This lecture is part of Berkeley Math 115, an introductory undergraduate course on number theory, and this lecture will be about calculation of the Legendre symbol. So I'll just start by quickly recalling what the Legendre symbol is. You remember the Legendre symbol is written as a sort of funny fraction with parentheses around it, and it's equal to plus 1 if a is a quadratic residue mod p, that means a is congruent to x squared for some x, a is not 0, and it's minus 1 if a is a quadratic non-residue, that means a is not congruent to x squared for any x, a is not congruent to 0, and it's 0 if a is congruent to 0. And last lecture we calculated whether or not 2 was a quadratic residue, so we recall that this was equal to plus 1 if a is congruent to 1 or 7, sorry p is congruent to 1 or 7 mod 8, and minus 1 if p is congruent to 3 or 5 modulo 8. And now we're going to calculate this for some, if we change 2 to some other values. So one easy case to do is to work out whether or not minus 2 is a quadratic residue of p, and this is very easy because we remember the Legendre symbol is multiplicative in the top, so minus 2 is minus 1 times 2, so we get this relation here. And now we know what both of these are, so we can calculate this as follows. Suppose p is congruent to 1, 3, 5 or 7 mod 8, then minus 1 p is equal to plus 1, minus 1, plus 1, minus 1 respectively, and 2p is equal to plus 1, minus 1, minus 1, and plus 1 by last lecture, so we just multiply these together and we find that minus 2p is equal to plus 1, here we get plus 1, here we get minus 1, and here we get minus 1. So minus 2 is a quadratic residue exactly when p is congruent to 1 or 3 modulo 8. Now we'll have an application of this, so you remember we showed that there are infinitely many primes of the form 8n plus 1 by looking at primes factoring x to the 4 plus 1, and using the fact that if p is an odd prime dividing x to the 4 plus 1, then it must be of the form 8n plus 1. What we want to do now is to show there are infinitely many primes of the form 8n plus 3, 8n plus 5, or 8n plus 7, so these are all special cases Dirichlet's theorem. And to do this we're going to use the quadratic residue symbol, so we recall that if p divides x squared minus 2, here we're going to take p odd, then p is congruent to 1 or 7 modulo 8, and that's because if p divides x squared minus 2, this implies 2 is congruent to x squared mod p, so 2 is a quadratic residue of p, and we've seen that occurs exactly when p is 1 or 7 mod 8. So that means we can find primes that are 1 or 7 mod 8 by factoring x squared minus 2, and the question is how do we get rid of the ones that are 1 mod 8? Well that's not very difficult. Suppose we take all the primes p1 up to pn, suppose they're all congruent to 7 mod 8, and we think of all the primes 7 mod 8 that we thought of, and we want to find a new one. What we do is we multiply them together, and then we square it, and we notice that this is now 1 modulo 8 because it's the square of an odd number, and then we're going to subtract 2 from it, and we notice that this is now 7 mod 8, because it's 1 mod 8 minus 2, and so now we take a prime p dividing this expression, and we see that any such prime p must be congruent to 1 or 7 mod 8, but not all primes dividing it can be 1 mod 8, because if all primes dividing this will 1 mod 8, their product will also be 1 mod 8, and this would have to be 1 mod 8, which it isn't. So there's at least one prime that 7 mod 8 dividing this number here, and it can't be p1 up to pn because then it would divide 2, which is just silly. So we can find infinitely many primes that are 7 mod 8 by just multiplying together all the ones we found squaring it, subtracting 2, and we've shown that at least one of the primes dividing this must be a new prime that's 7 mod 8. So let's actually try a few examples of x squared minus 2. So we have 3 squared minus 2 is 7, that's 7 mod 8, 5 squared minus 2 is 23, that's 7 mod 8, 7 squared minus 2 is 47, that's 7 mod 8, 9 squared minus 2 is 79, that's 7 mod 8. In fact, it seems to be a rather good way of producing, I mean, x squared minus 2 rather often seems to be prime, but when we get to 11 squared minus 2, this is equal to 7 times 17, and now we see that 17 is 1 mod 8, and 7 is 2 is 7 mod 8. So this isn't always prime, and some of its factors are not 7 mod, are not 1 mod 8, but at least one of them must be 7 mod 8. Of course, this isn't a new one, but that's because we didn't take 11 to be the product of all the prime 7 mod 8 that we thought of so far. So next we would take, you know, 7 times 23 and square it and subtract 2, and that would have a new prime factor that's 7 mod 8 and so on. So, well that does primes that are 7 mod 8. Now let's try and show that infinitely many primes that are 3 mod 8. Well, this is quite similar. All we do is we look at prime factors of x squared plus 2, and we notice that if we take x to be p1 up to pn, where all the p i's are congruent to 3 mod 8, then x squared is going to be congruent to 1 mod 8. So this thing will be congruent to 3 mod 8, and we also notice that if p divides x squared plus 2, then minus 2 is a quadratic residue. So p must be congruent to 1 or 3 mod 8, and just as when we look at primes that are 7 mod 8, they can't all be, so not all factors of x squared plus 2 are 1 mod 8, because then x squared plus 2 would be 1 mod 8, which it isn't, it's 3 mod 8. So we can get infinitely many primes that are 3 mod 8 by multiplying together all the ones we found, squaring it, adding 2, and looking for a suitable prime factor of that, that's 3 mod 8. So that leaves 5 mod 8. I'm not going to do p equals 5 mod 8 in detail. I'm just going to leave as an exercise. I'll just say you use 4x squared plus 1, where x is a product of a lot of primes, and this is always 5 mod 8, and all its factors must be 1 or 5 mod 8, and now you can sort of copy the proof we gave before. So that's given some applications of when 2 is a quadratic residue of p. The next case to do is to look at when 3 is a quadratic residue of p, so we want to calculate this Legendre symbol. And now let's recall Gauss's lemma. So Gauss's lemma says that if we take the numbers 3, 6, 9, up to p minus 1 over 2 times 3, then 3p is equal to minus 1 to the n, where n is the number of these, so it's the number of these things here, between p over 2 and p when reduced mod p. So what we have to do, we have to write these numbers down, and we get nought, and then we get p over 2, and then we get p, and we get 3p over 2, and if you write them out, they'll get 3, 6, 9 and so on, and they'll get all the way up here to p minus 1 over 2 times 3, which is slightly less than this. So what we want to do is to know how many of these numbers are in this region here. And that's easy to work out, because the number in this region here is the integer part of p over 3. Sorry, the number all the way up to here is the integer part of p over 3, and the number here is the integer part of p minus p over 2 over 3. So the number in this region here is p over 3 minus p over 2 over 3, where we take the integer parts of these, and we want to know is this even or odd, because this is the number n, and we want to know minus 1 to the power of n. Now in order to work out whether that's even or odd, what we need to do is to work out what p is modulo, well here we're dividing p by 6, so to work out whether the remainder is even or odd, we need to know what p is modulo 12. So let's just work it all out. We can have p is equal to 12n plus 1, 12n plus 5, 12n plus 7, or 12n plus 11, because any prime that's not 2 or 3 must be of one of these forms, and let's work out the integer part of p over 3, well here we get 4n, 4n plus 1, 4n plus 2, and 4n plus 3. So we're just dividing this by 3 and throwing away the remainder, and then we get p over 2 divided by 3, take the integer part of that, and there we're going to get 2n, and here we get 2n, and here we get 2n plus 1, and here we get 2n plus 1, and the difference is, well here it's even, here it's odd, here it's odd, and here it's even. So 3p is plus 1, minus 1, minus 1, plus 1. So this calculates the Legendre symbol 3p, it's equal to plus 1 if p is congruent to 1 or 11, mod 12, or you can write that as plus or minus 1, mod 12 if you want, and it's minus 1 if p is congruent to 5 or 7 modulo 12. Well, now working out whether or not minus 3 is a quadratic residue of p is of course rather easy because this is just minus 1p times 3p, and we've worked out both of these, so we just write it down, p can be congruent to 1, 5, 7 or 11 modulo 12, now minus 1p, the Legendre symbol is equal to plus 1, plus 1, minus 1, minus 1, and 3p is equal to plus 1, minus 1, minus 1, plus 1, so we just multiply these together and we find minus 3p for the residues 1, 5, 7 and 11 is plus 1, minus 1, plus 1, or minus 1. So we say that minus 3 is a quadratic residue, if and only if p is congruent to 1 or 7 modulo 12. Well, in fact, that's a bit of a clumsy way of stating it because you may notice this is actually equivalent to saying that p is congruent to 1 modulo 3, so whether or not minus 3 is a quadratic residue is actually easier than whether 3 is a quadratic residue because whether minus 3 is a quadratic residue only depends on what p is mod 3. As an application, what you can do is you can find that show their infinite number of primes congruent to say 11 modulo 12, and in order to do this you can look at prime factors of x squared minus 3, so any prime fact, any odd prime dividing this will have 3 as a quadratic residue if it's not 3, and therefore must be 1 or 11 mod 12, and now you can sort of copy the proof I gave for numbers mod 8, so I believe that is an exercise. It details because it's really very similar to the case we did for before, so the next application will be a test for where the firman numbers are primes, so you remember we have these firman numbers which are 2 to the 2 to the n plus 1, and we can ask is this prime? Remember the exponent here has to be a power of 2, otherwise this number definitely isn't prime, and it was prime if n is equal to 0, 1, 2, 3, or 4, and as far as nobody knows of any larger values of n for which this is prime, and what I'm going to do is I'm going to give a test that you can use for reasonably large values of n, maybe around 20 or so, where it becomes rather difficult to test this because that would be around 2 to the million or something, so you certainly can't just test all prime factors of this, and the criterion is that p which is 2 to the 2 to the n plus 1 is prime, if and only if 3 to the p minus 1 over 2 is equivalent to minus 1 modulo p, and that's very nice because we can work out this fast using the Russian peasant method of exponentiating, especially because this is a power of 2, so we just need to keep squaring it a few times, so we've got to show two implications, first of all let's assume p is prime, and we want to show this, well if p is prime we notice that p is equivalent to 5 modulo 12 because p is equivalent to 1 modulo 4 because that's divisible by 4 unless n is 0 or something, so we're going to ignore the first two or three small values, and we also notice that p is congruent to 2 modulo 3 because this bit is going to be congruent to 1 modulo 3 and so is that, and the Chinese remainder theorem shows that if p is 1 mod 4 and 2 mod 3 it must be 5 mod 12, well this implies that 3 is not a quadratic residue of p, and by Euler's criterion this is equivalent to saying that 3 to the p minus 1 over 2 is equivalent to minus 1 mod p, so that's just Euler's criterion for when something is a quadratic residue, so that shows that if p is prime then then 3 to the p minus 1 over 2 is congruent to minus 1, on the other hand if 3 to the p minus 1 over 2 is congruent to minus 1, so let's suppose 3 to the p minus 1 over 2 is congruent to minus 1 mod p, we notice that p minus 1 over 2 is a power of 2, and we also notice that 3 to the p minus 1 is congruent to 1, so the order of 3 divides p minus 1 but not p minus 1 over 2 which is 2 to the something, this is 2 to the something minus 1, well that means the order of 3 is exactly p minus 1 because the only factor of 2 to the something that doesn't divide 2 to the something minus 1 is 2 to the something, and so this means that z modulo pz star has order divisible by p minus 1, so since it obviously is order at most p minus 1 it has order exactly p minus 1, so p is prime because if p wasn't prime the order of this would be would be less than p minus 1, so this gives us a necessary and sufficient condition for a firm and number to be prime which is reasonably fast, it means you know that you kind of firm out primes with millions of did sorry firm and numbers with millions of digits and you can test them reasonably fast on a computer to see if they're prime or not. By the way we will just notice that if we compare 3p with p3 we see that this is congruent to 1 if p is congruent to 1 or 11 mod 12 and this is equal to 1 if p is congruent to 1 mod 3, so we see that these two things are actually the same if p is congruent to 1 mod 4 and different if p is congruent to 3 mod 4, this will actually be a special case of the law of quadratic reciprocity that we're going to discuss next lecture, we also notice by the way that minus 3p is actually equal to p3, so this is sort of hinting that there's some sort of relation between the Legendre symbol and the Legendre symbol turned upside down except it's the relation is a bit mysterious, we'll be seeing more about that next lecture. Now let's see what happens if we try a equals 5, so what does, when is 5 a quadratic residue of a prime p? Well what we need to do is to look at the multiples 5, 10, 15 and so on and we want to see that we want to work out the number n that are between p over 2 and p when reduced mod p, so here we have numbers going up from 0 to p over 2 to p to 3 p over 2 up to 2p up to 5p over 2 and here the numbers are going to go 5, 10, so all the way up to p minus 1 over 2 times 5 and we want to know the number in this interval plus the number in this interval, so n is going to be the sum of these two numbers and how are we going to work this out? Well you see the number in here that are here is is going to be 4p over 10 or the integer part of this and the number here is going to be the integer part of 3p over 10 and the number here is going to be the integer part of 2p over 10 and the number here is going to be the integer part of p over 10. So n will be equal to 4p over 10 minus 3p over 10 plus 2p over 10 minus p over 10. And what we want to do is to work out what this is and if you think about it a bit we only need to know whether this number is even or odd and all of these bits the even or oddness will only depend on what p is modulo 20. So what we do is we have this big table p can be 20n plus 1 or plus 3 or plus 7 or plus 9 or plus 11 or plus 13 or plus 17 or plus 19. Now we work out these four terms here. So 4p over 10 will go 0, 1, 2, 3. So we get 3 here because you know we 9 4s are 36 and when we divide that by 10 it's we get 3 plus a remainder. And I'm missing out the 20n divided by 10 because that's just that's always going to be 2n which is even and then we get 4, 5, 6 and 7. And if we subtract 3p over 10 we get 0, 0, 2, 2, 3, 3, 5, 5 and 2p over 10 we get 0, 0, 1, 1, 2, 2, 3, 3 and for this one we get 0, 0, 0, 0, 1, 1, 1, 1. And now we want to add or subtract all these numbers and work out whether the result is even or odd and it doesn't really matter whether you add or subtract them because that doesn't affect the even or oddness of them. So here we here the sum is even, odd, this is odd, even, even, odd, odd, even. So the 5, the quadratic where I'm not 5 is a quadratic where it goes plus 1, minus 1, minus 1, plus 1, plus 1, minus 1, minus 1, plus 1. So we see that 5p is equal to 1 if and only if p is congruent to what we've got, we've got 1 or 9 or 11 or 19 modulo 20. Well you see we don't actually need to work modulo 20 because this is just equivalent to p being congruent to 1 or 4 modulo 5. And in fact this is also the condition for p5 to be 1. So the squares modulo 5 are exactly the numbers that are 1 or 4 mod 5. So in fact we see that 5p is equal to p5. So it's another case of the quadratic reciprocity law which allows you to turn the Legendre symbol upside down sometimes. Well we could do this for 7 rather than 5 but already for 5 it's getting to be a little bit tedious and we really need a better method for working out where the numbers are quadratic residues and this will be the, we want to generalize this rule here into the law of quadratic reciprocity which will be coming up next lecture. Just to finish off this lecture I'll do one case that's reasonably easy which is when is 6 a quadratic residue of p? So we've done 1, 2, 3 and 5 and 4 is trivial because 4 is always a square. So 6 is the next one. Well that we can work out by saying 6p is 2p times 3p and now we know both of these and this depends on what p is mod 8 and this depends on what p is mod 12. So this should depend on what p is mod 24. So let's just work this out. Suppose p is congruent to 1, 5, 7, 11, 13, 17, 19 or 23 mod 24 then we just work out what is 2p. Well 2p goes plus 1, minus 1, plus 1, minus 1, minus 1, plus 1, minus 1, plus 1 because that's 1 for things that are 1 or 7 mod 8 and 3p we've worked out it goes plus 1, minus 1, minus 1, plus 1, plus 1, minus 1, minus 1, plus 1. So if you multiply these together we find 6p is congruent to plus 1, plus 1, minus 1, minus 1, minus 1, plus 1, plus 1. So 6 is a quadratic residue of the prime p where p is not 2 or 3 otherwise these cases are it's not for p equals 3 and not defined for p equals 2 so this is equal to plus 1 if and only if p is congruent to 1, 5, 19 or 23 mod 24. Okay so next lecture as I've said will be on the law of quadratic reciprocity which will make this process a lot more systematic.


We're going to kick off the final podcast of the year. Technically this will be episode 207, the final podcast of 2021. And we're going to go back to the most important conversation or discussion or thought that we can have in most areas of our life. And that is why. We're going to talk specifically about why calisthenics. And you know if you know who Simon Sinek is, you'll have probably come across is a very famous TED talk about knowing your why. So if we understand what our why is, then we can understand why we do what we do and we can probably get more enjoyment out of it. And when it gets hard, if we know what our why is, then we've got something really strong to lean on. So we're going to talk about why calisthenics today. And when we're talking about training, we're talking about calisthenics.  And there will be, you know, a whole host of things that you'll have seen. If you're like us, we get like excited about, right, I'm going to try and do these, do these things. And then it's like, right, we then start to potentially get frustrated when we don't see results straight away. Or we see results on a few things and we think everything can happen really quickly. And the reality is if you want to do some, some cool things with your body, you want to make some changes for your training. What's important is the invest time in making that process happen. And also we believe it's very important that we are following a process that we can trust. And that's what everything we've got inside our online training programs for online members is all about. And the annual offer that we're running is 25% off either our standard membership or VIP membership. The standard one makes it less than 75 quid for an entire year of training, giving you the opportunity to go right. I'm leaving. It's not about quick fixes for me anymore. I understand what the guys are saying. I'm on board and I've got to commit to this for at least a year. The investment is in time, not in the money. It's 75 quid for a whole year of training is not the challenge. The challenge is make that commitment in terms of the time you're going to invest in understanding your training with us and actually getting in and doing the work with us. But no, you've got a whole year to get through it. And there's far more in our online classroom than just a number of workouts that you often going to find of just plug and play kind of things, do this workout today and have absolutely no idea why you're doing what you're doing. See, it's on point where I'm going to have this conversation today. But we're actually, we're a school, right? So we've been, we've put a lot of time and effort into creating... We're not an actual school just for people that don't. No, no, no. We're not an institution if someone wants to. We did used to get that actually. We would get, how do I get a scholarship to your school from people from abroad? Anyway, but we did put a lot of content and like just snippets and little bits of kind of like soundbites that you can take throughout our training program. So as you go through our program, you're actually going to learn and get loads of really important, useful like tip bits that you can then apply to other parts of training. So when you're not doing a human flag program, for example, you probably understand why time on detention or isometric strength is going to be important. And you can take that into whatever else it might be that you're going to go and train, whether that be deadlifts or like rack pools or bench press or whatever it might be. They are just, it's full of training science. So if you're the sort of person that wants to train, wants to get really high quality workouts by expert coaches, but also wants to know what you're doing, there is probably no better platform for you in bodyweight training. Say what, Tim? Not only 25% off, how about we give them seven days as part of a free trial so you can test it out? Yeah, it's a way, come on. Right, seven day free trials to test it out for a week and then, you know, then you're in for a year. Let's talk about the why and then people can go on and start taking some action. So sit back and enjoy this chatting about why it's going to get philosophical, philosophical people on the movement, strength and play podcast. Roll back jingle. Listen, players. You're listening to the movement, strength and play podcast by the School of Calisthenics. Here are your hosts, Tim and Jaco.  Why Calisthenics? Why Calisthenics? And I think it's a super interesting question. It's a, there's something I think I totally agree with you in that like the understanding why is important, but understanding like what is it about it that's important to you. And then also, and this is what I want, I'd really like our conversations to go in the line of like, not just why it's important to me, but what does, what's the like iteration of Calisthenics to me? Like what, you know, Calisthenics means it comes from two great ways. Calisthenics means beauty and strength. And it's like, okay, so it actually doesn't, you know, like if you say football, like you can, it has a set number of rules. You go CrossFit. Well, there's a CrossFit as these things. And like whatever, whereas Calisthenics, it actually, if it means beauty and strength or movement and strength, like, well, then it could be, and, you know, by sort of by, we can chuck in body weight as a bit of a depth into that definition in a way of like, it's body weight training go. Well, so actually that could look like a lot of different things to different people. And I think that's one of the beauties of it and one of the whys of it for me as a, as a coach, because it's not like, okay, this is, this is, this is Calisthenics. And like you need to do it to the, to our members or the clients. It's like, no, this is Calisthenics.  And for someone else, it's these other elements. And then you can, you know, it's not, that's one of the things I love about it. It's so, it's so malleable. It's so, it's so wide that it really can let everybody find which bits of it resonate most with them, which bits of it their body needs the most. And it's not going to constrict us to stay in one, stay down one path. And I think for my own learning over, not just the last year, since we started trying to do some Calisthenics in like 2014, which started for me as like seeing Frank Madrano looking ridiculously rips doing like insane things that you're like, where's gravity gone? Why is this dude not playing by the rest of the rules that the rest of us believe by in science of like gravity? And that like just captured my imagination of like what the body could do. And it's probably been a bit of a ever growing iteration of that, but it's changed over the years from, and for me now it dows much more into like, what does my body need? And listening to my body and like Calisthenics for me doesn't have to be the same as what Calisthenics is for Frank Madrano. But there might be elements that we're the same and we're going to, there's going to be certain bits that we, that we all love the same and joins us all together. But understanding what it can give my body and how it's such a usable training method is where I've got to. And that's one of the things that I would encourage people to do is like think about not just the why is it important or why do I like doing it, but like what elements of it are going to serve me the most.  I'm going to go a level deeper with this one. There's the process of the five whys isn't there? So like if you say why do you like Calisthenics? You go, okay, because it makes me feel pretty good and it gives my body what it needs. You go like, well, why? And then you keep asking five times and then they say that you're going to probably get down to the crux of what it actually is. So I'm just thinking about some people, what is the attraction of Calisthenics? So if I think things I like, I like the flexibility of it like why? So I don't have to go somewhere all the time. I'm worried about a gym. I can take my gymnastics rings and I go, well, why? And you keep trying to drill it down. And I think that's the thing like if it gets down to a lot, I want more time. Okay, well time is a precious resource. Why do you want more time? Because I've got family and kids and I've got a business. Okay, why is that important? Well, those things need more time. I want to grow them. Like it's that whole thing is sometimes people are like, why do you want to go to a gym to train? Well, because it just gets me out of the way from those things. I need space for myself. So I think like just asking yourself what it is about why you do what you do. And if you say, well, I do Calisthenics because it makes me look a certain way. Or you like, I do Calisthenics because when I'm around people at a barbecue, I can do a handstand. Like, why is that important? Because you like attention. It's okay if you do. Like there's no, Jaco is perfectly on board with doing things for attention. Right, so. Why learn human play? Exactly. So I just kind of like reflect on why do I do Calisthenics? Well, it is a massively accessible. I love the fact that you can give me a set of gymnastics rings and I'll get as good a workout as anybody in a fully equipped gym. And then my rings cost me 30 quid. And the why behind that is not that I'm tight. It's just, it's like, we all like things that are good value. Like, and if you, and I really like the idea that I can just go and find a tree to hang on. Like that for me is like, it brings it into the what kind of physique and body do I want. What I want one, which is, it is functional. I was chatting to Andrew Tracy about this actually recently. We like going, the most functional thing that you can do is not being pain. Like, so when people kind of go, I'm doing functional training, but my knee hurts. Well, that's not that functional. Is it because what you're doing is now making you more dysfunctional in life. So I think the flexibility of what you get with Calisthenics, the management of your own body weight, particularly around the upper body. It's just like you are only ever able to play in the field or to a level of which you are able to play. So your body is going to tell you when you are strong enough to do something the next level. And you really have to earn that. And people will argue that you can do that in weightlifting as well. That you can only go and do the next weight when you've built the strength. But the thing that's different about Calisthenics is it respects the whole chain. There is a movement component to it. You can bench more weight and you can still have a horrible shoulder range of motion, inflection patterns.  But the thing with that thing around it is you just, it really is, you have to have, you have to move well. You have to be stable and you have to be strong. And when you can do those things, then you get to kind of go and do more. And I think like, I just see a point around like what it looks like for each individual person. Calisthenics for me over the last year particularly has not been about skills. Like I don't have a burning desire at the moment to learn new skills. I like playing around with the skills that I've got. But the focus for me has just been, and it's probably to do with just the pandemic and being training at home and the kind of the repetition of it. Whereas I just, skill training for me has become hard because I need to apply so much mental energy to it. So the level that I'm at now, like it's, and it's the same for people when you just started out, but you don't get those quick wins like you do when you're a beginner. So for me to go and learn something, I've really got to dial into that process because what I'm trying to do is actually like pushing, pushing my physical capabilities and limitations. Whereas just getting away with some relatively more basic volume training. Like I like how it makes you feel. I like how it makes me look at the book and Bendy program. And that for me is like that's ticking the boxes at the moment for me because I don't have the bandwidth for it. Why I don't have the bandwidth? Because life has been pretty mental for the last two years. So I think it's just interesting points and I like that. There'll be some really good comments on the community recently of people just going, yeah, I've also just taking a step away from smashing myself with plans training or muscle training because it's just feel like I need to break from it. That's perfectly okay. There was a time when my identity was completely wrapped up around calisthenics at least of being able to do certain things. It's not like that anymore. Yeah, there was a forget the forget the traps name. Certainly a message this week about it. Been working on the frog to handstand for quite a while. Sort of identified that needs probably a little bit more thinking to listen to the mastering the basics podcast around like a kind of get maybe I think I need to build my foundation strength a little bit more like just dial back. He was Nick or Dave have a little break from like the the specificity of the frog to handstand and is looking forward to the six week. But a basic program in in January to just have a like, okay, make a commitment or I'm going to do those six weeks build a base and then I'm going to come back to and use that basic strength back in towards the thing that I'm quite excited about doing. And it's my sort of like goal that I've set of the of the frog to handstand. And that's that's that's a process we've been through multiple times. I've had stuff where I've not done something for agents left it alone.  So long as we've in this periodization, right? We talk about that. We say that's periodization different phases of training to accumulate together to achieve something. Yeah, yeah. And it's and that's one of the things where even just for some people, even just having like a little bit more regulation around like D load weeks or rest weeks. You know, hence why in the periodization that the programs inside from my members that we've got where we have this like these modules that are four weeks long that build up progressively and then you know week week four is a D load or sort of like can be of could be a table rest week if you want to. But having that little bit of structure for us when we're trying to do some of these things is just massively helpful. Otherwise, it's far too easy. If you haven't got a coach or a trainer or you're not working as part of like a team within a system where it's like you could train every day if you had time and and never have a week off. And what you'll find is that the body just needs these sort of like peaks and troughs in terms of being able to then actually recover in and then perform again. That's an interesting question because we've had that a lot over the years of people who don't want to take weeks off.  D load week or rest week or whatever it is and ask yourself why you don't want to take a D load week. Well, because you've got an addiction to training, you don't feel good unless you train. We all have that. We need to learn to accept that rest is part of that. You go, why on that? And then that then you go, why am I okay? Because there's some, I don't know. You go, why am I addicted to training? Then then then you start asking yourself some some quite challenging questions. Yeah, for me, because I've talked about this before, I have this issue because I have a body with dysmorphia problem. If I don't train, I don't think I feel like I look good. Like that's why I often will keep training and keep pushing because that's where the root kind of comes from. I've got mental health. Basically, No, no, no, you're being honest. It can be lots of people out there the same and then you also go, if you don't, if you want to take a rest week, your why is that you don't understand training principles that well. So you need to educate yourself better because you cannot progressively overload and continue to get better. It is a physiological fact. You have to allow the body to recover. That's how it works. So if you don't D load, you're just grinding yourself down and actually you're stunting your own potential you're not allowing the opportunity for the body to adapt to stimulus and therefore move you forward. So there's so much within this. I've just spent a bit of time when I've been quiet since down a little bit and gyms are closed and you have a rest week and you're probably now sat here worrying it because you've eaten too much chocolate over the last thing you're like and feel terrible. I'm probably one of those people. But just I think there's a real value in just sometimes going why do I do what I do right now and you why can change right a little bit sometimes doesn't have to be the same all the time. That's there's there's I've got my I've got my two takeaways from today's podcast. One, understand your training enough as to why you need to have deloads and why that's actually going to help you progress. And then and what you said the ask ask why ask why five times or maybe I'm almost thinking actually ask why enough times until you start to feel uncomfortable because I think when you ask it you ask it enough times you'll get to a why of like now I feel uncomfortable myself that question of why because it's actually it's making me think far deeper than I previously thought about myself in the drama I've gone way below it becomes hard to answer like if I did that with you now to five things you probably get to two or three but I don't know I need to and then you're gonna do some work so you take an example you go why do you want to do calisthenics because I want to learn a handstand why do you want to learn a handstand because I think it looks cool okay so why is that important to you well I kind of want to feel what I can do with my body I want to teach myself something new and I probably quite like it when other people look at me and think I can do something cool and okay so like where's that coming from and that's okay to socially have that and to be different and to be able to do these things like that's that's not a problem it's just it's well for me you go like calisthenics. It's a point when you get back to stage you go where you're having that blip in your handstand training you hit that plateau which will happen where you have to grind through it knowing why you started to do a handstand in the first place will probably go a long way to serving you through to actually finishing the process we talk about that athletes all the time why do you swim because it's hard right you're in that mid-cycle point it's gruesome it's five o'clock in the morning when you're going to pull why do you do this and it's that which they got to lean on when things get hard to get you through to the end which when you can do a handstand for the first time or on request or you can do it as a showboat you are going to be absolutely loving it and thankful that you saw that process through in all terms from knowing why you started you just maybe think I didn't I wasn't going to make another point but just you made a really good point there that I want to just highlight of like Calis we've talked a lot about calisthenics being a great tool for like teaching you about your body you know what can you do with it what can't you do with it and why and then when you were talking when you were going through that series of like asking those questions why and you got down to like a point with that like handstand example of like like someone potentially I feel nice when people I'm perceiving that people think it's cool that why I can do this thing that I also think is cool and it's like why is that and it's it's then sort of teaching us something about our that process of asking those questions of why it's like teaching us something about ourselves and our sort of psychology if you dare go that deep of going like so what why is that why is my self confidence like that and what's what yeah you may not get to the actual bottom of that but I think it's it's useful to be vulnerable enough with yourself to like to at least go there and entertain it yeah yeah did the many dimensions let's leave that next yeah let's go and show I'm gonna keep it short this week because it's why I'm gonna go and think about why I yeah somebody we're gonna write why do I do well that's you that's cool go and find something else that fulfills you in and brings you that joy and excitement and happiness that's what it's all about right that's it when you get down to it like what do you enjoy yeah that's a pretty good thing to that why do I why did you do that because I enjoy it okay well what why do you enjoy it really annoying with it go and do enough work to a place where you feel a little bit uncomfortable and you've just found out something about yourself and and then just hold on to that when you go and when times get tough it will see you through and we just want to sign off by saying if if you're why it brings you down the roots are going like okay the thing I'm the thing I'm gonna try and do is probably gonna take me more than just a few weeks having that annual membership with us for all of our online programs for just less than 75 quid on the standard membership with that 25% off you can bag that with the code for the discount code for that is new year 25 so new year 25 all in capitals and the link is in the show notes so you can just click straight through to grab that offer and we look forward to seeing you online within the online community of amazing fate people that are part of our memberships and we look forward to seeing you in there perfect enjoy new year I believe this is going up between Christmas and new year so until we speak to you again keep exploring your physical potential with movement strength and play class dismissed until 2022


Tell me were they more males or females on that list? Well, it was a bit of a trick question. I think We had 48 Names in that list 48 faces 48 celebrities actually But of the 48 celebrities we had half of them were male and half of them were female and so There was no correct answer For the quiz but what we did do was we took the male faces Right, so the 24 males half of those 12 of those were famous celebrities So people might remember from the list. There was Brad Pitt Tom Cruz Steve Jobs right people things faces people that most everyone in the course would would recognize Now the thinking is here with this sort of experiment has been done time and time again But the idea here is that my bet would be that most people would have remembered the 12 males that were in that Original list, maybe not explicitly, but they may have remembered them And when they were thinking back to the list of males these people may have stood out now The female list the 24 of the females were all B grade celebrities for the most part So a few of them may they may have remembered but not many at all right, so when you're trying to make an estimate of Males relative to females in that list of faces Hopefully some of the males would have come to mind more easily And and the idea here is that it's that ease of processing that ease of sort of cognitive processing When thinking back to the males that went down a little bit easier and so people would would misinterpret That ease of processing for for the category being larger than the female category But are people actually remembering do they have a list of the celebrities in their head today? Do they do they remember Brad Pitt is that on the internal list not necessarily? I don't think it has to be explicit now There's another really good example that kind of highlights this what we can do is we can show students in the course these two letter strings Now the the letter string on the top and the letter string in the bottom and so what we want them to do is Estimate given these two letter strings how many words you can construct given the letters in these letter strings okay now even without Having any of those words come to mind People will quickly recognize that the letter string at the bottom will produce more words in the letter string at the top Okay, so it's exactly the same principle It's it's both of these examples are examples of the availability heuristic, right? Which is simply the number of instances that come to mind of that particular category and people Misinterpret the ease of processing Which could be due to any number of reasons as being indicative of the larger category I think this happens a lot in the media so we When we hear about deaths we tend to hear about shark attacks and and plane crashes and Terrorist attacks. We don't hear much about the people who died that night on the 6 p.m. News of asthma or heart disease So I'd be willing to bet that that people would pay Far more far higher insurance premiums to protect themselves From the things that they hear a lot in the media about how people die Versus what what what the base rates might actually show that people are most likely to die of now We have an example of this a couple of years ago in Brisbane There was a reasonably major flood and you and I had to to leave our homes and find someone else and it was a big mess And it took a couple of weeks to clean up, etc Now after that event I'd bet that the people who were directly Affected like we were by that flood and now far more likely to buy flood Insurance than people who are unaffected and I think they'd be far more willing to pay higher premiums for flood insurance But also over time as our memory fades of that event I think that willingness to pay those high premiums will will drop and people will be far Less likely over time to pay a lot for flood insurance Exactly, and so it's this kind of link between Availability the availability heuristic and risk perception has been demonstrated time and time again, and that's Absolutely the case and so in the media. Well, obviously the media is Extremely important in shaping our perceptions and the decisions that we make. I mean if all we hear about for example is As you said about homicides and about you know These sort of newsworthy deaths as opposed to things like heart disease or liver cancer and so on then we're going to have Enormously skewed sorts of perceptions about how common these types of deaths are and in fact That is the case if you were to ask people how likely it is that someone would die from a shark attack They would say that it's you know way more likely than it actually is At the expense of things like heart disease and so on which people really underestimate because they never hear about them You might hear about them with respect to family members or you know friends or something But when you're bombarded every night in the media by by these things. It's yeah exactly and so the media really shapes that and Danny Kahneman talks about this this idea of availability, but also the idea of availability cascades And so if there's a relatively minor event that happens say a tremor or something like that and and You have this news agency Who kind of blows it out of proportion makes it larger than it actually is then people start to freak out a little bit more Which then feeds more coverage Which results in people freaking out a little bit more and then it just kind of cascades and it gets worse and worse And in fact he talks about these Availability entrepreneurs these these people in these news agencies that kind of make a living out of doing this and so I mean that It's it's never been more important I think to to consider what's where we're getting our information and What sort of information we're exposing ourselves to because that fundamentally shapes the way that we see and perceive the world and if all we're doing is surrounding ourselves with kind of Second-rate news then we're kind of at the at the whim of these these types of agencies


From VOA Learning English, this is the Technology Report in Special English. Every year, IBM chooses five new technologies that the company believes will change the world within the next five years. The list is called Five in Five. IBM says it considers its own research and the new directions of society and business when identifying the technologies. This year, the list describes devices that will extend our five senses. Imagine looking for clothes online and touching your computer or smartphone to feel the cloth. IBM's Vice President of Innovation, Bernie Meyerson, says this technology could be available in the next five years. Touch is just one of the senses that technology will help to extend. IBM says smart machines will soon be able to listen to the environment and give us information about the sounds they hear. For example, Bernie Meyerson says an advanced speech recognition system will tell new parents why their baby is crying. Smart machines will also help identify medical conditions. If you sneeze on your computer or cellphone, the device will study the molecules that came out of your mouth. It will then tell you whether you need to see a doctor. Another prediction is that computers will be better able to not just look at images, but understand what they show and act on that information. Bernie Meyerson says IBM scientists are also developing a computer system that can examine and combine molecules to create the most popular flavors and smells. Mark Maloof is a computer scientist at Georgetown University. He says it will be exciting to see what young people will do with the mobile technology, networking, and computing power of tomorrow. For VOA Learning English, I'm Carolyn Presuti.


The Montero Medical Mission from the state of Virginia visited the island from the 16th to the 20th of September with the goal of providing human service to members of the public in need of dental services but who could not afford those services. While here the medical mission undertook a virtual dental van tour as it was called to all 77 primary schools. During that tour members of the delegation engaged the students during school hours via their Zoom platform with illustrations on proper tools brushing and a general oral hygiene. The mission also visited the Denver Hospital and Miku Wellness Center where they carried out dental procedures on both children and adults. We're offering a lot of services here for children that need it. We're doing cleanings, extractions and fillings for these children. We're doing the best we can for what we have and what we're using here. So I think there's a need here. There's a decay.  So there's a need for pediatrics here in this area and that's why we came down here to try to help. Dr. Martin is appealing to parents to cut back on the amount of sugar that a child or children consume. Sugar is just causing decay on children so it's the limitation of sugar and the actually the proper oral hygiene that help eliminate a lot of the problems. So that's we're doing instructions on dietary issues too as we see the children here. For Dr. Charles Johnson it was a rewarding trip to St. Lucia although he twice visited the island whilst on the cruise. This was his first work trip here. I have been all over the world. I've been the manager of the Paraguay, Ecuador but this was seen special for me because we are providing the service that the people were appreciative and they seem to want us to help them and the community, the Rotary and Alliance group, they really made it easier for us to work here and hopefully we can come back. The Virginia-based Montero medical mission was headed by its founding member and the president Dr. Juan Montero from the communications unit of the Ministry of Health, Wellness and Elderly Affairs to Lita Peter reporting.


I love dancing. We're the storytelling that we get to do. Alex came into the office. MRI indicated that there was an ACL terror. To not be able to dance was extraordinarily depressing. This is something that you can get through and get back to doing what you love to do. The care that I've experienced through this process, I feel so grateful for the opportunity to have like a second chance to pursue a dream career that I've dreamed about ever since I was a little girl.


Oh, I've been trying to record this several times now, and that's been getting frustrating. It seems like there's something wrong with the SD card and the camera that I've been using. So I pulled out my older one, because I know that definitely works, especially since there's no SD card, it's just the main memory. So, yeah, frustrating. Two days ago, when I meant to first record this, I had put out the article for the F-sharp Advent. I'll have a link down in the video description for that. If you've watched the, at this point, the only F-sharp video I've done on this channel, you've seen some of the content in there. I do provide some additional information and it'd be worth reading, even if you've already watched that. There is quite a bit of additional information. The update, the tutorial video that I have planned for Stringer that I've kept mentioning, I've run into so many ridiculous snafus on that. Oh my God. So, originally, as I had it planned, it was supposed to use a database and do just simple crud stuff where the text was validated on the client side using Stringer. So, set up the database file, connect to it using Visual Studio, get the connection string from Visual Studio and put that directly in. Of course, everything works. Those of you who are more familiar with using databases than I am, because I don't really do much that involves persistent data, you know what the issue is there. I didn't, but you do. You can't do that because the connection string winds up being a fully qualified local path. So, if you try to move that onto a different machine, unless you have the exact same username and checked the repo out in the exact same location, it's not gonna work. Like, all right. There's gotta be ways of making this path relative. So, I look into that and from quite a few answers on Stack Overflow, it seems like it is. I couldn't get it working. I don't know if that's a case of me not getting the stuff right or Stack Overflow answers being wrong. Both of them happen quite often. So, I don't know. I just know that that's not working. But then, as I'm looking into it more, realize that there's a bit of an issue if you don't have the libraries for working with that specific type of database installed on your machine. Oh, and like, all right, this, the point of this really isn't the database. The point of this is just that there's some kind of persistent storage and you're using what I developed for the input validation. You don't care how it's stored. This isn't an actual business app. It's just showing off how to do the validation so we can do a different type of persistence. Okay, no, XML is one option, but oh my God, is that crufty and hard to work with and you'd have to design a whole schema and I just know, there's another thing that XML stuff will apply to JSON and other stuff which are better to work with but you still have to supply your own schema. There's another approach which is definitely a very niche thing. I'll have some stuff to say about that in a second, but file system as a database. The idea that you're using the directories and files specifically in a way that it works like a database. It's niche. Please do not do that willy-nilly. Even if you look into research around it, you'll find that it's very niche. The benefits that it's able to provide are definitely edge cases. The way you do it properly is not at all obvious. There's a lot of areas to trip yourself up and there's good reason why you don't see FAD commercial products as it's just so niche. But for this kind of thing, you're not really concerned about how the data is actually being stored. It's not a business app. You don't care if you lose data. You don't care if things get malformatted sometimes. That's not the point. It's demonstrating something else. This kind of thing would work perfectly since obviously file system paths can be relative. I've never seen an operating system where they can't. So that works. I go and implement that and the whole thing's working. I'm ready to go. I'm getting ready to record it and the announcement for the release of .NET Core 3.1 happens. Now, I'm not gonna forget to record my video but I might forget to do that update. So I do that update before I forget because it's not gonna take terribly long. Go to record my video and it just stops working.  So I don't know if that means that there's a bug that was introduced in .NET Core 3.1. The runtime, I didn't update the project dependency but the runtime could have introduced a bug. I find that unlikely though. The other possibility is that I was depending on behavior that should not have been working and only was working due to a bug in the previous runtime. We're dealing with file operations and not any special kind of file operations. So I find that unlikely. But regardless, it's not working. I'm getting these exceptions about how a process is already accessing the file when I go to access the file and I can restart the machine and all sorts of other stuff to make sure nothing's happening. I can spin up a VM that is completely brand new that has nothing going on in it. Clone the repo into there and it says the same thing.  So I've been trying to figure out what is going on there for a while now and just yesterday towards last night I finally figured it out and it is so frickin' stupid. Remember how I said FAD has, it's very niche. It has a lot of pitfalls. There's a lot of things about it that are a bit weird. One of those things is that you cannot, under any circumstances, operate the thing with a debugger attached. What happened is I know that. So when I was doing the development originally, I made sure that it was always in release mode so that any time I went to run the stuff, the debugger would not attach and I could go in and check the files manually and it worked. It's a bit tedious but it works. I winds up not being as big of a deal as it might sound in practice because unit tests, make sure you have a lot of unit tests when doing it but again, it's very niche thing. Don't do it unless it's like a research thing. What are you doing? But what I realized is when I went to do the update, I sort of reset everything when I went to open up my projects again. It went back to debug mode which is the default. So I didn't notice that and kept trying to run the things with the fricking debugger attached. So yeah, it works now. I made sure that you can put it on different paths within the machine, on different machines, all sorts of shit as long as you do not run it in debug mode, it works. So I'll have a video on that fairly soon, especially since it works. I have some stuff to take care of today so I don't know if it can be done today but I will get that out as soon as I can. That's pretty much it, this little update. Again, if you are interested in that article for the F sharp advent, it's down in the video description. That's yeah, have a good one.


Mae'r arddur, yn ddod, i'r gweithio i'r ffrindiau a'r gweithiau, ac yn llwyddo ar gyfer ymddangos ystod. Mae'n gweithio i ddod i chi fod yn dweud ar gyfer yma yn yr awdurdodau o 2019. Felly, mae'r gweithiau yn ystod yn fawr o'r ddwybr. Fe'n gweithio i'r gweithiau ar gyfer y gwaith. Yn ceisio i'r gweithiau ar gyfer y gweithiau, ond mae'n gwneud ei ddwyf, ac mae'n gwneud ei ddwy i'r gweithiau. I had the opportunity to study modules that I wouldn't have had the chance to do at other universities and it allowed me to explore other regions of the world that I would never have had the chance to do. It's not only very interesting, it allows you to go into the workforce with the skills and the theory of economics and you are able to apply it to a specific region which is beneficial in many, many ways. Studying economics at SOAS has given me the confidence to argue and share my ideas. I feel the faculty members here actually care and make sure that you understand the things that you're supposed to understand. I remember when hosting an event called Culture Night, we collaborated with different societies. So that's East African, Somali, Yemeni, all different Sudanese, where different cultures collaborated showing different cuisines, different musical aspects. And for me, no other uni can offer that other than SOAS.