## Gaussian integers: an imaginary arithmetic

Gaussian integers are complex numbers with integer coordinates. Thanks to their norm, a kind of integer measure of their size, we can describe some of their arithmetic properties. In particular,

## What is a complex number? A simple geometric approach

There are various ways of defining complex numbers. The most direct way is to look at them as points or vectors of the Euclidean plane. Addition and multiplication are then

## 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