---
tag: 0B268GVX95
kind: definition
name: Gaussian integer
msc: [11R11]
defines: 074P7C85XH
sources:
  - url: https://en.wikipedia.org/wiki/Gaussian_integer
    locator: Basic definitions
---

## 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|00HPT88ZX0]] $N(a + bi) = a^2 + b^2$ links them to [[sums of two squares|04NPWJMA6H]].

## 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.