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Maybe my question is very trivial. I would like to have the definition of "number". Can anyone advise me some documents online?

thank you very much

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Try starting here? –  Old John Nov 27 '12 at 12:59
In the first place, a number is an element of some number system. –  Christian Blatter Aug 9 '13 at 9:42
A number is a theoretical concept that encodes universally intrinsic data in a form that is profoundly meaningful. (This actually might be one for a philosophy forum) :D –  S Valera Aug 12 '13 at 23:39
Cognitive scientists Lakoff and Núñez wrote a book called "Where Mathematics Comes From" on this topic among others. –  Andrey Sokolov Aug 13 '13 at 1:14
An alternative to set-theoretical definition of numbers is provided by homotopy type theory homotopytypetheory.org/book. Although, the answer it gives is not trivial. –  Andrey Sokolov Aug 13 '13 at 1:22

7 Answers 7

up vote 9 down vote accepted

As some people already pointed out, the letters "1", "2", "3" and so on are just labels for a mathematical concept. However, this mathematical concept can be made precise, of course, by mathematical set theory. This is an overview of the precise definitions of numbers:

The natural numbers. What are the numers 0,1,2,3,4,...? Mathematical set theory gives the answer: \begin{align} 0 &= \emptyset \\ 1 &= \lbrace \emptyset \rbrace \\ 2 &= \lbrace \emptyset, \lbrace \emptyset \rbrace \rbrace \end{align} and so on. More precisely, if $x$ is a set, then we define the operation $'$ by \begin{equation} x'=x \cup \lbrace x \rbrace \end{equation} and say:

A set $x$ is a natural number if and only if $x$ is element f every set, that contains the empty set and is closed under $'$.

That means: $0$ is just a label for the empty set. $1$ is just a label for the set that contains the empty set. $2$ is just a label for the set that contains the empty set and the set that contains the empty set, and so on. This is the precise notion of a number. The existence of these sets is guaranteed by the axioms that mathematics is build on: the axioms of set theory.

The integers and rationals. Now what about the numbers $-1,-2,-3,...$? Now it gets slightly more involved. First (by the recursion theorem, that also follows by the axioms of set theory) you have to define $+$ on the natural numbers. The integers are then defined by "equivalence classes of ordered sets of natural numbers". More precisely, two "ordered sets" $\langle a_1,b_1 \rangle$ and $\langle a_2,b_2 \rangle$ are equivalent if $a_1+b_2=a_2+b_1$. Here we imagine the ordered set $\langle a_1,b_1 \rangle$ to mean the difference $a_1-b_1$. One then has to show (by complicated techniques) that the set of natural numbers is isomorphic to a subset of the integers.

Similar, one finds the rational numbers as equivalence classes of ordered sets by $$ \langle a_1,b_1 \rangle \simeq \langle a_2,b_2 \rangle \Leftrightarrow a_1 b_1 = a_2 b_1$$ after defining multiplication on the integers (again by recursion).

Again, e.g. $-4$ is just a label for the equivalence class of $\langle 2,6 \rangle$ using the first equivalence relation and $\frac{1}{3}$ is just a label for the equivalence class $\langle 1,3 \rangle$ using the second relation.

The reals. Now what about numbers like $\pi$ and $e$, that are not fractions? This is even more complicated. We first define $$ R = \lbrace f ~|~ f: \mathbb{N} \to \mathbb{Q} \rbrace,$$ i.e. the set of all sequences in the rational numbers. Furthermore, we denote by $C$ the set of all Cauchy sequences in $\mathbb{Q}$. The reals are defined by the equivalence classes $$ x \in R \simeq y \in R \Leftrightarrow x-y \mbox{ null sequence}.$$ So $\pi$ is just a label for the equivalence class of all Cauchy sequences that converge to $\pi$.

There are even more classes of numbers. The complex numbers can be found by similar techniques. The cardinal numbers are "infinite numbers" and can be introduced to measure different "grades" of infinity.

To make all the notions above precise you can consult any good book on set theory, e.g. this one.

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What is a dog? I guess it's a three leter string, where we can permute two letters to make it into a god. What is a number?! No, I'm just trying to be a douchebag, good to have the information you provided in your thread ;) –  NikolajK Aug 9 '13 at 7:34
Defining the Real line using Cauchy sequences is slightly 'loopy'. Better to consider a real number as a ray (a bounded-from-above subset of the rationals with the property $t\in X\wedge s<t\Rightarrow s\in X$). –  Jonathan Y. Aug 9 '13 at 23:25
@user142526 No, it's not. You consider Cauchy sequences of rational numbers, that you introduced beforehand. –  mjb Aug 10 '13 at 8:28
Hence I said "slightly", because you can get around it in the manner you described (but then, we'll defined Cauchy sequences of rationals separately, disregarding general metric spaces). A matter of preference, I guess (orderings or lack thereof seem more integral to field theory). –  Jonathan Y. Aug 10 '13 at 8:43

Number means different things to different people (and even to different mathematicians). When we count objects, we number them, so in that context, a number is a label or word attached to the idea of multiple things, so $1, 2, 3, \cdots$ are numbers. It takes a small leap of faith to include the absence of objects as a numerical concept, which gives us $0$. When someone writes "463", we understand that to be a number, but it is written using three numerals (which are often just called numbers themselves for simplicity's sake). But that "463" could be a number written in red in the debit column of my budget, in which case, I know that it was \$463 spent rather than received, and so we can label numbers by their "direction," whether positive or negative. Thus $-463$ is also a number, but it's important to realize the expression simply labels some concept that was useful (debit of money, for example).

We also write numbers on a line at equal distances from one another (a ruler), and this object can be used to compare lengths of objects. Of course, most objects will have a length falls between whole numbers. The ancient Egyptians (among others) solved this conundrum by thinking of parts of a whole number, that is, fractions. So for example, $1$ has 3 equal parts, each of which has "length" $1/3$. These fractions $1/n$ are also considered numbers. You get the picture here....

Of course, number usually comes along with operations, such as addition, multiplication, and their inverses. Two whole numbers added or multiplied together give another whole number ($\mathbb{N}$ is closed under $+$ and $\times$). However, one needs the negative numbers to have inverses for addition. One needs to include all fractions $m/n$ in order to have inverses for multiplication (of course, we still can't get around the fact that $1/0$ is undefined, and so we don't usually consider $1/0$ to be a number).

At any rate, this post is getting much longer than I intended, and I'm in danger of clouding my point.... A number is just a label for a concept that may not have a definite definition... which starts as counting objects but is extended by the needs of various people and/or mathematicians. As a final example, in my field of mathematics (algebraic topology), the objects that we play with are spaces (think of surfaces, spheres, the torus, etc. etc. etc.), and in many ways, we can think about adding spaces, multiplying them, and doing many other things that seem to be operations on numbers. So, in a highly strained analogy, spaces can be also considered numbers.

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A definition of "number" should express this notion in terms that do not depend on it and are, in a sense, more primitive. The only such definition that I know of belongs to Francis Lawvere and is based on category theory. It does presuppose that we know what a "set" means.

A "number" object (in the Set category) consists of a set $N$, with a distinguished element called zero $(0\in N)$ and an endomorphism $h:N\rightarrow N$, which is has the following property. For every structure $e\in X\xrightarrow{g}X$ exists a unique function $f : N\rightarrow X$ such that $f(0)=e$ and $f(h(n))=g(f(n))$ for each $n\in N$.

This can be expressed graphically by saying that there exists a unique function $f$ which makes the following diagram commutative: $$\require{AMScd} \begin{CD} 1 @>0>> N @>h>> N \\ @VV{id}V @VVfV @VVfV \\ 1 @>e>> X @>g>> X \end{CD}$$

Here $1$ is just a convenient notation for a one-element set.

From this definition also follows that $N$ is determined uniquely up to isomorphism.

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says Wiki. I like it to think of a number $n$ as a tag you put on a bag containing $n$ things.

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I would agree with "label" and perhaps even "measure," but if $n$ is just a tag for bags containing whole numbers of things, then this could be construed to mean a number must always "count" things. –  Shaun Ault Nov 27 '12 at 13:18
I was thinking of $\Bbb N$atural $\Bbb N$umbers... –  draks ... Nov 27 '12 at 13:27
A number green is a tag you put on a bag containing green things ;) –  Jostein Trondal Dec 29 '13 at 21:48
now that X-mas is over, my bag is pretty much an empty set... –  draks ... Dec 29 '13 at 21:50

The general concept of number resists definition. Set theory includes rigorous definitions of two types of number - cardinal numbers and ordinal numbers. (Cardinal numbers are probably closer to your intuitive idea of number.) Frege spent a lot of effort with the general concept of number, but ultimately failed.

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The definition of "number" cannot be separated from history and development of language. Why do you call "1" a "1"? Why do you call a "tree" a "tree"? I am not an expert in history, but I will try to use common sense to reconstruct the history of numbers.

Natural numbers are so called because they are natural in counting identical objects. Let us try to count sheep by drawing lines. Suppose there's only 1 sheep. You draw 1 line on a sheet of paper. Suppose there are two sheep. Then you draw 2 lines on a sheet of paper. When developing symbolic language, you arbitrarily decided that a sheep is isomorphic to a line. You may have very well decided it is isomorphic to a circle. It is more convenient to draw 1 vertical line for each sheep than it is to draw a sheep for each sheep. You can use it to represent stones, coins, fishes, etc. Anything. A vertical line is simply a representation of a real world object belonging to a certain class. This class of objects may depend on the context. For example, when you are counting stones, the class is all varieties of stones. If you are counting stars, the class is the set of all stars in the sky. 1 star maybe written as "|". 9 stars maybe written as "||||||||". Why choose a vertical line to represent a number? Because our finger can be a vertical line. We can represent them without using papers, by just showing our fingers. Thus it is very natural to use "|".

You can already see that "||||||||||" is hard to instantly see that there are 10 stars. Hence, a more convenient notation will be to write "||||| |||||" to represent 10. This number system will have sufficed to count 4 deer, or 7 wolves. Maybe, if you and I raise our hands together, we can count upto 20 using our 20 fingers.

However, as soon as writing developed and we started dealing with more numbers, we wanted to be able to represent numbers even more succinctly. We just draw one vertical line for each object, but even that is proving to be cumbersome. We wanted to save time. Hence, a new system of roman numerals were developed. We have I=1, V=5, X=10, etc. Thus we have I, II, III, IV, V, VI, VII, VIII, IX, X, XI, XII, .... LV, etc. Thus V=|||||, a letter to the left of a higher letter removes a lower letter, X=VV, L=XXXXX, C=LL, etc. Ultimately, they all get converted into a bunch of "|", where each "|" is a representation of an object. The object in consideration depends on the context, as discussed already. Wikipedia has reference to some other number systems used by Egyptians and Mesopotamians.

The Roman Numeral system would've worked fine then, but notice that we have to continually invent new symbols: I=1, V=5, X=10, L=50, C=100, D=500, M=1000, etc. As long as we only want to receive 1000 gold coins and no more, this system is fine. However, as soon as human kind grew to need even more numbers it became impractical to keep introducing and memorizing new symbols. It is more natural to count using fingers and we needed to preserve that. Thus the concept of place value was introduced. We can put a number of boxes from left to right. We can fill the vertical lines "|" in any of the boxes. However, to interpret the number, we need to move all the "|" to the right-most box. We also have a moving rule: When a "|" moves from a left box to an adjacent right box, it makes 10 copies of itself, as "||||| |||||". Also, to represent a number succinctly, we can group them into "||||| |||||" patterns, and move each such pattern to a higher order box as a single "|". Thus the agreement is that a "|" in a box is isomorphic to the number of "||||| |||||" in the box immediately to the right of it. Of course, we will try to minimize the number of fingers "|" in order to represent any given number, by moving "||||| |||||" to the higher box. When we can no longer reduce the number of sticks, we would only have 10 patterns which we now know as 0,1,2,....9. These are all the symbols needed to represent each box. Thus the decimal system was born. It would take a while before the mathematicians understood the meaning of 0. Nor were they smart enough to prove the uniqueness of representation back then.

Then man grew even further, and he decided he will exchange gold with his fellow men. They needed to be able to do addition and subtraction. Simple: If you give 27 gold coins to someone, you just remove 27 "|" from your number (and then simplify it according to place value). However, you may end up losing all your sticks. Or you may only have 14 "|" in which case you will need to represent the fact that you owe 13 coins as opposed to having 14 coins. Thus the concept of 0 and negative numbers needed to be introduced.

Multiplication was introduced when there arose a need to give 232 gold coins each to 121 people.

Then man had the need to split an apple into two halves and give 1 half to each person. This lead to fractions. The denominator will represent how many parts form the whole and numerator will count how many of those parts you own. It would be intuitive to agree that 1 of 3 parts is the same as k of 3k parts, since each of the 3 parts can be split into k parts. This allows addition of fractions without a common denominator. It would take a while to understand gcd and lcm clearly and formally, but that would not stop man from doing trade and splitting pieces fairly. For the medieval man though, the decimal system would work fine with an additional decimal '.'. To split 7 apples among 8 people, he would start with 7 "|". Then he moves them to the right to obtain 70 "|". 64 dots can be given out, and 6 remain. So he moves them further to the right (split each piece into 10 more pieces) to obtain 60 "|". Then he gives away 56 and has 4 remaining. These 4 become 40 pieces, and he ended up with the decimal notation 0.875 for the fraction. He can repeat this procedure and see that the decimals either terminate or repeat in a cycle. No way of formally proving it until some geeks come together to define gcd, lcm, etc. and formalize these concepts.

As someone else pointed out in the comments, mathematicians wanted to define '+','-','','/' etc. Whether it was because the society demanded those to be defined or whether it came about because mathematicians are geeks is like a Chicken and Egg question: "What came first: the chicken or the egg?". Here's my take on it: People wanted to show they owe money, so they invented negative numbers. People wanted to split apples, so they invented fractions. People don't care if '+','-','','/' are closed in their set of numbers. They will throw away 1 in 10000 parts of an apple. We know however that mathematicians care. So they had to define the set of rationals formally and prove closure.

Now the rest of the story is geeky. We prove that there is no rational number whose square is 2. Consider the set of rationals whose square is <2 and those whose square is >2 and there is a gap. So geeky were we that we went ahead and defined real numbers, much to the annoyance of the public. Then some dude had to ask what is the square root of -1. There we go, we have complex numbers.

Well, so here is the definition of number. The natural number 1 is simply a "|" that is used to represent a real world object. A natural number is a collection of "|" such that each "|" represents a unique real world object. A representation system for natural numbers based on place value is obtained by placing contiguous boxes and saying that a "|" in a box represents "||||| |||||" in the box immediately to it's right. Then we design the decimal system by noting that a minimal representation will only need symbols 0,1,...9. Then the rest of the numbers (negative, rational, irrational, algebraic, complex, etc.) that we know can be defined by the geeks! Copy-paste!

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Many languages have words for "one of something" where the word for "one" may change depending on what that something is. In Danish we have the archaic "tvende" for "two of something", but one of the more extreme examples comes in Japanese. See for example this clip from Nihonjin no shiranai Nihongo - or just this list of counters. –  kahen Aug 13 '13 at 2:26

What you really want is The Princeton Companion to Mathematics; while not an online source, I think you'll find an enormous amount of breadth and depth in it, thus justifying its cost.

Here's an excerpt from Section 1.3, Some Fundamental Mathematical Definitions:

1 The Main Number Systems

Almost always, the first mathematical concept that a child is exposed to is the idea of numbers, and numbers retain a central place in mathematics at all levels. However, it is not as easy as one might think to say what the word "number" means: the more mathematics one learns, the more uses of this word one comes to know, and the more sophisticated one's concept of number becomes. This individual development parallels a historical development that took many centuries (see FROM NUMBERS TO NUMBER SYSTEMS [II.1]).

The modern view of numbers is that they are best regarded not individually but as parts of larger wholes, called number systems; the distinguishing features of number systems are the arithmetical operations – such as addition, multiplication, subtraction, division, and extraction of roots - that can be performed on them. This view of numbers is very fruitful and provides a springboard into abstract algebra. The rest of this section gives a brief description of the five main number systems.

It then goes on to discuss the natural numbers, the integers, the rationals, the reals, and the complex numbers. That covers slightly more than 2 of the book's 1000 pages. If you buy it, I would definitely recommend an eBook format (Kindle is currently $55); it's an enormous book and had it been available digitally when I bought it, I most assuredly would have preferred that format.

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