Root of unity

Definition

Let $n$ be a positive integer. A complex number $z$ is an $n$th root of unity when $z^n = 1$. A complex number $z$ is a root of unity when it is an $n$th root of unity for some positive integer $n$.

Remarks

The same definition works in any field, or in any commutative ring. The $n$th roots of unity in $\mathbb{C}$ form a group under multiplication, written $\mu_n$. For example, the $4$th roots of unity are $1$, $i$, $-1$ and $-i$. Each root of unity $z$ satisfies $|z| = 1$, because $|z|^n = |z^n| = 1$.

Explanations

L1 human

A root of unity is a number that gives $1$ when you multiply it by itself often enough. The number $-1$ is one, because $(-1)^2 = 1$, and so is $i$, because $i^4 = 1$.