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,
Category
Number theory
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
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