## 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,

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# Number theory

## Gaussian integers: an imaginary arithmetic

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

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

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

## What is an integer ? A crafty representation

## What is a natural number? Defining or axiomatising

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

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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,

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

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

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

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

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

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