Introduction

What are numbers? In this post, I try to give the rationale how numbers where first defined. Perhaps it is also interesting as an insight into how mathematicians think. I even give the classical description and try to not involve category theory.

Natural numbers

You know all of them. They are $0, 1, 2,...$ and they are usually denoted by the letter $\mathbb{N}$.
How do we define them? There is a function called the successor, which i will denote by $\operatorname{s}$. You can think of it as adding one. Then we have the following axioms:

1. $0$ is a natural number.
2. For every natural number $n$, its successor $\operatorname{s}(n)$ is a natural number.
3. There is no natural number which has zero as its successor.
4. If two natural numbers have the same successor, then they are equal.
5. If a set $X$ contains zero and for every natural number its successor, then the natural numbers are contained in $X$.

This should all be fairly trivial but perhaps unfamiliar. The last one is minimally more complex. It basically says, that the natural numbers are the smallest set which contains the zero and every successor. It can be expressed as a ‘no noise’-condition.

Integers

In school they said to you, that if you only have four apples, you cannot eat five of them. They lied to you. Of course you can. Mathematicians found a way to warp reality such that this is possible. They invented negative numbers.
For this i have to explain the concept of an equivalence class. We call $\sim$ an equivalence relation on a set $X$ if for $x,y,z\in X$

1. $x\sim x$ for all $x\in X$
2. If $x\sim y$ then $y \sim x$
3. If $x\sim y$ and $y\sim z$ then $x\sim z$

For example equality of two numbers is an equivalence relation. If you do not trust me then you can try to compute it yourself.
Equivalence relations partition the set $X$ into disjoint sets, in which all elements are equivalent. Those sets are called equivalence classes and if we pick an element from them, it is called a representative for the respective equivalence class.

For example we can write three as $3=7-2$ or as $3=8-3$. So we build up pairs $(7,2)$ and $(8,3)$ and somehow want them to represent the number $3$. If they represent the same number, then they should be equivalent.
So we define $(r,s)\sim (u,v)$ if and only if $r+v=s+u$. We do not write the complicated stuff with minus, because we cannot subtract each number from the others. At least not up to now. You should be able to verify for yourself, that this is indeed an equivalence relation. As said above each equivalence relation is a partition into disjoint sets. In this case we partition $\mathbb{N}\times\mathbb{N}$. If we observe the equivalence classes, then we have exactly the integers, often denoted by $\mathbb{Z}$.

Rationals

In school they said to you, that you cannot divide three apples among five kids. They lied to you. Of course you can. Mathematicians found a way to warp reality such that this is possible. They invented rationals.
The construction of the rationals, often denoted by $\mathbb{Q}$ takes place in a very similar way to the construction of the integers from the natural numbers.
We define for $r,s,u,v\in \mathbb{Z}$ and $s,v\neq 0$ that $(r,s)\sim (u,v)$ if and only if $rv=us$. There is a slightly more general construction called localisation, which explains why you have to exclude the zeroes, but you can look that up, if you want to. Then we do the usual construction of taking the equivalence classes. Normally one does not write $(r,s)$ for representatives of an equivalence class, but instead $\frac{r}{s}$. This does not intuitively makes sense, but is more familiar.
Now you can see why we took this special equivalence relation, since $\frac{r}{s}$ can also be written as $\frac{rt}{st}$ for every $t\neq 0$, $t\in\mathbb{Z}$.

Real numbers

In school they said to you, that you cannot take a squareroot of two cookies. They lied to you. Of course you can. Mathematicians found a way to warp reality such that this is possible. They invented the real numbers.
Real numbers are a lot harder and there are at least two constructions that lead from the rationals to the real numbers. One is via dirichlet cuts and the other is via cauchy sequences. I will only brush cauchy sequences. Cauchy Sequences are sequences, where for each number $\epsilon$, there is an element in the sequence, after which the distance of two elements is smaller than $\epsilon$. So intuitively they should converge. In general this is not true in $\mathbb{Q}$ since one can construct Cauchy sequences which approximate $\sqrt{2}$ arbitrarily close, but since $\sqrt{2}$ is not in $\mathbb{Q}$ they do not converge.
The remedy here is again to take equivalence classes over Cauchy sequences over rational numbers. The equivalence relation is defined to mean that the two sequences ‘converge to the same point’ whatever that means exactly in $\mathbb{Q}$.
Real numbers are often denoted by $\mathbb{R}$.

Complex numbers

In school they even said, that you cannot take roots of negative integers. This is of course also a lie. Since we want each nonconstant polynomial of degree $d$ to have exactly $d$ roots, we added an element, called $i$. This $i$ is one of the roots of the polynomial $x^2+1$. At least it behaves like it would be a root. If we define two polynomials over the real numbers to be equivalent if they are equal up to a multiple of $x^2+1$ and then take the equivalence classes, we get a representation of the complex numbers. Another construction is that we take the algebraic closure of the real numbers, which is unique up to isomorphism. But in the end we get the same thing. At least up to isomorphism. The thing we get is mostly denoted by $\mathbb{C}$.

Conclusion

I hope you had fun. Remember that if there is a problem with reality then you can bend it to suit your will.

15 October 2014

Lesson