to someone who asked James whether or not he was thinking about doing some violent action.]
I'll admit that I'm still wary of the impact of a sudden change precipitated by me just factoring some really large number so I'd prefer buy-in from the cryptographic industry ahead of that event.
But make no mistake, if this impasse ends with me, say, factoring a large enough number to show this research must be viable, WITH the growing history now of an improper response from the cryptographic community then the likely impact will be a sharp loss of confidence in that industry.
But that industry would next be tasked with resolving the issue along with the high tech community so that secure transmissions could continue.
So the first major step would be just figuring out which of you are in on this cover-up or not, and even if you're not part of this particular cover-up, do you actually have real mathematical skills or are you a fake?
That could mean a snarl on looking for solutions, so yes, my putting out the theory now and talking about the research in-depth is methodical AND important.
Posters challenging me to just factor an RSA number are pushing the snarl, when if the research is viable the theory would show lots of indications that I could do so with plenty of time for proper industry action.
Given what I know about the current corruption in the mathematical community I'm not surprised by the behavior, but I'm still hopeful that there are some members of the cryptographic community who are legit.
Otherwise the snarl is what we will see down the line with the world—and I'm sure all major world leaders—facing the big issue of trying to figure out what to do when RSA encryption goes away, probably literally overnight, as it's potentially broken now by others, but we don't know that so it still works if only out of faith in it.
Tuesday, February 26, 2008
JSH: In the neighborhood
Oddly enough to me the most fascinating find from surrogate factoring which has created the means to end the impasse is a remarkably simple result that follows from a relatively simple equation:That is the equation that comes from letting 2αx = k + pr_2, and z = x + αk, when
z^2 = y^2 + nT
and considering k = k_0 + pj, to see how
nT - (1 + α^2)(k_0^2 + 2pjk_0 + p^2*j^2)
behaves as you increment or decrement k with j.
So actually I just kind of expanded out the traditional difference of squares.
Um, that's what they call thinking out of the box.
And you have trivially that as j increments OR decrements, r_2 will tend to be negative to compensate, so
nT - (1 + α^2)(k_0^2 + 2pjk_0 + p^2*j^2)
will have a minima and change around that value as j increases which is just an incredibly powerful result as it allows you to to get an idea of the value of z.
So the approach to the factoring problem is really tackling finding how to get z, when
z^2 = y^2 + nT
and all those variables are just helpers in that task.
You're just trying to get in the neighborhood.
And it can be shown that if x, y and z are rational
k^2 = (1 + α^2)^{-1}(nT) mod p
so you can go looking for z by looking for k, where you can get k's residue modulo p, an odd prime.
Qualifications are few. Yes, for a given choice of p, x, y and z rational may not exist such that all equations are satisfied but you can try different primes. Um, there are, after all, a LOT of primes.
So you have prime numbers as helpers that disappear after helping you to factor, and you have a surprisingly simple result with a parabolic minima, and you get quadratic residues, and it's the factoring problem and I've been talking about this latest research for days, and still math society waits…
Um, could REAL mathematicians wait?
Would Gauss or Euler or Fermat? Archimedes?
My place in history is secure even though I know a lot more than most of you clearly know so I also know that there may not be much history left! Not human history at least.
But not understanding is what this situation is all about, as some people didn't understand that lying about math would invite the retribution of the math because they didn't believe in mathematics itself.
The poor field was overrun by people who hate math but found a way to work the system by lying. That's all. Nothing more.
But without advancement in mathematics, humanity has no future, so the Universe will just kill off the species as no longer of further use.
By stopping mathematical progress, these people removed a key element in the purpose of the continued existence of the entire species so the clock is ticking down faster than any of you can imagine because you're too dumb to realize that if YOU lied and got things wrong, why couldn't others have?
Guess at how many years are left, and I'm sure you'll be wrong.
Yup. The test of humanity was a subtle one but it was very fair.
It was all about mathematical absolutes.
[A reply to someone who described what James had written as "meaningless garbage".]
Actually it is easily derived.
Let z^2 = y^2 + nT, where T is the target to be factored.
Further let 2αx = k + p*r_2, where I use r_2 for historical reasons since the full theory also has an r_1, and where p is an odd prime of your choice.
Then you let z = x+αk, and substitute, and finally you let
k= k_0 + pj
where j is an integer.
As I explained in my initial post you get a remarkable result that k_0 will be near the maximum k such that abs(nT - (1+α^2)k^2) is a minimum, which you can prove rigorously withso the circle is complete.
It IS a simple result that has profound consequences, but we live in a complex world, so the debate continues as I face people who have learned to just fight for one more day.
Their strategy is always just to fight for one more day, fooling the world, and each day they keep people from the truth is a victory for them.
I am just one person fighting against a society around the world that is firmly entrenched that has betrayed the public trust.
At this point smarter people can exploit the mathematics but unfortunately I am sure that there are people who will try to hide exploits.
So yes, if say, a bank gets invaded by hackers who are breaking RSA at will, I fear that will be hidden. If you lose all your money as a result they will tell you it's your fault and you will not get a penny back.
If you protest you will be ignored.
And then you will understand how powerful they truly are.
I wish I knew a better way. Some way to save innocents from the fall-out.
But with a betrayal of trust on this scale I am at a loss for a better answer.
They will fail with a big collapse I fear, when they can no longer hide the security breaches. And can no longer explain away the collapses in security.
Florida in the United States lost power today. Is it yet another demonstration that will be explained away by powerful people fighting to keep their control? Or are the official explanations given correct?
I don't know.
My main task is to preserve civilization. IN order to do so I am empowered to sacrifice whatever needs to be lost. There are probably already lost.
Unheard. Unappreciated, except by me. I will honor their memory even if no one else understands.
Monday, February 25, 2008
JSH: Here comes alpha
Given a target composite T that you wish to factor, it can be shown that if you have
z^2 = y^2 + nT
where n is a non-zero integer, then there exists an integer k such that
k = 2a*z/(1+2a^2)
where 'a' is alpha, though I just use 'a' for text postings, and it is a non-zero integer.
Further z = x+ak, and 2ax = k.
And finally,
k^2 = (1+a^2)^{-1}(nT) mod p
where p is an odd prime.
Also k will be near the maximum value of k such that
abs(nT - (1+a^2)k^2)
is a MINIMUM, which is the powerful bit of mathematics which makes this very likely to be a solution to the factoring problem. That is the most crucial finding.
So if you've noticed me posting a lot on this subject you may have seen postings where I said z should be divisible by 3, that is because if alpha is coprime to 3, then 1+2a^2 is divisible by 3, so z must be, if that value of alpha works.
And it is about finding an alpha value that works as some value WILL work, if you have non-zero integers z and y such that z^2 = y^2 + nT.
So the first and most likely factor of z is 3, the next is 9, and the next is 19 when a=3, so yes, you can have z coprime to 3, followed by 33 when a=4, and 51 when a=5, and 73, when a=6.
Those must all be factors of z, where again
z^2 = y^2 + nT.
So alpha=1 is the most likely, and then you have other possible values for alpha since z is set for each non-trivial factorization so the question is finding alpha and k.
Finding k is about looking near the maximum value of k such that
abs(nT - (1+a^2)k^2)
is a minimum, and the most likely alpha is 1, but it may be others.
And you use
k^2 = (1+a^2)^{-1}(nT) mod p
where p is an odd prime of your choice, to get the residue of k modulo p, where you pick an odd prime and go looking.
Most likely for your prime, alpha=1, but it can equal the other values though the probability is less.
Those forced factors of z, again are
3, when a=1, 9, when a=2, and the next is 19 when a=3, followed by 33 when a=4, and 51 when a=5, and 73, when a=6.
So, for instance, if for your picked prime a=5 works, then z does not have to be divisible by 3, and it will have 73 as a factor as
k = 2a*z/(1+2a^2).
I went to the factoring problem to end an impasse where modern mathematicians around the world as it has taken a good bit of effort by these people from what I've seen, have been successfully blocking knowledge of major mathematical finds that overturn results that have a lot to do with their careers.
And they're blocking knowledge of certain key things, like that Andrew Wiles did not prove Fermat's Last Theorem, as the research that I have that does, also shows that a crucial error slipped into the mathematical field around the time of Dedekind—over a hundred years ago.
Those of you who know a bit about math know that an error can allow people to make "proofs" which are in fact, not mathematical proofs as mathematics does not tolerate error, so this error allows these people to create an ever growing body of useless and wrong research, indefinitely.
But they have to stop real researchers, so they have turned to various tactics as they claim they are keeping the field "pure", including interestingly enough casting doubt on mathematical proof itself so that people talk of delicate proofs.
As this answer to the factoring problem shows you, real mathematical proofs are not delicate.
These people are con artists who got into an area where they could lie with apparent impunity until they made a mistake, which was the introduction of a technique for information security based on factoring supposedly being a hard problem.
They have nowhere to go from here so do not expect them to tell the truth.
For them, now it's just about waiting until the world catches up, and they face the consequences of their actions.
JSH: All on my math blog
Now I've been arguing still with posters who are playing the same games that worked to block the world from knowing that I found a proof of Fermat's Last Theorem--over five years ago.
And even a freaking math journal dying did not make a difference.
So I turned to the factoring problem.
Some of you are parasites. Like parasites in nature it is not in your nature to consider your own life when it comes to behaving as you are genetically or environmentally programmed.
You are human parasites so you behave as parasites and parasites can kill their host.
Here the host is human civilization as we currently know it.
Now the parasites would not be parasites if they were brilliant.
They established themselves in academic areas as the idea of "pure research" allowed them to get away with doing nothing of value if they gained critical mass so there is no telling how much of what we think of as valid knowledge is just parasitic waste product.
They just say that the research of their own is correct. Easy for them.
They needed to "publish or perish" to keep up their parasitic activity.
Ok, so with the fate of the human race in the balance my job was to stop parasites who could block acceptance of a proof of Fermat's Last Theorem, of my prime counting function, of my definition of mathematical proof, of my basic logic axioms and just about anything else I came up with including my open source project and I think I even gave Google a business plan for YouTube and even that didn't matter.
So the factoring problem was it.
After arguing with myself for years about the ethical issues, while searching for the answer I concluded, especially as global warming grows as a problem, that the parasites were capable of driving the human race to extinction not because they want to, or because they're completely evil, but because they're stupid.
They are parasitic.
Do you think some math professor doing fake math really thinks he could end civilization as we know it?
No. No, but if you lose your family, or you are months away from a place where starvation is something you are facing as a reality when you're in a vibrant London today, or a healthy Amsterdam today when in a few months it can be like something out of a horror movie, the entire city, then once you are there, there is no going back, and there is no thinking to yourself that you should have listened.
Death is the greatest reality check of them all.
I can assure you of one thing, if what I call the Math Wars continue for much longer then many of you will learn that yes, there are such things as parasitic humans who have almost completely infested academia by your resignation as you face your own death, probably by starvation, though it may be by disease or lack of clean water.
We are so close to the brink here. And I would warn those that would exploit the solution of the factoring problem that they need to take stock and consider that maybe it's not such a good thing to steal a lot of money if their actions can just end everything as we know it.
I'm not sure how long parasitic math people can hold the line on this one.
But they sure are trying as hard as they worked to block my previous research.
So your life may be one of the sacrifices for the future of humanity.
But I had to make the calculation. And I decided that our entire species could be made extinct if the threat were not faced head on with force-on-force.
So I decided and here we are. You may die as a result, but consider if you lived so that everyone would die later in a world where parasites had simply removed the ability for people to solve problems for real, versus fantasy.
Even if it lasted hundreds of years before humanity finally just died out, everything before would have been for nothing. Our entire history, for nothing because at the end we couldn't handle the enemy from within.
So the real war has finally started. And if you die as a result, you have my apologies and my sympathy, but better you than everyone.
Saturday, February 23, 2008
JSH: How they do it
To me it's still rather neat to find this surprisingly useful relationship from considering
abs(nT - 2k^2)
and the maximum k for which that value is a minimum, when T is an odd composite to factor, and n is just there to force nT mod 3 = 2, so you can use 5, when T mod 3 = 1, but 1 otherwise.
Then that k is at or near a value such k is a solution when z = 3k/2, and
z^2 = y^2 + nT
and you have a non-trivial factorization of nT.
Now that follows from doing something out of the box as I deliberately looked at
2x = k mod p
for quite some time just so that I could complete a square, as I was speculating about factoring T by using the factorization of some other number.
So getting the full result is just about using
2x = k + pr
where r is just some integer and substituting with z = x + k, into
z^2 = y^2 + nT
and doing some basic algebraic analysis and I explain all of that to emphasize the simplicity of the approach, and many of you can do it yourself to verify or read my other recent posts to see more detail.
And it's a fascinating result for many reasons, but one of them is that the p kind of vanishes after helping you factor by helping you find the correct k.
I can also show that k^2 = 2^{-1}(nT) mod p, so you can just go looking for large primes—as the bigger the prime the quicker you can factor—when you have a large composite T, so you actually get to just go find some primes and know about where to look and can factor.
So that should be a big deal as a nice bit of interesting mathematics that could have impact on the use of RSA encryption and it's just a neat result, so why am I still the "crackpot" on newsgroups instead of celebrated in the mainstream?
Well I'm looking at posts in reply to me and not seeing much new which is why I wanted to explain to you how they do it: how they keep the truth from being mainstreamed.
I have other mathematical results and watched the same process with them.
That process is to deny, mostly ignore, or claim ignorance of the result or of the behavior around the result.
So some posters will keeps posting nastily in reply and just claim I'm wrong, while others will post puzzlement that I'm even posting! While others will post puzzlement that anything I have ever said was ever ignored as they claim none of it was and that in fact I get plenty of attention so why am I complaining.
But they'll ignore any current result. If pressed they will simply quit replying.
Years ago when I first faced this process, yes, I went off Usenet, and got a paper published. I'm sitting back waiting for math society to acknowledge that remarkable result—but monitoring Usenet just to see what happened—and some poster posts that I'm published on the sci.math newsgroup and the group erupted in fury.
Never paused for a moment to consider that maybe publication meant something.
There is what people say they believe and there is what they demonstrate they truly believe. Publication. Hmmm…
Now you can see how they react with the factoring problem.
Yes, I've said for some time that I had solved it only to be wrong, but I know this process better than you do, and my strategy is based on game theory.
And you can see what I'm up against if you understand that simple approach I outlined before about using abs(nT - 2k^2) and understand it, as well as the implications, and see how math society is STILL to this point reacting.
Lying is as old as humanity. People lie to get things they would not get otherwise.
Some of you may be going to math classes to listen to professors who would not be there if they told the truth. It's that simple. It can be easy to talk about the fate of the human race and the importance of knowledge and progress in the abstract, but if you're some middle-aged man with a mortgage, a wife who thinks you're brilliant, and other perks that may come with being a math professor (yes there are some perks for some maybe not many but some) considering accepting not being so brilliant and losing what you have, then lying can seem to be just about survival.
Perspective is an amazing thing.
Some of these people may figure that humanity is such a big thing that there's no way it can really matter if they stop intellectual progress in mathematics for a while. So they did. They stopped most progress in "pure math" areas for the entire human race across the planet.
So I went to the factoring problem.
Now then, yes, humanity is kind of big to us, and it can seem easy to think that getting your little piece today is worth blocking its progress for a little while, especially to stop some annoying guy—just some guy after all—who keeps going on and on about his mathematical research.
Because history makes the legends—so I can't be one to you now—so it's later that students will read about you and find it incomprehensible that you could have even paused with the fate of your own species in the balance, and what could have been so important to you that you'd actually deny progress for people who were just lying?
But they would also know how it's in the balance. Some of them might contemplate not getting to be born because of what you are doing now—if you weren't stopped.
But they aren't in your shoes now. Perspective.
Legacy is a word that gets tossed around, but thinking that maybe down the line your legacy can be that of THOSE people, who didn't do the right thing, who held on even when it was clear that they were wrong, and broken later, could only fumble out rationalizations or apologies and accept their branding for life, is an exercise for those who aren't thinking about their bills and just trying to have a life.
If you were brilliant you wouldn't be in this situation.
Make no mistake, the world is not a nice place when you get on the outside and are looked at as a renegade versus being one of us.
You are inside now, and if you are a math professor, then you are very inside, but think about what happens if you keep up this nonsense and you are completely out.
I'm a problem solver. I brainstormed my way through some important and difficult math problems, got the answers, and found a society that couldn't or wouldn't live up to expectations so I am working on solving that problem as well.
My mistakes are many, but I own up to them. I have been wrong many times before, said things I know I'll regret later, and often wondered how this situation is even possible—until I remember—perspective.
No criminal ever thought first, long and hard, about getting caught, enough to not do the crimes.
JSH: Sorry, but exasperated
It's been over five years since I found a proof of Fermat's Last Theorem.
I still believed enough in modern math society that I questioned and questioned and questioned my own result as people fought successfully against it being accepted.
I even kind of wondered still when I pulled out a piece of it and got that published, only to have math society fail completely when the math journal went against formal peer review, pulled my paper and later died.
You people may have heard the story but may not have realized that I have been right, dealing with people who I continually saw shifting their tactics as I explained as they fought a political battle to make sure no one believed me.
One thing that kept me from simply giving up was that I knew they wanted it. They begged for it, literally.
After all, if I weren't around to champion my ideas then they could keep them suppressed.
Then Andrew Wiles could keep credit for something he didn't do.
And undergrads could keep getting taught crap math which would never work not because it is "pure" but because it's wrong, but not easily testable in a way that can show it's wrong.
The perfect trap.
The way the fight has gone against my research has evolved as my strategies evolved and as the people doing it had to handle my moves in other areas, as I looked desperately for ANY way to prove that I was right and that these people were deliberately lying.
They turned success into a perception of failure.
I've said it's like winning the Olympics and being booed and the gold medal going to someone who didn't even run the race.
They turned everything on its head.
So I turned to the factoring problem.
It still gets to me though, after so many years of dealing with these people to see posters STILL trying the same games, the same ways to distract and deny with an argument so simple it stunned me.
Who knew? Turns out there's this neat thing with abs(nT - 2k^2) where you look at a simple integer minima with k maximal.
Not even I thought the entire world could turn on a result that simple, but turn it will.
Blocking "pure math" is one thing but posters fighting now are trying to stop knowledge of a result that can mean your actual physical life could be in danger, or your financials.
Lying here can mean the ruin of some of you, and isn't it ironic, don't you think?
They will try. As what have the got left now?
Their battle was an all or nothing war against the latest (maybe the last) major discoverer.
They have nothing left to do but fight until the bitter end.
There have only been a handful of people like me in all of human history.
And I wish that I had not been born to face this mess of what humanity has become.
JSH: Test factorization
I've modified one of my existing programs to start testing out the latest surrogate factoring research though I haven't yet optimized it, so it kind of dumbly just looks for solutions around k approximately equal sqrt(nT/2) where n=1 if T mod 3 = 2, and n=5, if T mod 3 = 1.
Here's an example factorization:
T = 1342517983, k = 58480. surrogate = 127230885, which factors as
(3^3)(5)(449)(2099)
and T factors as (27893)(48131). The prime factors of the surrogate are of interest here and since T mod 3 = 1, the program multiplies it by 5, so
k^2 = 2^{-1}(5T) mod p
where you can check it for each prime. I haven't bothered to check.
Because the program is dumb, it took 183 checks of k's looping up from k approximately equals sqrt(5T/2), skipping over odd k's or k's divisible by 3.
Notice that k/2p approximately equals 14 using the largest prime, so you'd have a roughly a 1/14 chance of finding a solution, if you did it the smarter way, but that gave a good enough chance that even the dumb way stumbled across the factors.
It's not a complicated idea here, which is why it's amazing to me there are still people trying to argue over some rather basic algebra.
It just so happens that if you let k = 2x, and z = x+k, when
z^2 = y^2 + nT
then the maximum k that will give the minimum value for abs(nT - 2k^2) will tend to be close to the correct k, which must exist if z is divisible by 3 because
z = x + k = x + 2x = 3x.
Figuring that out just requires using
2x = k + pr
where p is some prime and r is an integer, and the substituting out z, with z = x+k, to get
if you substitute out z and simplify a bit you have
x^2 = y^2 + nT - (2xk + k^2)
so you can substitute out 2x, and get
x^2 = y^2 + nT - 2k^2 - kpr
and now let k_0 be the value for which r=0, so you can let
k = k_0 + 2pj
where j is an integer and substitute, and you have
x^2 = y^2 + nT - 2(k_0^2 + 4pjk_0 + 4p^2j^2) - (k_0 + 2pj)pr
and x, y, nT, k_0 and p are all constant, so as j varies, the j^2 term will dominate and the r variable will tend to be negative to counterbalance it.
If you need it all multiplied out to help you with this basic point:
x^2 = y^2 + nT - 2k_0^2 - 8pjk_0 - 8p^2j^2 - (k_0 + 2pj)pr
where with k_0 positive (as why have it negative?), you'll notice that while
k_0 <2pj
the negativity of 8p^2j^2 can't be overridden by -8pjk_0 no matter what the sign of j, but y^2 + nT - 2k_0^2 is constant as is x^2, so r must be negative to compensate until k_0 = 2pj, which is when k=0 anyway.
So it's trivial mathematics that k_0 will tend to be near the correct answer for k, when it is the maximum value such that abs(nT - 2k^2) is a minimum.
That amazing bit of mathematics puts the factoring problem within reach, just like that just because the j^2 is always positive.
Trivial algebra gives you the range as k should be equal to or greater than k_0, and j should be negative and greater than -k/2p.
Figuring out that k^2 = 2^{-1}(nT) mod p, is a little more complicated but if you were paying attention when I was babbling on about factors mod p, I explained it exhaustively and ad nauseum.
So, oddly enough, to tackle a composite T, you just need to get a a prime for which k exists that makes it very likely that you will find k quickly.
And it's all trivial algebra.
Now if you people wish to argue on still and wait until I or someone else is motivated to fully implement the trivial algebra, then fine. But don't come crying later when I say you people don't really know math, as then you clearly don't.
Easy algebra ignored when it's the factoring problem does not make you brilliant.
It does still annoy me and I still wonder how I let some of you bother me, that you can pretend to give a damn about mathematics and come out with sophistry to attack a beautiful and simple argument, proving how much you hate math, but having the gall to keep at it as if you can just fool people one more day, that's all that matters.
I think some of you every day you post arguing with me just tell yourself, to just try to get people to believe wrong things mathematically one more day, and you win, as you make humanity as a whole lose. One more day yesterday you people won.
Did you win today? Is humanity still being fooled by you?
[A reply to someone who asked which constraint should one use to pick k when programming.]
Oh please. Like what you do matters.
If this result is correct then someone in the world will pick it up.
Your input is irrelevant as is the input of everyone else on these groups.
I'm mostly just talking to myself anyway.
When the real storm hits, your voice will disappear as nothing you will say at that point will make any difference at all.
I'm kind of just appreciating the calm before the storm, before the world knows fully who I am, and that I am the next great discoverer, of a long line of discoverers.
That I am the next legend—living, breathing and solving mega problems in the here and now.
Not just some person to be read about, but someone that can be asked important questions, which is what I truly dread. When people get smart enough to ask me the real questions.
Friday, February 22, 2008
JSH: Fairly straightforward
So yeah, some guy going on and on about mathematical arguments that are wrong can be annoying, but what if he gets something right?
Then who cares? Right? If you care about mathematics for real then you take the good, and you take the bad and just accept what has to be true.
And THAT is what was so devastating for me, years ago, when I realized that there are so many people in the mathematical community who don't think that way.
They pick and choose.
And they think if they don't like a person they can justify ignoring what the person discovers because of what that person can get from those discoveries.
So they care more about the gain.
There is no way that math people give a damn if I get famous or make a lot of money from my mathematical discoveries if they aren't obsessed over getting famous and making a lot of money—probably figuring that it's just something they won't get, but maybe…
If they just care about the mathematics, then who cares what I get?
If money means nothing, and if fame means nothing, and the mathematics means EVERYTHING then all that matters is getting the knowledge and marveling over the mathematical reality.
But you people pick and choose, now don't you?
And you do so calculating what you think I'll get based on how you react.
Ergo, you don't really care about the mathematical truth.
You cannot.
If you did then you would behave as people who do.
As Forrest Gump might say, people are as people do.
I feel like Forrest Gump—the guy who did what a group of people didn't think was possible but then they showed that they didn't give a damn about their own area, but only what they thought it could give them.
Intelligence is as intelligence does.
Loving mathematics is about loving what mathematics is.
Not what you wish it were.
x^2 = y^2 + T - 2(k^2 + 12j + 36j^2) - 3(k+6j)r
That's a mathematical equation. It just is. What it tells is about mathematical truth.
If x, y, T and k are constant, then as j varies, r must vary to counterbalance it and it will tend to be negative as j increases either positively or negatively.
That can hurt though as it shows a way to factor when you generalize to p odd prime.
Using just 2x = k + pr, and z^2 = y^2 + nT and z = x + k.
Beautiful mathematics in its conciseness but possibly repulsive to some of you because you see it as validating me and my approaches, and what I call Extreme Mathematics, and arguing with people and getting a lot wrong just for WHAT YOU GET RIGHT.
And hating the process and hating a person can be everything to you because you do not love the mathematical truth, so why later should you be allowed to stay in the mathematical field and claim to be a mathematician?
Why?
Because people like you and feel sorry for you?
Because even if you can't do real mathematics you really, really, really want to believe that you can?
[A reply to someone who called James "narcissist".]
So? Even if I were, so? I don't go around trying to find people to verbally assault claiming it's their fault. You do.
I work at hard math problems.
I get a lot wrong.
But I admit it. I talk about the process, about brainstorming.
And I know that it can take a lot out of you to do the effort to get something right.
Then I consider creatures like yourself who think that it is my job to just sit back and be nice when I get it right and think about how much damage someone like you can do.
What if instead I try to make it harder for your type to operate?
Why shouldn't I?
You made your bets, right? You put yourself out there, made your posts and that is about what you committed yourself to doing.
Why shouldn't I make certain that you get the full consequences based on reality?
Nothing any of you do here is truly anonymous.
All of you as wanted will be tracked down and known.
To me justifying my support of that activity is all about how much time I've spent thinking about the kind of people who make an effort to try and find other people as prey.
I've pondered those who look to try and find vulnerable people that they think they can feed on in some way, so yeah, to such parasites a guy they think is just giving wrong answers all the time when math society says he's wrong could be one of these vulnerable humans.
So I studied that behavior. Contemplated it. Pondered what it meant about the creatures who displayed it.
But what if he's a great discoverer when no one really knows what one is like as it has been so long since one was here and none have been around in the Internet age when you could talk to one repeatedly in a direct way?
Why if you got it so wrong should I not let you feel the full consequence of that failure?
The answer is, I should not.
It is not in my nature to do so, so it must be then that you will discover reality is not as simple as you thought, and that human prey can turn out to be more than you ever thought possible because being a parasite is about what was.
While I am about the future and what will be.
No major discover has emerged before in the Internet age.
No one has been around for you to know exactly what a person like me is like, so you have no clue what is coming.
Thursday, February 21, 2008
JSH: Finding k
Surprising answer with surrogate factoring that focuses on finding k, and leverages a rather intriguingly simple little result to factor.
As consider
2x = k + 3r
when
z^2 = y^2 + T
where T is the target to be factored, is odd and coprime to 3, and T mod 3 = 2, as then z must have 3 as a factor, so
z = x+k
gives
x^2 + 2xk + k^2 = y^2 + T
which is x^2 = y^2 + T - 2xk - k^2, and I can substitute out 2x, to get
x^2 = y^2 + T - 2k^2 - 3kr
and that's where a nifty thing pops in, as, you want r=0, but in general, r will be NEGATIVE if you start with the optimal k when r=0 and move about modulo 6, as that k will be even. That's because you'd have
x^2 = y^2 + T - 2(k+6j)^2 - 3(k+6j)r
and if you have positive k (no reason to use negative) and try to move with positive j, then r will be negative, and even if you move with negative j, the 36j^2 term will tend to dominate, forcing r to be negative to compensate.
So r = 0 should be near the value at which abs(T-2k^2) is a minimum.
Trouble is, if you continue the analysis you find that k at that point is the minimum that might work—remember k is positive—but the actual k MUST be within k/6 steps from that value.
But it gives you a sense of what is possible, just with p=3.
I generalized to p odd prime though, but found that it's still important to have z with 3 as a factor, but then you have as a crucial requirement:
k^2 = (nT)(2)^{-1} mod p
so you need a prime for which k exists, and then you find a maximal k modulo that prime, as the same argument above works, except now you'd have
2x = k + pr
and you have a solution within k/(2p) steps from the maximal k.
If T mod 3 = 2, then you'd use n=1, else you'd use n=5 or maybe 2, I'm still not sure, to force nT mod 3 = 2, as there is a crucial requirement that
z = 3x
so z has to have 3 as a factor.
Oddly enough then for me, after over four years of trying to find an alternate factoring method with a concept I call surrogate factoring it all depended on this little thing that with
x^2 = y^2 + T - 2(k+6j)^2 - 3(k+6j)r
the j^2 will tend to dominate as you move around the correct value, which forces r to be negative to compensate.
That is just so amazing to me. How such a little thing is so important.
Without that, there'd be no clue about how to get close to k, and best you could loop through searching modulo p, which is what I worked at before, but the other crucial thing was realizing that
k^2 = (nT)(2)^{-1} mod p
was the REALLY important way to go as generalizing I used a variable I call α, but it turns out that with the generalization
z = (1+2α^2)x
and it's just easier to force z divisible by 3 than it is some of the other values that can be, like 19, or 73.
All easy math, and easy to play with thankfully. Something of a subtle result though that requires you just do this odd thing of looking at
2x = k + pr
along with z = x+k, and z^2 = y^2 + nT, and then it's easy, as long as you do all those things and can argue it out for months until it all comes together.
Tuesday, February 19, 2008
JSH: Goes to my worries about factoring
So now with the full surrogate factoring theory, results are coming fast and furious and I'll admit being very, very, very surprised that an RSA number might be factored by p=3 and a fairly simple technique.
Those looking over the argument may recognize that there is only one area where it's even maybe kind of looking like I didn't feel in the blanks which is with how you find k.
But just try it. Factor a few numbers and you'll get that weird, giddy out of this world feeling like maybe you stepped into the Twilight Zone.
Reading over posters ranting and raving in reply to me is kind of weird now. It's like there is something oddly wrong with them, but I can't quite put my finger on it.
The challenges to factor an RSA public key though, seem to be answerable now, and I'm mainly just absorbing the latest and the sense of profound oddity of it all.
You can factor an RSA public key, if that key is 2 modulo 3, and if with
z^2 = y^2 + public key
z is divisible by 3, and the math will just do it and kind of wink at you as if it wasn't even hard.
And you get k by finding k such that abs(public key - 2k^2 ) is a minimum and k is even, and you have two possibles k = 1 mod 3, or k = -1 mod 3.
Then x = k/2, and z = 3x, and you get y and factor the public key.
Just like that.
Yup, I had reason to worry about factoring. Wacky. Factoring a public key with p=3. Who would have thought the RSA system would crash so profoundly?
Still seems so weird though. So easy. All that work people did for all those years and the answer is so easy.And it's trivial math, like I explained in a previous post about helper primes.
The primes were there to help all along.
The prime numbers were there to help all along.
They step in and they step out.
I really don't think that my research was ignored by accident or honest mistakes in considering it. I got the one paper published in SWJPAM and the damn journal editors pulled it under SOCIAL pressure from sci.math'ers. Newsgroup people influencing math editors.
And then the freaking journal died.
Mathematicians went on the run, that's all. Rather than accept that they had things wrong they thought they could just lie and rely in me not being believed.
Think about all those undergrads taught crap math deliberately when they could have been taught real mathematics.
Over five years of undergrads.
Deliberately taught wrong.
And I've contacted Ribet, and I've contacted Mazur, so who knows what Wiles knew. He did not find a proof of Fermat's Last Theorem.
His research fails with a simple logical fallacy.
How gone do people have to be when they can't be moved by the fate of the human race? When they can claim to be at the pinnacle of mathematics when they're teaching wrong information and blocking the correct?
Friday, February 15, 2008
JSH: Frustrated.
Maybe I should say more after going on and on about an approach that I now accept was just useless for factoring, but I keep thinking about why I'm so desperate for something in the factoring area anyway, which is the blocking of my "pure math" research where math people don't follow their own rules.
I NEED a practical math result because one dead journal shows how locked down math society has it now.
Maybe we'll take each other down before all of this is over and I'll get my way of convincing the world that mathematicians routinely lie and your society will find a way to get me in return.
And it is a sad testimony to the true reality of modern academia.
It is a medieval system. And you end up in medieval crap with it, like what I call the Math Wars.
Some of you seem to think the Math Wars are just fun and games or just some silly "crackpot" mouthing off, but it is about me finding a way to do things like end tenure, reduce funding for academics and convince the world that objective measures, rather than just letting academics say which of their buddies supposedly did something great, are necessary.
So sit back. Think you have it all handled as I remain frustrated, and venting in futile anger at a feudal system of academia, because I need that practical mathematical result to break a broken system, but remember, it only takes one result for me to then go back and use every moment like this to emphasize to the public why it needs to do as much as necessary.
Our modern world exists today because discovery was cherished.
But parasites have turned things upside down for a few dollars and some empty accolades because they can't appreciate the value.
If humanity loses then it loses down the line, and the extinction of our species whenever it occurs as we probably won't manage to get off this planet, may trace back to a tragic shift, when major problem solving was lost, and pretend took over.
Call me crazy. But then nothing I do will work in the real world, right?
But then you won't find funding drying up, and you won't find an increasingly skeptical public demanding more than just your say-so, right?
Call me crazy, and if I am then the solutions I find to convince people that your society lies in the real world won't really be solutions, right?
But I see the end of the Math Wars putting most of you in other areas of work outside of mathematics. But my vision is against your will to stop me.
JSH: Enough venting, but funding is an issue
Ok, so enough venting out my frustrations with the math community. They're liars, so what? But funding for academia is increasingly a concern as I think, yup, that we're spending too much given the returns.
Yeah, I know, supposedly all this activity out there is going to lead to something big down the line and there are people doing fantastic research that is pushing our technology ever forward but I really feel that most of you are duds working the system.
And I think that about academics in general.
So the trendline that I'm pushing, for real, not just venting, is reducing funding and looking for survival of the fittest.
I don't buy the line that academics doing real research can't justify why the public should dole out money to pay their bills.
IF you can't explain why your research is worth the public dime, I think the public has a right to take it away.
It is our money.
The exceptions in my mind remain, materials science, medicine, biology, including anything to do with genetics, and, um, I think there was something else but I can't remember right now.
As for the rest, I want survival of the fittest and I want a lot more pressure on universities with big endowments—far, far, far more than you're starting to see now.
I want justifications across the board for monetary expenditures as I look to the world to weed out parasites: people who just are playing the game and not doing anything of value.
Thursday, February 14, 2008
JSH: What if no one believes you?
What do any of you have if no one believes you? Or better yet, if society doesn't?
I burn credibility because I'm right.
I don't need it, but in looking at my solutions people feel that I'm right so I can go out here and burn the belief in certainty itself.
Maybe it's all shades of gray, right?
What if NO one is right?
What if all the academics really are just saying stuff, that if not verifiable could just as well be something else?
What if you people do not matter?
Who listens to you now anyway?
I took from you what you didn't even know you could lose, and now you have no way of getting it back, without even accepting that it's gone.
End goal is simple: news people will not publish "pure math" results, at all.
That is the end goal.
I talk to a lot of people behind the scenes about politics, the economy, business as I cover a lot of territory with answers that work in the real world.
And I tell people that you people lie. That academics lie. That universities have endowments that are too big, and that people doing valuable research should be able to prove it, so we should slash funds and get survival of the fittest.
We must slash funding across the board and reduce expenditures at universities and colleges around the world is the message.
And it doesn't matter what you say, as I want you to disagree with people who bring arguments to you when you don't even accept who the source is.
JSH: So it's not a solution
It's just depressing. The math people can't be beaten unless I solve the factoring problem because they lie about proofs so I need something they can't lie about, and that's it.
So I'm stuck. They'll win.
Yeah I did get published. In SWJPAM after nine months when I even told them before publication I was an amateur.
I despise mathematicians.
And never ever again tell me that publication matters as it doesn't if the "experts" just decide to ignore it, or even break it, like those sci.math'ers did, getting that journal to pull my paper after publication.
They are scum. They lie and they know it and there's no way to stop them.
NEVER ever again tell me that publication matters.
It doesn't.
Mathematicians are the scum of the earth. They lie and they know they lie and there's no way to stop them because they are beneath contempt.
They are beneath contempt.Ranted a bit, then I went back to wondering why this latest idea wasn't working. And I figured it out.
So, um, all the previous about potential negative impact applies.
Looks like only I had that deep down gut feeling that two primes could be used in this way, so if the math people tried what I had before, they might have noticed it didn't work but didn't realize that they should puzzle out why as it SHOULD work.
So they just went back to whatever they were doing supposing I'd failed, I guess.
I got upset, got depressed, and then just went back to problem solving as that's what I do.
JSH: Scary situation, getting scarier
So I solved the factoring problem by doing some simple things and it's an easy proof and it's easy to check to verify with some simple numbers to see that I have a solution that must work, but that was days ago.
How could mathematicians not report this major find?
Well let me tell you a story. Years ago I pioneered a technique in mathematical analysis which turns a lot of established ideas in mathematics upside down and even got a paper on it published in a peer reviewed mathematical journal.
I've talked about that many times before, but the rest of the story is that I didn't just give up when the now defunct journal SWJPAM went belly-up a few months after withdrawing my paper AFTER publication after pressure from the math community against it.
But rather than elaborate on what I faced I'll point out that out of the blue a math grad student from Cornell University sent me an email offering to help. He claimed to be interested in having me explain to him my ideas and that the upside to me could be support from someone at Cornell.
So I sent him a beginning argument of a few pages. He sent back questions which I dutifully answered, but I noticed he took longer and longer between question so that a simple math argument of mostly algebra was taking this grad student MONTHS to work through.
After one long pause in his email he talked about long walks in the early morning hours like around 3 a.m. or something and I knew he was nearly gone.
Coming to the final pieces of the argument—remember I sent him beginning stuff—he finally replied back that he needed to get another math text, and that was it.
I've had a mathematician go on an immediate sabbatical when a colleague tried to help me out by having him look over some of my research on prime numbers.
A six month sabbatical.
When he returned he claimed he had never been asked about it, but refused to look at it.
I had one math professor just tell me that an equation that I knew worked—as it counted prime numbers—could not work, and he refused to be dissuaded.
They snap.
Now the factoring problem is solved. It's an easy solution but people who thought they were brilliant are wrapped up with people who are just con artists and neither of them are doing the right thing.
You physics people need to wake up to what can happen to you as well as everybody else.
The math people have gone bye bye. There is a solution to the factoring problem and it is TRIVIAL.
Hey, guess what? You could lose your funding! Your entire university could find itself stopped in its tracks if mad hackers go wild on your systems!!!
DO SOMETHING you people. Or while you sit twiddling your thumbs civilization as we know it can go up in flames and you won't be doing research on advanced computer systems but maybe writing things out on whatever scraps of paper you can find between foraging for food and dodging wild dogs.
JSH: Simple matching, factoring versus math politics
Playing around with various approaches to the factoring problem I noticed that if I did something as simple as consider T = 9 mod 11 and T = 2 mod 13, I would find that the minimum positive number for T that would work would be 119, which is a number I like to use in my examples and it occurred to me that the primes were forcing something.
So I started thinking about what information prime numbers could give about factors as if the two primes were forcing T to be 119 or greater, then they were also forcing the factors to be certain values.
So I expanded out a factorization with primes, what if you GUESSED using f_1*f_2 = T mod p_1 and g_1*g_2 = T mod p_2, so you'd know the f's and the g's?
And I realized that then you could use reduce to a solution for d_1.
d_1 = (f_1 - g_1)*p_2^{-1} mod p_1
and then you also need d_2, so
d_2 = (f_2 - g_2)*p_2^{-1} mod p_1.
So suddenly I had it!!! Information from the intersection of the primes.
The two prime numbers were now telling me something about a key variable in the expanded factorization!!!
And now you can go back toNow then, does this work?
Well, is there an answer for d_1 mod p_1 and d_2 mod p_2?
Of course, yes. If I have the correct residues for the f's and g's then will that answer be given by
d_1 = (f_1 - g_1)*p_2^{-1} mod p_1
and
d_2 = (f_2 - g_2)*p_2^{-1} mod p_1?
The answer is, of course, yes.
Now perturb them. Shift f_1 or g_1 and will you change d_1 and d_2?
YES!!!
So then, this method will work to tell you when you have them correctly so the answer to the key question of factoring is available: what is f_1 mod p_1 and f_2 mod p_2?
As you answer that question prime by prime you can factor the target, as a minimum positive value for f_1 is forced, like T = 9 mod 11 and T = 2 mod 13 forced a minimum positive value for T.
Now that is easy math. Trivial algebra.
Hating mathematics is about hating truths that you don't like, and I know that feeling so I can understand why so many of you despise mathematics, even if you claim to be mathematicians.
You despise it because you do not control it.
That's why your society came up with "delicate proofs" and "logical contradictions" as you found a human solution to an inhuman discipline.
The math does not care. What's true is true even if you can't make a living in the field telling the truth.
So most of you learned to lie as otherwise, you'd have to make your living some other way, and if society let you, why not live the fantasy?
Why not just pretend? Why not act? Actors are heroes in this society right?
Like Melanie Griffith. Or Brad Pitt. Or Morgan Freeman. Or Claire Forlani.
Actors make big bucks, and get accolades so why not just act with mathematics too?
Tuesday, February 12, 2008
JSH: Problem solving techniques
I use modern problem solving techniques. Those techniques recognize failure as just part of the process and brainstorming is one of the most known where lots of failures are just expected.
But the modern math world is corrupted.
So posters use those failures to try and hide the successes, even when it's the factoring problem—or a proof of Fermat's Last Theorem.
There were Catholic priests who turned out to be pedophiles (and nuns). Enron collapsed dramatically. And "pure math" mathematicians lie.
The world goes on.
Now as to why they lie it's simple: math is hard.
There are too many people who are supposedly mathematicians doing valuable research in the world today.
It's just so hard to do real research in mathematics that there is no way all those people are.
There just isn't that much discovery possible for a species at our level.
So most of what they're doing is fake as you can make a living as a math professor and produce papers, and have a job where most people haven't a clue what you're doing, so they don't know it's fake.
But you have one problem: every once in a while these pesky discoverers come around who want to do REAL mathematics as if that stuff is valuable, and they have this annoying tendency to want to tell the truth about mathematical ideas!!!
So your professors came up with a system to stop them, which involves ignoring answers or insulting them a lot, like calling them insane.
Not a bad system, and I discovered how potent it is, but it has one fatal flaw: discoverers are problem solvers.
So a mathematical discoverer at a certain level would just consider their system another problem to solve and figure out a way to dismantle it.
It's a challenge. I like challenges. And I'm good at solving problems.
So I decided your group was just another challenge, another problem to solve as a measure of how good I am. Neat.
So they just gave me a challenge. That's all.
Oh yeah, they ARE fakes. From what I've seen, not much real mathematics is being done in "pure math" areas today.
JSH: It's a hard life
People think that life is easy now when most of humanity is struggling out there, while a few people—percentage wise—are living large. So they think something has changed because they personally are desperate for attention versus eating, or having clean water.
But all that has changed is the illusion of a change has gotten stronger.
The difference now is that our mistakes can lead to the extinction of the human species.
And that would kind of suck.
Problem solving is not just a way to get attention. Science is not about impressing people with how smart you are. And publish or perish is about the survival of our species, when it's right.
I can assure you that many of you are lost. You are lost souls who think the world is a giant piggy bank with endless funds and it doesn't matter if you lie, cheat and steal because SOMEONE will save the day down the line, or you just don't believe in much of anything so you think that nothing matters, except what you can take.
Except I can tell people exactly what you take and that you don't think anything matters except what you take.
The window is closing for us.
There is only so much time to get things together before we close the door on life on this planet which is what the small-minded people out there think can't happen, so they make it inevitable.
Yes, we can kill life on this planet, and in doing so, end our own as well.
But out of the tragedy that I think is rapidly closing, whether you can see it or not, or believe it or not, if it happens nothing you think can stop it, or even pause it, and your stupidity won't save a single living thing, maybe some life can survive, and with it, a soaring future can be realized.
Our sister planet is Venus. She is our twin. We are heading towards her fate.
It's time to get ready to go. Humanity will have to leave planet earth, and soon.
Monday, February 11, 2008
JSH: Factoring IS stupid simple
Ok, sorry, as I hate it to some extent when a problem turns out to be harder to solve than I thought when some part of me must kind of know the answer but it takes a while for the rest of me to get it.
Here's the correct answer to the factoring problem where I just kind of diverged a bit before.
I was puzzling about the latest failed factoring idea wondering where I went wrong, as I really felt that there was information wrapped up in the intersection of the residues modulo T of two primes.
So I pondered some more.
Before too long I was writing out things along these lines what if you GUESSED using f_1*f_2 = T mod p_1 and g_1*g_2 = T mod p_2, so you'd know the f's and the g's, and I realized that then you could just solve for the c's by using I realized you could solve for everything and reduce to a solution for d_1.
That's from before, but after that point, before, I went in the wrong direction as it is so OBVIOUS what you should do next, which is solve d_1 modulo p_1 (maybe I was sleepy when I first typed that last).'s stupid simple.
I just took a little time to figure out exactly how the damn thing worked.
But that's problem solving for you. Some part of me must have known the answer but the rest of me was stupid, until simple won.
[A reply to someone who asked James why doesn't he just guess p1 such that T/p1 is integer.]
Funny. Ok, yes, I admit it. I went on and on about having solved the factoring problem through an intersection of primes when I hadn't shown it.
So I had a strong feeling that there had to be a way to do it, and then got waylaid in figuring out exactly how, but this latest result is actually consistent with the original idea while what I had before was not.
Here you DO test, and the guess just gets you SOME information, so you need a series of primes.
The approach IS straightforward and it is stupid, simple, so get mad and upset about the problem solving process but that just shows you have no clue how it really works.
Mostly it's about hard work, lots of misses and being willing to keep beating on that hunch until it pays off, and later let the freaking historians re-write history and talk about what a genius you are.
It's a big mess until you're right.
During the process most of the time you feel like a freaking idiot.
[A reply to someone who said that the issue is to find an efficient algorithm.]
If the approach holds i.e. if the values for d_1 mod p_1 and d_2 mod p_1 are provisional values that work only when you guess correctly, then this approach DOES lead to an efficient algorithm.
It turns out that it is well-known that if you have f_1 mod p_1, and f_1 mod p_2, … f_1 mod p_n, where n is a sufficient number of primes then you have f_1 explicitly.
The problem though has been, how do you find f_1 modulo successive primes?
I contemplated this issue and hypothesized that when you have two primes you have some kind of intersection that will give the answer, and I went looking for that intersection.
My first approaches went awry as I tried to actually solve for d_1 exactly, but thinking about why those approaches didn't work, I looked at the equations again and saw:
d_1 = (f_1 - g_1)*p_2^{-1} mod p_1
and I had everything I needed where BOTH primes gave input, so I had my intersection.
So this approach IS a valuable one as long as the hypothesis holds true and the value of d_1 modulo p_1 found above is invalid unless you've guessed f_1 and g_1 correctly, as those are residues, where
f_1*f_2 = T mod p_1 and g_1*g_2 = T mod p_2
and are NOT the full solutions. Maybe I could have picked other variable letters.
In any event, the point is an intersection between two primes with a guess at residues and checking to see if the guess is right.
If you can do that then factoring is trivial as you need less primes than m, where T/m! < 1.
So you have factorial working for you here, and that is a big over-estimate for m.
If this approach holds water then factoring an RSA public key of any size possible on modern desktop computers would be trivially done in minutes if not seconds.
The factoring problem is solved by using the intersection of prime numbers, in an elegant, precise, and remarkably short solution.
JSH: Factoring problem solution, update
I stumbled across a remarkably simple solution to the factoring problem, which exists because of what is called the floor() function. That function just means to drop any fractions or decimals, so like floor(3.1415) = 3.
It is crucial to the solution to the factoring problem.
Here is the full solution.
It suffices to determine variables to fulfill the factorizations:where T is the target to be factored and p_1 and p_2 are primes to be picked, as this method works because you're using primes in this way which is why you need so many variables.
Note thatSo the r's and k's are easily calculated and the only remaining variables are c_1, c_2, d_1 and d_2, and you guess at values for the f's and g's. So guessing is crucial in this method.
And it can be shown that solving for the factors reduces to finding integer solutions to a family of 4 equations and 4 unknowns, which are an intermediate solution is forced as all of the equations will not then be interdependent, and then you can solve for the c_1 in the first equation and d_1 in the second to get
c_1 = (k_1 - m_1 - c_2*f_1)/(f_2 + c_2*p_1)
and
d_1 = (k_2 - m_2 - d_2*g_1)/(g_2 + d_2*p_2)
and divide through by the denominator to get to a number that is factored to find an integer c_2, and an integer d_2, where that cannot be done with the original set of equations as the numerator is T itself, so that is equivalent to factoring T.
But here, crucially, the m's break you from directly getting T, so that
k_1 - m_1 - c_2*f_1 does not equal T
and
k_2 - m_2 - d_2*g_1 does not equal T
(curious readers can substitute the m's out wrongly with m_1 = f_1*f_2/p_1 and m_2 = g_1*g_2/p_2 and see what happens—you get T back).
So m_1 and m_2 are crucial to the solution as floor() is a discrete function.
If you have guessed the right f's and g's then your integer solutions for the c's and d's will give you a factorization of T, otherwise
(f_1 + c_1*p_1)(f_2 + c_2*p_1)
and
(g_1 + d_1*p_2)(g_2 + d_2*p_2)
will equal some other number.
If you have selected the correct f's and g's then you will get integer solutions, otherwise you will not, so if you do not then you shift to another set, so there are
(p_1 - 1)(p_2 - 1)
MAXIMUM total checks without regard to the size of T.
If you have fewer prime residues available than factors of T, then you will not be able to solve for the c's and d's exactly but must use the approach mentioned above, but if you do have enough residues you will have 4 independent equations and can just solve for the variables, and will get integers when you have the correct residues.
It is a fantastic solution where the floor() function does all the heavy lifting, creating a logical situation that provides for a trivial solution to the factoring problem.
If the solution is ignored then civilization as we know it may end.
And it can do so within a few days, my analysis indicates.
Which would be the end of humanity's halcyon period.
Many of you might simply die of starvation, if you're lucky.
[A reply to someone who asked James why should anyone care about the value of floor (3.1416).
Because I found a solution to the factoring problem, but the academic world is corrupted so I couldn't present it in a way designed to have the least negative impact.
Which means that the solution can be exploited and end civilization as we know it if mathematicians continue to do what I think they'll do.
Didn't you read the post The Art of War? Didn't get it?
If these math people are as powerful as I think they are, then no way would they just leave it to chance that some genius could come in and wreck their system, so they had to believe they were safe.
But they know I found a proof of Fermat's Last Theorem so I maybe could figure out the factoring problem, right?
So why am I still here?
They must have something up their sleeves so I declared all these solutions to the problem that turned out to be bogus and then got lucky and found the right answer.
If it's correct, then, for instance, the entire Internet could be on its knees in a few days and you wouldn't have to worry about what I or anyone else posts as you wouldn't SEE any posts.
Yup, for the rest of you, no more posting for anyone. Get it? Concerned yet? Willing to help?
(I think that might have more of an impression than the possibility of starvation. God knows some of you have to post.)
If I'm right, the world as you know it is going to change.
Just because of some math wars the entire Internet could be in flames in a few days, if I'm right.
Read that Art of War story.
[James replies to his own first post, up to the point where he wrote "m_1 = floor(f_1*f_2/p_1) and m_2 = floor(g_1*g_2/p_2)".]
Well I started out correctly but from there went down the wrong path.
But the intuition that I was following was that the intersection of primes gives information about factoring, which was such a strong gut feeling that when I realized I'd screwed up, I went back over to see where I might have gone wrong, and, well, found the trivial solution.
So I backtracked a bit and then found the correct path.
Turns out you next need to solve for d_1 modulo p_1: is a key in the lock technique. When you have the right residues of factors modulo each prime then
d_1 = (f_1 - g_1)*p_2^{-1} mod p_1
and
d_2 = (f_1 - g_2)*p_2^{-1} mod p_1
are the right keys in the lock so that you get
k_2 = d_1*g_2 + d_2*g_1 + d_1*d_2*p_2 + m_2.
For those wondering if maybe it's wrong, imagine you have the WRONG f's and g's, then the d's should be different, right?
So when they're right…
Easy solution but a lot of inertia I'm sure because of so much invested in thinking factoring is hard.
Of course, if too much time is taken then, well, I've harped enough on the potential negative consequences.
I just have to hope that there are adults around.
[A reply to someone who wrote speed is of the essence.]
I'm using the factoring problem.
If as I fear math society IS corrupt then it possibly knows that potentially I could solve the factoring problem to prove that, so crooks within that society must feel they have a backup.
But what?
My guess is that they have protocols in place with the Bush administration and with agreements between nations where such a solution would be sealed away, along with the discoverer.
But how can they seal away what they don't know exists?
And that's where you came in, you and other posters on the newsgroups.
You helped provide the distance—the gap—between when they might know and when they wouldn't.
So everything now is about how effective you've been.
You see, I don't want you to believe me now.
I want the opposite.
That gives the solution time to propagate and force the situation, and take away the final solution from unethical governments, like the Bush administration.
Same plan in place for those willing to help: work in concert, with a display that takes away any notion that governments can seal this away and hide it, along with me.
JSH: Factoring problem solution, latest objections
Some posters have STILL been attacking the simple solution to the factoring problem so I will answer their claims and first point out a crucial step that I caught one of them doing which was removing m_1 and m_2.
You'll notice in what follows that m_1 = floor(f_1*f_2/p_1) and m_2 = floor(g_1*g_2/p_2) where all the variables will be explained. What's important here is that the floor() function creates an integer requirement, and is key to solving the factoring problem.
Without it, there is no solution.
So a poster sneakily taking out m_1 and m_2 is removing the very basis for the solution.
With that said there IS a situation when you can not find specific solutions which I need to address, so it's worth addressing that as well and I'll do so below.
The full system again isset T to your target, and pick two primes p_1 and p_2.
NowAnd it can be shown that solving for the factors reduces to finding integer solutions to a family of 4 equations and 4 unknowns, which are
c_1, c_2, d_1 and d_2: the math has no choice but to kind of loop on itself, and not allow you to solve for the c's or d's, BUT you can still solve in that case:
Those are just incredible equations, as they give you something you can't have without the floor() function as otherwise the m's are fractions.
Stupid simple is what the answer to the factoring problem is.
Stupid simple.where you can factor the numerators in each case to find integers that will work, which is an approach that cannot work, you'll notice with the original equations as you are then just factoring T itself!
Here it works because
m_1 = floor(f_1*f_2/p_1)
and
m_2 = floor(g_1*g_2/p_2)
though if the f's and g's are wrong when you substitute back in before, you won't get T.
So m_1 and m_2 are crucial to the solution as floor() is a discrete function.
If you have selected the correct f's and g's then you will get integer solutions, otherwise you will not, so then you'd shift to another set, so there are
(p_1 - 1)(p_2 - 1)
MAXIMUM total checks without regard to the size of T.
Modern mathematicians unfortunately lie. Their system collapses if they are caught so they CANNOT celebrate a fantastically simple proof as I'm telling them that I'll inform the world that they lie all the time.
So it's a fascinating situation. I'm curious how long the world will let them lie about the factoring problem since they can collapse civilization with the lie.
Seems a few days already, which makes you wonder how much rich people really value their billions.
Maybe they don't really care if they end up being poor, soon.What continues to amaze me though are the students.
Getting taught crap ideas from people who really are kind of dumb would upset me.
My guess is that it's about not wanting to accept having wasted so much of your lives with junk ideas, so you take in more junk.
It's like after being told you were eating crap for caviar you stuff yourself with more out of denial.
They did it on purpose.
I just want you to know that they had to know they were wasting your time, your life, and your mind, and they did so because they are parasitic.
Just con artists masquerading as mathematicians.
Nothing more.
[A reply to someone who told James that it was time for him starting to delete posts.]
Let's see how confident you are tomorrow.
The math is in the post.
The m's would be fractions without the floor() function.
Your objections all centered on removing the floor() function.
Sigh. If I'm wrong, then so what? It's just another mistake of mine.
But if I'm right, then the destabilization of society is just around the corner.
And then it won't matter what dreams you used to have as the civilization necessary to realize them, will no longer exist.
Sunday, February 10, 2008
JSH: Art of War
I am a fan of the book THE ART OF WAR, and there is a great story in it about this great general who wanted to conquer this heavily fortified city so he encamped near it and prepared to attack, but he had no intentions of going through with it at that time.
Still as his army approached the city had no choice but to prepare so it closed its gates and its citizens rushed to their defensive positions.
At the last moment the general had his armies wheel away from attack, return to their camp and bed down.
He did this routinely for months.
The same drill each time, his troops would assemble for the assault, charge the city, the city would close its gates on go on the defensive, and then the general would wheel away.
So guess what happened?
The city grew complacent, and used to the drills, so one day, when the general came charging, they didn't even bother to close the gates and no one went to defensive positions as they thought it was just another drill.
But this time the general completed the assault and conquered the city.
JSH: Sorry for ranting and raving
You'd think I'd just be happy with the realization that I DID solve the factoring problem, but I felt a lot of rage. I HAVE been right all along, and it IS true that academics have worked to block my research for years now.
And solving the factoring problem just scares me as well, as who knows how things will play out now.
But I think the realization that there are academics in this world who not only do not do real research but they deliberately work to block major discoveries to HIDE that they do not do real research just infuriated me.
How many of you are fakes?
How much "scientific research" are you doing that you know is crap, where you also work together to attack scientists doing legitimate research?
I think the problem is extremely widespread.
My case is a dramatic example but again I'm reminded of Dr. Halton Arp, a distinguished scientist who was an assistant to Hubble himself, who has the distinction of being both a major figure in his field—and supposedly a crackpot.
I don't agree with a lot of what he says, but I don't call him crazy, and I am flabbergasted at the major reason his colleagues do.
It seems to be all about cosmologists wishing to ignore strong evidence that their theories about red-shift are oversimplistic.
So they can claim that quasars and galaxies are further away than they really are and that we know more about the universe than we really do.
They squashed Dr. Arp's funding in the US and he had to flee to Germany.
The parasitic academics are a scourge on humanity.
They have no morals. They work to get money for bogus research and flawed research ideas and fight legitimate research because it endangers their cash flow.
So now we have a simple solution to the factoring problem and I'm afraid the same type academics in the math field will try to ignore it or hide it.
But this time that can mean a direct impact on every single one of you, so there is no just turning a blind eye hoping it's for the best, or thinking there is nothing you can do as if you do nothing then you may find it hard to eat in the coming months. You and your families.
Yup, you can help crash civilization as we know it, which is what parasitic organisms do: they destroy their hosts.
That these are human parasites does not end their impact but may make it worse and yes, you may not be able to eat, literally, in the coming months if they are not handled here.
So I'm back to ranting and raving a bit but it's such an impossible situation!!!
Mindless and immoral human beings who can't be bothered to care about humanity or its future lie mindlessly to make a few buck and to stop them civilization itself has to be endangered?
The work of countless people around the world who trusted society could be blown away in days?
Entire countries could bite the dust? Because some academics lie for a living?
When you understand the enormity of the situation I think you will think I'm being amazingly calm. |
Maths Prodigy Gains Admission Into the University at Age 12
A 12-year-old boy who got A* in his maths GCSE aged eight, has now become Britain's youngest university student.
Xavier Gordon-Brown may still be too young to join Facebook, but is already studying for a degree in maths.
He manages to fit his university studies into his free hours between full-time school, practicing the clarinet, piano and violin, learning four languages and playing football with his friends. When Xavier, from Haywards Heath in Sussex, passed his GCSE in 2009 he was the youngest ever student to gain an A* in maths, and could recite 2,000 digits of Pi.
He is now studying abstract structures, vector calculus and Newtonian mechanics in his spare time, but since he only turned 12 last month he has to be accompanied to his Monday evening lectures by his mother.
Despite being the youngest student by far, his proud mother Erica said he is still top of the class. Erica said: 'He absolutely loves it.
"It's one of those things we thought about for a while, but seeing how much he enjoys it has made it all worthwhile.
"People make a lot out of keeping children in their age group, but it is good for him to be amongst his intellectual equals too."
The youngster's interest in numbers started at an early age. Xavier knew his times tables before he was four and could do double-digit mental arithmetic before starting school. A year after his GCSE success he passed his A-levels.
Xavier, who lives with housewife Erica and father Michael, travels with his mother to Open University lectures in East Grinstead two to three times a month. "When he's with kids his own age he's fine, but when it comes to maths he needs to be with people on his intellectual level," she said. She added that Xavier loves playing football with his friends and that, despite his busy schedule, he never falls behind in his studies.
Erica said: "He's not falling behind. If anything he's top of the class and everyone gets on with him really well. When I go I absolutely don't understand a word of what's going on."
Although Xavier is an extraordinary talent, it seems to be a family tradition, both Xavier's older sister and brother took GCSEs at a similar age. As well as football, Xavier enjoys chess, Taekwondo and trampolining and on top of French and Spanish lessons at Oathall Community College, he is teaching himself Latin and Mandarin at home |
Given Table of Numbers Problemwww Given a m * n rectangle, Numbers Problem, place all numbers from 1 to mn that minimizes the sum of the products of rows and columns (both in Spanish and English). Table of Numbers Problem.
At Mathematics Museum (Japan)www At Mathematics Museum (Japan) you would be surprised how interesting mathematics is. Museum (Japan). You will find exhibition rooms produced by Japanese researchers and educators. Mathematics Museum (Japan).
A The Eugène Strens Recreational Mathematics Collectionwww A special collection at the University of Calgary, Eugène Strens Recreational Mathematics Collection, including the archives of Martin Gardner. There is a searchable online index. The Eugène Strens Recreational Mathematics Collection.
A The Aesthetics of Symmetrywww A brief digression into how people perceive symmetrical patterns -- what makes them boring, Aesthetics of Symmetry, interesting, or overly intricate The Aesthetics of Symmetry.
An Mudd Math Fun Factswww An archive of interesting math facts for use in the classroom or just for fun. Math Fun Facts. Browse by subject, difficulty, keywords, or try the "random" feature. Based at Harvey Mudd College. Mudd Math Fun Facts.
A Maze Classification and Algorithmswww A short description of mazes and how to create them. Classification and Algorithms. Definition of different mazetypes and their algorithms. Maze Classification and Algorithms.
New The Diamond 16 Puzzlewww New version of the classic puzzle using row/column/quadrant permutations to display symmetries of graphic designs. Diamond 16 Puzzle. Has link to a site on the underlying mathematics (Diamond Theory). The Diamond 16 Puzzle.
A Math Forum: 2002 Mathematics Gamewww A contest where the contestants have to write all integers from 1 to 100 using only the digits 2, Forum: 2002 Mathematics Game,0,0,2 and arithmetic operations. Math Forum: 2002 Mathematics Game.
In A Mathematician's Aestheticswww In his classic A Mathematician's Apology, Mathematician's Aesthetics, G. H. Hardy likened mathematics to poetry and painting. This site elaborates on Hardy's remark with quotations from Stevens, Klee, Fry, and Focillon. Links to related sites are given. A Mathematician's Aesthetics.
A HAKMEMwww A collection of problems from MIT. HAKMEM. Work reported herein was conducted at the Artificial Intelligence Laboratory, a Massachusetts Institute of Technology research program. HAKMEM.
An Sir Roger Penrosewww An article about him and his interests and contributions to recreational mathematics. Sir Roger Penrose.
An Who Can Name the Bigger Number?www An essay by Scott Aaronson on the quest for ever-bigger numbers, Can Name the Bigger Number?, from exponentials to Busy Beavers. Who Can Name the Bigger Number?.
Includes Roman Numeralswww Includes a introduction to Roman numerals including a translation of the digits used and a converter which can convert decimal to Roman numerals and vice versa. Roman Numerals. |
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The Golden Ratio: A Constant Beauty
Disclaimer #1: the WordPress formatting has gone strange and no matter how many times I try, this comes out in all caps. I apologize.
Disclaimer #2: there are those who believe this to be totally hooey. This is because the Golden Ratio, usually simplified to 1.6180, is actually an infinite decimal, like Pi, meaning that it never ends, it extends into infinity. Therefore, some claim, nothing can truly measure up, so to speak, to the Golden Ratio, because said ratio can never be measured.
Now, I am not a mathematician; I am an artist. So my reaction to this argument is to shrug and say, "Meh, close enough." If that sort of blasé attitude offends your mathy sensibilities, you should probably stop reading now.
The above picture is the simplest example of the Golden Ratio. Two quantities fit the ratio if the larger part, divided by the smaller part is equal to the whole length divided by the larger part. For more detail, try the Math is Fun [ explanation.
This ratio, (also referred to as PHI and the Divine Proportion) can be found all over the natural world: in honeybee hives, the females outnumber the males by 1.618 to 1. The ratio of each spiral of a chambered Nautilus' shell is 1.618 to 1. Sunflower seeds grow in opposing spirals –the ratio is the 1.618 to 1; the same is true for the spirals of pinecone petals, rose petals, leaf arrangement on plant stalks, insect segmentation… the list goes on.
Why is this? What does it mean? There is no definite answer. Some say it's just a simple pattern, and simple patterns often repeat, much like convergent evolution, which is when two distinct species evolve similar traits. For example, bats, birds, and insects all independently developed modes of flight. Others choose to take a more spiritual approach and read deeper meaning into the ratio scribbled across the universe like a creator's signature. Whatever your view, you cannot deny the wondrous beauty of this simple, repeating pattern.
PHI is stamped all over the human body as well. Da Vinci was one of the first to notice that each segment of fingers between the joints on the human hand matches the ratio. The distance from your wrist to your elbow and from the tip of your middle finger to your wrist: PHI. The distance from middle fingertip to elbow, divided by the distance between elbow to shoulder: PHI. The ratio is all across the human body, not only the arms, but I don't have time to list them all. This graphic will have to do.
When it comes to measuring beauty in the human face, the more universally attractive a person, the more closely his or her face fits the ratio. The length of the face divided by the width, the length between the lips and the eyebrows and the length of the nose, the length between the distance between the pupils and the distance between the eyes… again, the list goes on. Click here if you want to see how your face measures up.
Classically beautiful women of history: Greta Garbo, Queen Nefertiti of Egypt, a woman from a Botticelli painting I have yet to identify, sincerest apologies
Even your computer follows the divine proportion. Look down at your keyboard; the distance between the shift key and the Z key and the distance between the shift key and caps lock is PHI.
PHI is found in architecture, logos, and numerous other man-made structures.
Now, all this also ties in with something called the Fibonacci Sequence, but that is a post for next week. |
Mathematics as a Social Science
Most people (both mathematically versed and far from any math) would agree that mathematics is a science. Some of them, however, are apt to believe that this is a very special kind of science that lies in the basis of any science at all, providing the necessary pre-requisite for cognition as such. Finally, there are those who treat mathematics as an innate ability, the supreme knowledge embedded in the human animals in a mystical way, as a primordial touch of consciousness.
There are, indeed, serious reasons for all the above. Apparently, mathematicians behave like other scientists; at least, they talk the same way. On the other hand, it may be rather difficult to say what exactly they study in all those formal theories, whose practical significance finds recognition many decades (or even centuries) later, if ever. For an outer observer, mathematics is like a swarming ball of protoplasm that would suddenly sprout in one direction or another, to give birth to a new theoretical science, in the regular sense. To put it bluntly, a science must study something objective, that is, lying outside that very science and taken as an external source of facts and an application field. In this relative objectivity principle, we account for the whole hierarchy of indirect research, with higher-level sciences built upon a number of other sciences, however abstract, serving as a kind of empirical foundation. Like in any hierarchy, the "up" and "down" directions are readily interchangeable, and one could encounter the situations when two sciences lie in the empirical background of each other.
With mathematics, one cannot get rid of the impression of arbitrariness, inherent emptiness of any discussion, since one can easily develop a formally consistent theory starting from a collection of random assumptions, which are all equally acceptable, and there is no obvious reason for choice.
This was not that way in the ancient times, when mathematical knowledge came from immediate practical experience and satisfied people's everyday needs. Social differentiation and division of labor have detached the skill from its applications; however, up to the end of the XIX century, there was a hope for a "natural" foundation of mathematics, the common root of all the further abstractions. Multiple geometries, the algebraic revolution and computers have dispersed that illusion, and now, we are left face to face with our strange ability of constructing imaginary worlds that no one will ever inhabit.
But did the character of mathematical knowledge really change? Despite all the formal games, the basic feeling of a number and a shape remains intact, while the alternative notions of rigor manifestly continue the same line of causal arrangement that has always inspired technological progress, from the troglodyte magic to the modern robotized industry. If we do so and so, we are bound to finish as expected, unless some external circumstance (including the operator's blunders) abruptly modifies the operational environment.
This brings us back to the scientific status of mathematics. The object area of this peculiar science could be elicited using the hints from its early days, when its practical origin was yet evident enough.
Human activity is always aimed at producing some changes in the world. Each typical mode of such change gives impetus to developing a special science, with the object area related to the objective organization of the prototype activity. The hierarchy of sciences reflects the hierarchy of common activities. However, there is a fundamental distinction induced by the universal organization of any activity at all, which implies a conscious subject to take a portion of the world for an object and intentionally transform it into some product. As a result, each product can be characterized from two complementary aspects, as a kind of object (the material product) or as a representative of a certain way of action (the ideal product). This inner complexity of each product has eventually resulted in two complementary branches of science: the material aspect of activity is targeted by the so-called natural sciences, while knowledge about the modes of action is aggregated by social sciences. The latter name is appropriate since there is no individual that could exercise conscious activity outside any society at all, and the very idea of consciousness is only meaningful in the social context. That is, the subject of any activity is hierarchical, and any individuality belongs to that hierarchy along with the numerous forms of collective subject, from the family of two up to the humanity as a whole.
Now, does mathematical knowledge refer to any material things? No, it doesn't. There are very few people who would consider mathematical constructs as self-contained things, existing on themselves (this philosophical position, known as objective idealism, is usually associated with the name of Plato). Intuitively, mathematical knowledge is rather about some common properties of things; but, for us, the only relevant properties are those that are significant for using things in our activity. Mathematics is, therefore, to study certain modes of human activity, and hence it must belong to the class of social sciences. In other words, mathematics brings us knowledge about ourselves, just as any other humanitarian research. This perfectly explains the apparent arbitrariness of mathematical theories, since social sciences take the world under a subjective angle, including the freedom of choice. In our everyday life, we have to decide on the appropriate modes of action; this, in particular, is reflected in the versatility of mathematics. Philosophical materialism, however, holds that no choice can be entirely arbitrary, and that the variety of available options is always determined by the objective organization of the world, by the nature of things. This circumstance is responsible for the apparent rigidity of the mathematical method complementing the apparent arbitrariness of the premises.
One could expect that new modes of mathematical thought would come in response to significant cultural shifts; however, the humanity has not yet (at least on the memory of civilizations) experienced revolutions of that scope, and we are still quite comfortable within the existing paradigms. Nevertheless, some hints to the open possibilities might be drawn out of the several methodological turnovers known in the history of mathematics. In any case, no science can ever reach the state of permanent completion; though some sciences (including certain mathematical theories) seem to have exhausted their creative impetus, their abandonment is of an entirely local significance, as there is always a chance of running into an interesting feature that has earlier been irrelevant, or just overlooked.
During the periods of relative stability, natural sciences develop an inner organization that drives then away from nature, to resemble the humanities in the very occurrence. A typical mathematical (or physical) paper is packed up with metaphors and allusions, using the regular language (indispensable in any discourse, however formal) in a very loose manner, mentioning thousands of names (which is intended, but fails, to reference earlier introduced ideas), lacking conceptual and theoretical consistency, as a clear exposition of the matter is impossible without a range of assumptions and preliminaries beyond any tractability. Professionalism gets almost entirely reduced to mastering the parochial slang, while logical transparence is sacrificed to erudition. Everything is made to impress the public rather than educate it.
A novice will find modern science almost incomprehensible, since an individual life is not enough to get just acquainted with all the parental work, nothing to say about a critical examination. Considering the chaotic character and inevitable circularity of references, there is practically no way to check the logical consistency and factual substantiation of any special report. The validity of reasoning is no longer a matter of proof, but rather a kind of common consent, prejudice or academic fashion, so that the whole of science virtually develops from one level of belief to another, rather than from truth to more truth. In this context, credulity and good memory are much more important for a student than inquisitiveness and prehension.
As a result, the overall structure of mathematical knowledge remains utterly conventional, just like in social disciplines similar to law, or accounting. The absence of a natural organization makes it impossible to establish a standard reference frame, to make mathematics searchable. This is a dictionary with no alphabet, sorted by random criteria, like keys and the number of strokes in Chinese and Japanese hieroglyphic dictionaries (or abstract hash values in computers). Eventually, there are too many characters to learn, and the whole thing splits into a number of traditional areas poorly communicating with each other.
Well, every cloud has a silver lining. In its chaotic mass, mathematics just cannot come too restrictive in natural sciences, leaving more room for metaphorical usage and losing the aura of magic that led many scientists to overestimating the role of formal manipulation and starting to toss phantasies instead of studying nature. Given the limited accessibility of advanced mathematical methods, we have to organize knowledge according to the structure of the objective area rather than stretch observations to an arbitrary formalism; this may reveal uncommon structures that could eventually push forward our mathematical thought.
Typically, a working scientist (e.g. in physics) has an individual mathematical toolkit, a store of standard components to reuse in any new theoretical model. However, when it comes to a drastic conceptual change, the already available forms are no longer sufficient, and a mathematical description has to be invented from scratch, since it is almost impossible to find the relevant pieces in the body of modern mathematics (counting out the always-possible random encounters). In case of success, mathematicians would assimilate some of such handicraft to the earlier introduced constructs, or add yet another ad hoc theory to the rest, to keep on with piling up formal junk. The portions of mathematics that penetrate other sciences are nothing but the coming back of their own inventions reformulated and "refined", spiced up with a scent of "rigor". This, again, resembles the situation with the humanities: to gain an official status, a new teaching needs authoritative support, a formal assignment and right to compete; later on, each authorized discipline can play itself a role of official authority for the newcomers. The pretense of mathematics to the absolute dominance in science is a neat replica of the superpower image of the USA on the political and economic stage. The inner discrepancies of the American society leave enough room for the other nations to break the dictate and develop on their own, thus influencing the development of the USA as well. Someday, science will probably abandon the idea of scientific ranks and forget interdisciplinary competition, to grow a new hierarchy of knowledge that would not distinguish natural sciences from anything "unnatural" or "supernatural". This must obviously follow the overall democratization of the world order, annihilating the market economy, class society and any kind of competition, throughout both the human culture and the world. |
Skeptics in the Pub - maths
answer unique |
Mandelbrot Orbit Traps
Like the previous category of images, these are images of the Mandelbrot Set. However, these images use a special coloring technique called "orbit traps". Conceptually, an orbit trap is an object that is "placed" into the same mathematical environment as the fractal itself. Due to the nature of the mathematics, the orbit trap object is copied many times, while being scaled, rotated, and stretched in all sorts of ways. So a single orbit trap object can result in an image containing what appear to be a great many objects.
In more mathematical terms, an orbit trap is a test that can be applied to complex numbers. One possible test, for example, might be "is this complex number within 0.5 units of distance from the point 0.75 + 0.25i or not?" This example orbit trap test would correspond to an orbit trap object that is a disc of radius 0.5 with its center at (0.75, 0.25). If the test is "true" for the point z at any time as it travels around the complex plane over the course of the iteration of the Mandelbrot Set's generating function z ← z2 + c, then the point z is considered "trapped" by the orbit trap. The original point c, which through iteration became the point z that was trapped, is colored in some way associated with the orbit trap. The coloration is based not only upon which trap was hit, but on "how true" the trap was (how far from the center of the orbit trap disc the point z was, for our example), what iteration the trap was hit on, what other traps were hit in the course of the point c's full iteration, and other factors. |
Mathematician essay
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Mathematician essay
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Euclid was a famous mathematician a greek mathematician, euclid is believed to have lived around 300 bc (ball 50) most known for his dramatic contributions to. Essays - largest database of quality sample essays and research papers on famous mathematicians. What is mathematics - about mathematics current students faculty and the reference to the essay by david garcia was added sometime around 2000. Read this essay on famous mathematician come browse our large digital warehouse of free sample essays get the knowledge you need in order to pass your. This free history essay on essay: history of mathematics is perfect for history students to use as an example. |
"It is important to learn more mathematics.
The contest itself wouldn't be that important, but it
creates new enthusiasm.
From this point of view it is important.
Also it has a good effect: it stimulates interest
in mathematics..."
(Erdos was widely regarded as the
greatest living mathematician until his recent death; a bright
high school student in Hungary, he excelled in math
competitions and went on to a brilliant career as an itinerant
mathematician known by many simply as"Uncle Paul". His
life and work is a testimony to the value of mathletics in
identifying, encouraging, and training young mathematical
talent.) These pages host
information and resources for Manitoban mathletes and their
teachers and provide an online home for the University of
Manitoba Mathletics teams.
MAP
Current problem sets and
solutions, a schedule of our activities, our archives and
various information of use to our own mathletes. If
you are interested in participating in any of our
activities, visit this page and contact the current
coaching staff.
This contest is locally run
and written concurrently with the Pascal/Cayley/Fermat
contest each year to (i) discourage age-advancing -- the
contest is that it is for graduating seniors, so a student
may write it only once; and (ii) facilitate
invigilation of both contests by a single teacher. This year's contest will be
written Feb. 24, 2011. See also the
first "News and Announcements" item below. Watch
this space for up-to-date information, an archive of
recent prizewinners.
Held early in every year,
during the lead-up to the high school math contest season,
100 or so students from the Winnipeg area converge on our
campus for four Saturdays to experience training in
problem solving. Up to 6 students from each
school can be mentored in labs by our own
university-level mathletes, learn mathematical games, be
exposed to sample contests and compete for prizes in an
internal competition we hold during the workshop.
Each week a dynamic speaker opens up some area of
mathematics to these eager young minds. This year's
workshop will be heldon:
Jan. 22, 29 and
Feb 5, 12 2010.
Olympiad-Level High School
Mathletics Training (no link yet)
In the Spring of 1998, over
a 5 week period, 12 of the top Mathletes in Manitoba from
grades 8 through 12 participated in Olympiad-level
training sessions at the University of Manitoba, a pilot
project for a proposed permanent program in which our top
Mathletes can receive appropriate instruction in
Olympiad-style and invitational competitions such as the
CMO, IMO and APMO. The program involves instruction
in tools and techniques and exposure to problems and
solutions from these contest. We are contemplating a
year-round training program in which elite
university-level mathletes from around the world would
mentor the best and brightest in Manitoba. Watch
this spot for updates. In the meantime,
students hoping to prepare for competitions at this level
are strongly encouraged to participate in OLYMON,
the national-level training correspondence program run by
the CMS, which is open to all students who show promise at
this level.
Everything you might need
to know about various competitions at all levels, for
which Manitoba Schools may be eligible. This
includes links to central sites for the various contests
as well as summaries of: deadlines, target grade
levels for each, and the style of each competition
(multiple choice, short answers, full solutions, etc.),
any registration fees or special conditions. CURRENTLY OUT OF DATE --
Watch for Fall 2010 update.
Updates about current and
upcoming Mathletics events and opportunities in Manitoba
will be sent several times per year to High Schools around
the province. This list is continually under
construction, and it is currently incomplete -- YOU CAN
HELP: check out the list of schools for which we
have no current contact and have someone who is willing to
play this role get
in touch with us.
News and Announcements
Get regular mathletics
updates by email: If your school is currently
active in mathletics or hoping to become more active it will be
very helpful to be apprised of events, deadlines and
opportunities throughout the year. In our experience
surface mail to schools often goes astray, or ends up on the
wrong person's desk. To avoid this problem we are trying
to move to electronic updates, by keeping a current email list
of mathletics contacts at every Manitoba school teaching grades
8 through 12. Often things change mid-year, particularly
for the workshop, and it would be very helpful to have a means
of rapid contact to all interested parties---that is the sole
purpose of this list; it won't be used or distributed for
anything beyond Manitoba mathletics-related information.
If you are (or know who should be) the mathletics contact at your
school please let us
know by email, the sooner the better. Thank you for
helping us stay in touch.
The winner of the 2009 UMOMC
(annual beginning-of-year internal Contest at U of M) was Todd
Sierens. Todd will receive a cheque for $100 and a book
prize ... not to mention fame, glory and personal satisfaction!
The winner of the 2008 UMOMC
was Ruiqang (Richard) Liu, an engineering student.
In spring of 2008 we piloted a brief training program at the
University of Manitoba for students showing aptitude for
competition at National and International levels.
Sometime in the Fall or winter of 2009 we hope to reprise this
with a more permanent program involving training and mentoring
by past mathletics champions, and in conjunction with the
national OLYMON correspondence training program. Direct
enquiries to R.
Craigen, |
The Joy of X: A Guided Tour of Mathematics, from One to Infinity
-By Steven StrogatzHow should you flip your mattress to get the maximum wear out of it? How many people should you date before settling down? How does Google search the internet? Why does the stock market swing so often, and so wildly? In this book, the author explains the great ideas of maths - from negative numbers to calculus, and fat tails to infinity.
read more...
Award-winning Steven Strogatz, one of the foremost popularisers of maths, has written a witty and fascinating account of maths' most compelling ideas and how, so often, they are an integral part of everyday life. How should you flip your mattress to get the maximum wear out of it? How many people should you date before settling down? How does Google search the internet? Why does the stock market swing so often, and so wildly? Maths is everywhere, often where we don't even realise. Award-winning professor Steven Strogatz acts as our guide as he takes us on a tour of numbers that - unbeknownst to the unitiated - connect pop culture, literature, art, philosophy, current affairs, business and even every day life. In "The Joy of X", Strogatz explains the great ideas of maths - from negative numbers to calculus, fat tails to infinity - with clarity, wit and insight. He is the maths teacher you never had and this book is perfect for the smart and curious, the expert and the beginner. It is suitable for readers of Ian Stewart's Professor Stewart's "Cabinet of Mathematical Curiosities", Alex Bellos' "Alex's Adventures in Numberland", Marcus de Sautoy's "The Number Mysteries".
read less... |
The Babylonians discovered a strange form of trigonometry
Enlarge/ The 3,700-year-old Babylonian tablet known as Plympton 322 turned out to be a trig table, expressed in ratios of the lengths of the sides of the triangles, rather than angles. (credit: UNSW/Andrew Kelly)
The Babylonian civilization was at its peak roughly 4,000 years ago, with architecturally advanced cities throughout the region known today as Iraq. Babylonians were especially brilliant with math, and they invented the idea of zero as well as the base 60 number system we still use today to describe time (where there are 60 minutes in an hour). Now it appears that the Babylonians invented trigonometry, almost 1,000 years before Pythagoras was born.
University of New South Wales mathematicians Daniel Mansfield and Norman Wildberger discovered this after a breakthrough analysis of an ancient cuneiform tablet, written between 1822-1762 BCE in the Babylonian city of Larsa. Long a mystery, the tablet shows three columns of numbers. Describing their work in Historica Mathematica, the researchers call the tablet "a trigonometric table of a completely unfamiliar kind and… ahead of its time by thousands of years."
Mathematician Daniel Mansfield explains the Babylonian system for doing trigonometry.
What made it hard for scholars to figure this out before was the complete unfamiliarity of the Babylonians' trigonometric system. Today we use the Greek system, which describes triangles using angles that are derived from putting the triangle inside a circle. The Babylonians, however, used ratios of the line lengths of the triangle to figure out its shape. They did it by putting the triangle inside a rectangle and completely circumvented the ideas of sin, cos, and tan, which are key to trigonometry today. |
Mathematical musings for every maths lover!
Greater than one and less than infinity
A snippet from the IB Economics textbook: "greater than one and less than infinity". Isn't everything less than infinity?!
UPDATE: Apologies for sixth-former-me, I now realise that infinity is not a number, but a limit, and so while every number in the reals is less than infinity, a limit (including the PED, I suppose) may be infinity. |
Magical Maths Show
Maths and philosophy were the order of the day for an exciting and interactive talk given by Ben Sparks at the latest community science and maths event hosted by Glebelands School.
Ben Sparks is a maths teacher and speaker, tutor and co-ordinator for the Further Maths Support Programme.
"He delivered an inspiring talk that asked the audience to consider whether numbers were created or discovered, reflecting and discussing the history of maths and indeed the conflicts maths has caused over the years" said the school's head of maths, Dee Gardner.
"During the school day he entertained and amazed our Year 9 and 10 students with 'mathemagic tricks', allowing them a glimpse of how beautiful maths can be when he talked about fractals showing some truly amazing images. Our students left the hall inspired and astounded at how maths can be used in the real world," she said.
Last Thursday's evening event was attended by the then deputy Mayor - now Mayor - of Waverley, Cllr Patricia Ellis, councillors, school governors, families and members of the local community. "The feed-back was very positive with several people indicating they would have liked to have listened for longer," said Ms Gardner, adding "We are very pleased with another successful Maths event and are looking forward to our next community science show in October". |
My latest column for Quanta Magazine is about the recent classification of pentagonal tilings of the plane. Tilings involving triangles, quadrilaterals, and more have been well-understood for over a thousand years, but it wasn't until 2017 that the question of which pentagons tile the plane was completely settled.
Here's an excerpt.
People have been studying how to fit shapes together to make toys, floors, walls and art — and to understand the mathematics behind such patterns — for thousands of years. But it was only this year that we finally settled the question of how five-sided polygons "tile the plane." Why did pentagons pose such a big problem for so long?
In my column I explore some of the reasons that certain kinds of pentagons might, or might not, tile the plane. It's a fun exercise in elementary geometry, and a glimpse into a complex world of geometric relationships.I'm excited to announce the launch of my column for Quanta Magazine! In Quantized Academy I'll be writing about the fundamental mathematical ideas that underlie Quanta's stories on cutting edge science and research. Quanta consistently produces exciting, high-quality science journalism, and it's a tremendous honor to be a part of it.
My debut column, Symmetry, Algebra and the Monster, uses the symmetries of the square to explore the basic group theory that connects algebra and geometry.
You could forgive mathematicians for being drawn to the monster group, an algebraic object so enormous and mysterious that it took them nearly a decade to prove it exists. Now, 30 years later, string theorists — physicists studying how all fundamental forces and particles might be explained by tiny strings vibrating in hidden dimensions — are looking to connect the monster to their physical questions. What is it about this collection of more than 10^53 elements that excites both mathematicians and physicists? |
Sunday, February 28, 2010
I've been reading Steven Strogatz' ongoing series in the New York Times about the basics of math. In this post about Division, he referred to the above exchange between George Vaccaro and a Verizon customer service rep about an error in his bill. Vaccaro was supposed to be charged .002 cents per kilobyte but was charged .002 dollars per kilobyte:
About halfway through the recording, a highlight occurs in the exchange between Vaccaro and Andrea, the Verizon floor manager:
V: "Do you recognize that there's a difference between one dollar and one cent?" A: "Definitely." V: "Do you recognize there's a difference between half a dollar and half a cent?" A: "Definitely." V: "Then, do you therefore recognize there's a difference between .002 dollars and .002 cents?" A: "No." V: "No?" A: "I mean there's … there's no .002 dollars."
It's a common joke between a few of us in the office that We Hate Math. What we mean is that, as lawyers, we think in words and not in numbers but, as corporate lawyers, we are constantly asked to do math. For instance, when we are doing a recapitalization someone needs to perform the math to convert the current number of current shares into the number of shares that will exist after the conversion and then convert each individual shareholder's shares into the new number. It's not difficult, except that often it ends up with fractional shares. Sometimes it gets more complicated. And you'll hear someone muttering as they take a break and walk down the hall, "I HATE math."
So part of me feels for the Verizon rep.
By the way, since this is a math post I should point out that this marks my 300th post on my blog. That would 2 times 150 posts. Or 1/3 of 900 posts. Or … well, you do the math |
Tuesday, September 17, 2013
A Fun Mathematical Game
Hi folks!
So I know that the class blog is technically supposed to be about physics, but I've been having some fun with a bit of light number theory and I wanted to share it with y'all, especially considering that any self-respecting physicist should have a healthy respect for number theory.
So this is a simple mathematical game that produces a sort of numerical fractal.
The first iteration is just 1.
The second iteration is based on the first iteration: 11. The third iteration is based on the second iteration: 21.
The fourth is: 1211.
Fifth: 111221
Sixth: 312211
Seventh: 13112221
Eighth: 1113213211
etc.
So, have you figured out the pattern? Every iteration looks at the previous iteration and writes down in numbers what you might describe in English words. So the second iteration looks at the first and says, "There's one one." The third looks at the second and says, "There's two ones." The fourth looks at the third and says, "There's one two and one one." The first looks at the fourth and says, "There's one one, then one two, then two ones." So on and so forth. You might notice that these numbers come in pairs. I'm going to call the first number of the pair the "coefficient" and the second number the "descriptor".
Well that isn't too horribly exciting, although deriving this is kind of relaxing if you're having a stressful day or are profoundly bored in class. (Hello all fans of ViHart!) Well, since we're all super smart, let's start to look at some properties of these numbers. First of all, it looks like the highest number we see in any given place is "3". Is there any point in the pattern where we could get a "4".
The answer is no, and here's the proof:
Assume that there is an iteration in which a four exists. In that iteration (n), we have "4 x". Four must first be a coefficient because the only way for four to be a descriptor is if it appears in a previous iteration. That means the previous iteration (n-1) had "x x x x". The iteration before that (n-2) must have had x appear 2x times. So "x x x x" would be an incorrect description for the n-1 iteration. The correct description would be "(2x) x". Which means on the nth iteration, we would have "1 (2x) 1 x" because 2x=x only when x=0 (we'll prove that x!=0 in a moment). so the iteration that could lead to "4 x" would always lead to "1 (2x) 1 x", which produces a contradiction.
Okay, so now we have to prove that no zeroes can appear in this sequence. Well, we'll assume the same thing: that we have "a b 0 x c d" in the nth iteration. That means that in iteration n-1 we have b appear a times, then d appear c times. This means that we would write the nth iteration as "a b c d", which is a contradiction. Now lets try "a b x 0 c d". That one gets a little messier, but it's basically the same. Somewhere in the pattern, zero must appear in a coefficient spot eventually, which produces this same contradiction.
It seems like I'm being pedantic in proving all of these things, but it's important to know that you can trace everything tightly using the rules you created. While we could probably all figure these things out intuitively, intuition can be either wrong or at least imprecise and when you use imprecise intuition to derive other elements of a system, you can get some wildly inaccurate ideas, even if the intuition you initially used was sound in theory but in need of some refinement.
All right folks. Well that was super fun, but there's more work that can be done here. You can take this pattern with you to your most boring class and see what sort of things you can prove or disprove. Here's some ideas:
What is the relationship between the iteration number and the length of the string?
Does the infinitieth iteration have infinite length?
Is there a way that you can have a zeroth iteration or negative iterations?
Decrement all numbers in the iterations by one and throw them into a ternary number system. Do these numbers have any fun properties? Maybe they produce a bunch of primes?
Is the a relationship between n and the corresponding ternary value? Can we write a formula for that function?
What if you string all of these ternary numbers you created together after a decimal point; would that number be normal (good luck with this one; proving numbers are normal is nigh on impossible)? |
FRACTALS
What's new in mathematics?
Fractals are.
Source=
Most mathematics that
you learn about in school was developed over 300 years ago. Most
of the mathematics relating to fractals, was developed within the last
10-25 years. Think about that! These were studied
in your life time. What else in school can you say
that about? Not much!
What is mathematical about fractals? Why are they so colorful? How can you create one? |
The subtitle to this book is The Art of Educated Guessing and Opportunistic Problem Solving and Mahajan sums up his philosophy in the very first paragraph of his preface:
Too much mathematical rigor teaches rigor mortis: the fear of making an unjustified leap even when it lands on a correct result. Instead of paralysis, have courage—shoot first and ask questions later. Although unwise as public policy, it is a valuable problem-solving philosophy, and it is the theme of this book: how to guess answers without a proof or an exact calculation.
This book weighs in at a scant 135 pages including the index. The title comes from a short course of the same name taught by Mahajan at MIT. The makeup of the class ranged from first-years to grad students in a wide variety of academic programs. The course was designed to focus on techniques for using math to solve real-world problems.
This may seem odd at first because we already use math every day to solve real-world problems.
Don't we?
Well, yes. But, as pointed out above, there is a time and place for mathematical rigor (as taught in traditional classes) and at other times you need to be able to feel the answer. It's kind of hard to explain but Chapter 4 ("Pictorial proofs") suggests what the author is trying to achieve:
Seeing an idea conveys to us a depth of understanding that a symbolic description of it cannot easily match.
Make no mistake. This is not a book simply listing short-cuts or "cheat codes" that we can use to avoid math. This is instead a different approach to mathematics by way of mapping math to our life experience and giving us a way to internalize it, giving us a sort of sixth sense.
Let me give an example. A bookshelf that is out of alignment and unstable can be described by a series of geometric expressions, involving lengths, widths, angles and so forth. However, an experienced carpenter can tell if a bookshelf is out of alignment and unstable by looking at it and touching it. Now the carpenter certainly knows the mathematical properties of a well made bookshelf but she has internalized the math so that now she can 'sense' when something is wrong. It doesn't add up, so to speak.
The book covers a wide range of topics, from economics and Newtonian mechanics to geometry, trigonometry and calculus. Yet it's very reader-friendly, written in a clear, engaging style with plenty of examples and even some sample problems. There are only six chapters and you can dip into the material in almost any order. The book has been published under a Creative Commons license so there is a PDF version that is freely available for download. If you're like me, however, you'll also spring for the paperback edition.
Street-Fighting Mathematics is a unique book that will reward you with hours of thought-provoking (and practical) reading. |
Maths, Music, and More
It's been a while, but I have something interesting to show for it. (for those who don't know, a pythagorean triple is 3 integers that fit the format a2+b2=c2). I've mentioned my work on pythagorean triples before hand, and now I have some more to show for it. To this point, there are 2 interesting tidbits I have found from my work.
1: all natural numbers larger than 3 can be part of at least one pythagorean triple
2: I have found a method for finding all the pythagorean triples a given integer can fit into (as 'a', not 'b' or 'c'). This method is a little complicated, so I made a python program for it. The link is here for those interested.
Basic rundown of the program's main functions:
alltriples(n) will give you a list of all the triples that n fits into as an 'a' value.
Most() starts running through a list of integers, and marks the ones that are in the most triples so far.
MostTriples(n) lists the number of triples that n fits into.
The rest of the functions are mainly there to support those three.
I am planning on doing a proper write up of my idea, so I will put that here when I'm done. |
Caption: Leonardo Fibonacci. Portrait of the Italian mathematician Leonardo Fibonacci (1170?-1250?). Fibonacci introduced the "Arabian" numeral system to Europe through his Book of the Abacus (1202). He also discovered the Fibonacci series, a sequence of numbers in which each number is the sum of the previous two: 1, 1, 2, 3, 5, 8, 13 and so on. This curious series is often found in nature, for example in the pattern of leaf growth. |
Welcome to the great, Math-ropolis, the place where the Number-family could make every kid say "I love math!". Sounds weird, now, isn't it?
I want you to know a little secret.
This place really existed.
A place where the king of all Numbers, King Googol the first, ensured the happiness for the math-ropolis citizens. Speaking of citizens, let's meet them, shall we?
One had oh, so many friends, Two, was the little sunshine for everyone's morning, Three, heh, he planted so many trees, flowers, and flew with the bees … Oh, wow! Look, there, in the skies, it's a bird, no, it's a plane, no… it's astronaut Four, coming back from another star-exploration. And thank God, for Five, the fireman, cause he sure knows how to stop the fire that Chef Six starts by cooking the weirdest pizzas in the world. Miss Seven was the cutest florist in the city, where cowboy Eight, saved you from trouble in a jiffy… And you might think where, how, who started this magical world? Well, a magician of course, in a little tent, owned by Nine. Quite a place, heh?
It is… well, actually it was, because no further than a few years ago, a terrible thing happened. The Letter-kingdom, wanted all the kids, to hate the numbers, and love the letters. You might ask, why not both? Why can't everyone just love letters and numbers, animals and plants, the moon and the sun equally, like you love your mother and father. Selfishness? Jealousy? Nobody knows, and as for now nobody actually cared either but the Letters wanted all of the love only for them, so they went to every Number, and took only one thing from them, the one thing that made them happy. As so, every number got sad, shutting himself from the outside world, being shy, or rude, or disrespectful… Seeing this meant that King Googol lost the one thing that made him happy, the joy of the city. It was this easy to go from "love math" to "hate math" because slowly, every kid, started to get as far from numbers as they could. The city was lost… ShortPING |
Download An Episodic History of Mathematics: Mathematical Culture by Steven G. Krantz PDF
An Episodic historical past of Mathematics offers a sequence of snapshots of the background of arithmetic from precedent days to the 20th century. The reason isn't to be an encyclopedic background of arithmetic, yet to provide the reader a feeling of mathematical tradition and historical past. The booklet abounds with tales, and personalities play a powerful function. The e-book will introduce readers to a few of the genesis of mathematical rules. Mathematical heritage is fascinating and worthwhile, and is an important slice of the highbrow pie. an exceptional schooling involves studying varied tools of discourse, and positively arithmetic is among the so much well-developed and demanding modes of discourse that we have got. the focal point during this textual content is on getting concerned with arithmetic and fixing difficulties. each bankruptcy ends with an in depth challenge set that might give you the pupil with many avenues for exploration and plenty of new entrees into the topic.
Casimir, himself a well-known surgeon, studied and labored with 3 nice physicists of the 20 th century: Niels Bohr, Wolfgang Pauli and Paul Ehrenfest. In his autobiography, the intense theoretician shall we the reader witness the revolution that ended in quantum physics, whose impact on glossy society grew to become out to be again and again greater than the 1st atomic physicists may have imagined.
Additional info for An Episodic History of Mathematics: Mathematical Culture through Problem Solving
Example text
The next regular polygon in our study has 192 sides. It breaks up naturally into 192 isosceles triangles, each of which has area 2− 2+ A(T ) = 2+ √ 2+3 . 4 Thus the area of the regular 192-gon is 2− A(P ) = 192 · 2+ 2+ 4 √ 2+3 √ = 48 · 2− 2+ 2+ 2+3. 14103 . 3 Archimedes 37 This new approximation of π is accurate to nearly three decimal places. Archimedes himself considered regular polygons with nearly 500 sides. His method did not yield an approximation as accurate as ours. But, historically, it was one of the first estimations of the size of π.
19. But now we may use the Pythagorean theorem to analyze one of the triangles. 20. Thus the triangle is the union of two right triangles. 3 Archimedes 27 hexagon—is 1 and the base is 1/2. Thus the Pythagorean theorem tells √ us that the height of the right triangle is 12 − (1/2)2 = 3/2. 20, is √ √ 1 1 3 3 1 = . A(T ) = · (base) · (height) = · · 2 2 2 2 8 Therefore the area√of the full equilateral triangle, with all sides equal to 1, is twice this or 3/4. Now of course the full regular hexagon is made up of six of these equilateral triangles, so the area inside the hexagon is √ √ 3 3 3 = .
Thus the area is A(T ) = = 1 · (base) · (height) 2 1 · 2 2− 2+ 2+ √ 3· √ 2− √ 2+ 3 . 4 The polygon comprises 48 such triangles, so the total area of the polygon is √ 2− 2+ 3 √ A(P ) = 48 · = 12 2 − 2 + 3 . 1326 . This is obviously a better approximation to π than our last three attempts. It is accurate to one decimal place, and the second decimal place is close to being right. And now it is clear what the pattern is. The next step is to examine a regular polygon with 96 sides. The usual calculations will show that this polygon breaks up naturally into 96 isosceles triangles, and each of these triangles has area 2− 2+ A(T ) = 2+ 4 √ 3 . |
Back in the 1700s, a city known as Königsberg, in what is now Russia, consisted of two islands on the Pregel River and the surrounding land. The city wasn't known for anything in particular, or really anything at all. But because of some interesting geography and two creative mathematicians, Königsberg has gone down in the history of mathematics as a key city in the development of a whole new genre of mathematics: Graph Theory.
The story starts with a mathematician named Carl Gottlieb Ehler. Ehler noticed that the islands of Königsberg only had seven bridges going from the mainland to the islands and one bridge going from island to island. This caused Ehler to wonder if there was a way to make a trip such that you traveled across each bridge only once. The great video above, made by TED-Ed, explains how Ehler, with the help of the mathematician Leonhard Euler, came to a conclusion to the "Königsberg Bridge Problem" and at the same time invented a whole new type of math |
Reading the Comics, November 19, 2016: Thought I Featured This Already Edition
For the second half of last week Comic Strip Master Command sent me a couple comics I would have sworn I showed off here before.
Jason Poland's Robbie and Bobby for the 16th I would have sworn I'd featured around here before. I still think it's a rerun but apparently I haven't written it up. It's a pun, I suppose, playing on the use of "power" to mean both exponentials and the thing knowledge is. I'm curious why Polard used 10 for the new exponent. Normally if there isn't an exponent explicitly written we take that to be "1", and incrementing 1 would give 2. Possibly that would have made a less-clear illustration. Or possibly the idea of sleeping squared lacked the Brobdingnagian excess of sleeping to the tenth power.
Exponentials have been written as a small number elevated from the baseline since 1636. James Hume then published an edition of François Viète's text on algebra. Hume used a Roman numeral in the superscript — xii instead of x2 — but apart from that it's the scheme we use today. The scheme was in the air, though. Renée Descartes also used the notation, but with Arabic numerals throughout, from 1637. (With quirks; he would write "xx" instead of "x2", possibly because it's the same number of characters to write.) And Pierre Hérigone just wrote the exponent after the variable: x2, like you see in bad character-recognition texts. That isn't a bad scheme, particularly since it's so easy to type, although we would add a caret: x^2. (I draw all this history, as ever, from Florian Cajori's A History of Mathematical Notations, particularly sections 297 through 299).
Zach Weinersmith's Saturday Morning Breakfast Cereal for the 16th has a fun concept about statisticians running wild and causing chaos. I appreciate a good healthy prank myself. It does point out something valuable, though. People in general have gotten to understand the idea that there are correlations between things. An event happening and some effect happening seem to go together. This is sometimes because the event causes the effect. Sometimes they're both caused by some other factor; the event and effect are spuriously linked. Sometimes there's just no meaningful connection. Coincidences do happen. But there's really no good linking of how strong effects can be. And that's not just a pop culture thing. For example, doing anything other than driving while driving increases the risk of crashing. But by how much? It's easy to take something with the shape of a fact. Suppose it's "looking at a text quadruples your risk of crashing". (I don't know what the risk increase is. Pretend it's quadruple for the sake of this.) That's easy to remember. But what's my risk of crashing? Suppose it's a clear, dry day, no winds, and I'm on a limited-access highway with light traffic. What's the risk of crashing? Can't be very high, considering how long I've done that without a crash. Quadruple that risk? That doesn't seem terrifying. But I don't know what that is, or how to express it in a way that helps make decisions. It's not just newscasters who have this weakness.
Mark Anderson's Andertoons for the 18th is the soothing appearance of Andertoons for this essay. And while it's the familiar form of the student protesting the assignment the kid does have a point. There are times an estimate is all we need, and there's times an exact answer is necessary. When are those times? That's another skill that people have to develop.
Arthur C Clarke, in his semi-memoir Astounding Days, wrote of how his early-40s civil service job had him auditing schoolteacher pension contributions. He worked out that he really didn't need to get the answers exactly. If the contribution was within about one percent of right it wasn't worth his time to track it down more precisely. I'm not sure that his supervisors would take the same attitude. But the war soon took everyone to other matters without clarifying just how exactly he was supposed to audit.
Mark Anderson's Mr Lowe rerun for the 18th is another I would have sworn I've brought up before. The strip was short-lived and this is at least its second time through. But then mathematics is only mentioned here as a dull things students must suffer through. It might not have seemed interesting enough for me to mention before.
Rick Detorie's One Big Happy rerun for the 19th is another sort of pun. At least it plays on the multiple meanings of "negative". And I suspect that negative numbers acquired a name with, er, negative connotations because the numbers were suspicious. It took centuries for mathematicians to move them from "obvious nonsense" to "convenient but meaningless tools for useful calculations" to "acceptable things" to "essential stuff". Non-mathematicians can be forgiven for needing time to work through that progression. Also I'm not sure I didn't show this one off here when it was first-run. Might be wrong.
Saturday Morning Breakfast Cereal pops back into my attention for the 19th. That's with a bit about Dad messing with his kid's head. Not much to say about that so let me bury the whimsy with my earnestness. The strip does point out that what we name stuff is arbitrary. We would say that 4 and 12 and 6 are "composite numbers", while 2 and 3 are "prime numbers". But if we all decided one day to swap the meanings of the terms around we wouldn't be making any mathematics wrong. Or linguistics either. We would probably want to clarify what "a really good factor" is, but all the comic really does is mess with the labels of groups of numbers we're already interested in |
Then why aren't you posting concrete and specific references to mathematical material? (If you don't understand that question, please let us know).
While learning theory is a good thing, its value without any practical application is questionable. For example, you asked about fields and rings. Can you apply your knowledge and give an example of each field axiom for, say, the field of real numbers?
associative, 0, 1, and other things.
See?!
I know it in hebrew, I don't know the termination in English and yes I know what is ring and what is field.
They are have axioms.
If you want to build a theory on number it is useful.
See? |
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