Gaussian integer

Definition

A Gaussian integer is a complex number $a + bi$ such that $a$ and $b$ are integers. Here $i$ is a complex number with $i^2 = -1$. The set of Gaussian integers is written $\mathbb{Z}[i] = \{ a + bi : a, b \in \mathbb{Z} \}$.

Remarks

The set $\mathbb{Z}[i]$ is closed under addition, subtraction and multiplication, so it is a subring of $\mathbb{C}$. It is an integral domain. The units of $\mathbb{Z}[i]$ are $1$, $-1$, $i$ and $-i$. In the complex plane, the Gaussian integers are the points of the square lattice. The norm $N(a + bi) = a^2 + b^2$ links them to sums of two squares.

Explanations

L1 human

Gaussian integers are the complex numbers whose real part and imaginary part are both integers, such as $3 - 2i$. They behave much like the ordinary integers: you can add them, multiply them and factor them into primes.