## What is a real number ? Cauchy’s fantastic construction

The real numbers are all the "quantities" that we can order, and we can "construct" them in various ways thanks to set theory. "Numbers govern the world." Pythagoras The real numbers idealise

## What is a rational number? Quotients of numbers and sets

1.The intuition of rational numbers Rational numbers, i.e. "fractional" numbers, such as $$-\frac 1 2, \frac{27}{4}, \frac{312}{-6783},\ldots$$, form an intuitive set which we note $$\mathbb Q$$. It is an extension of

## What is an integer ? A crafty representation

Integers are an extension of the natural numbers where the existence of subtraction provides a more appropriate framework for certain questions of arithmetic. They can be described axiomatically, but can

## What is a natural number? Defining or axiomatising

Mathematical science does not seek to define the notion of a natural number, but to understand the set of natural numbers. "Natural numbers have been made by God, everything else is

## An infinity of prime numbers : Euclid’s theorem

The prime natural numbers are those which have no divisors other than 1 and themselves. They exist in infinite number by Euclid's theorem, which is not difficult to prove. 1.Prime numbers 1.1.Divisors
A finite set is a set that can be counted using the natural numbers $$1,\ldots,n$$ for a certain natural number $$n$$. But what is counting ? And then, what is