---
tag: 02PETK9PQ4
kind: statement
role: proposition
name: Sum of the roots of unity
msc: [11R18]
uses: [0J817RTVG6, 03X8V7D64M]
sources:
  - url: https://en.wikipedia.org/wiki/Root_of_unity
    locator: Summation
---

## Statement

Let $n \ge 2$ be an integer and let $\zeta$ be a [[primitive root of unity|0J817RTVG6]].
Then
$$\sum_{k=0}^{n-1} \zeta^k = 1 + \zeta + \zeta^2 + \dots + \zeta^{n-1} = 0.$$

## Remarks

Proof: $\zeta \neq 1$, so the [[geometric sum formula|03X8V7D64M]] gives $\frac{\zeta^n - 1}{\zeta - 1} = 0$.
Thus the sum of all the $n$th roots of unity is $0$ for $n \ge 2$. For $n = 1$ the sum is $1$.

## Explanations

### L1 human

The $n$th roots of unity are the corners of a regular polygon with center $0$.
By symmetry, their center of mass is $0$, and so their sum is $0$.
For $n = 4$: $1 + i + (-1) + (-i) = 0$.