General Roots of Unity

General Roots of Unity

## Definition / Concept

The $$n$$th roots of unity are the solutions of:

$$z^n=1$$

## Formula

$$z_k=e^{2\pi ik/n}$$

where:

$$k=0,1,2,\ldots,n-1$$

Equivalently:

$$z_k=\cos\left(\frac{2\pi k}{n}\right)+i\sin\left(\frac{2\pi k}{n}\right)$$

## Properties

$$z_k^n=1$$

The roots are equally spaced on the unit circle.

The sum of all $$n$$th roots of unity is:

$$\sum_{k=0}^{n-1}z_k=0$$

For $n>1$:

$$1+z+z^2+\cdots+z^{n-1}=0$$

for any non-real $$n$$th root of unity $z$.

## Usage

- Used in equations involving roots of unity and cyclic powers.