---
tag: 057BH3JC9S
kind: statement
role: theorem
name: The nth roots of unity
msc: [11R18]
uses: [0S61N75H45, 03MTBVG3VE, 0BRXFNQ5JV]
sources:
  - url: https://en.wikipedia.org/wiki/Root_of_unity
    locator: Trigonometric expression
---

## Statement

Let $n$ be a positive integer.
A complex number $z$ is an $n$th [[root of unity|0S61N75H45]] if and only if
$$z = e^{2 \pi i k / n} = \cos \frac{2 \pi k}{n} + i \sin \frac{2 \pi k}{n}$$
for some integer $k$ with $0 \le k < n$.

## Remarks

Proof: by [[de Moivre's formula|03MTBVG3VE]], each such $z$ satisfies $z^n = e^{2 \pi i k} = 1$. Conversely, $z^n = 1$ forces $|z| = 1$ and $n \arg z \in 2 \pi \mathbb{Z}$.
For $n = 3$, the roots are $1$ and $-\frac{1}{2} \pm \frac{\sqrt{3}}{2} i$.

## Explanations

### L1 human

The $n$th roots of unity sit on the unit circle at equal angles of $2\pi/n$, starting at $1$.
For $n = 6$, they are the six corners of a regular hexagon.