# Definition:Root of Unity/Complex

## Definition

Let $n \in \Z_{>0}$ be a (strictly) positive integer.

The complex $n$th roots of unity are the elements of the set:

$U_n = \set {z \in \C: z^n = 1}$

### Primitive Root of Unity

A primitive (complex) $n$th root of unity is an element $\alpha \in U_n$ such that:

$U_n = \set {1, \alpha, \alpha^2, \ldots, \alpha^{n - 1} }$

### First Root of Unity

The root $e^{2 i \pi / n}$ is known as the first (complex) $n$th root of unity.

### Order of Root of Unity

Let $z \in U_n$.

The order of $z$ is the smallest $p \in \Z_{> 0}$ such that:

$z^p = 1$

## Illustration

From Complex Roots of Unity are Vertices of Regular Polygon Inscribed in Circle the complete set of complex $6$th roots of unity can be depicted as the vertices of the following regular hexagon:

where $\alpha$ is used to denote the first complex $6$th root of unity.

## Examples

### Complex Cube Roots of Unity

The complex cube roots of unity are the elements of the set:

$U_3 = \set {z \in \C: z^3 = 1}$

They are:

 $\displaystyle$  $\, \displaystyle e^{0 i \pi / 3} \,$ $\, \displaystyle =\,$ $\displaystyle 1$ $\displaystyle \omega$ $=$ $\, \displaystyle e^{2 i \pi / 3} \,$ $\, \displaystyle =\,$ $\displaystyle -\frac 1 2 + \frac {i \sqrt 3} 2$ $\displaystyle \omega^2$ $=$ $\, \displaystyle e^{4 i \pi / 3} \,$ $\, \displaystyle =\,$ $\displaystyle -\frac 1 2 - \frac {i \sqrt 3} 2$

The notation $\omega$ for, specifically, the complex cube roots of unity is conventional.

### Complex $4$th Roots of Unity

The complex $4$th roots of unity are the elements of the set:

$U_n = \set {z \in \C: z^4 = 1}$

They are:

 $\displaystyle e^{0 i \pi / 4}$ $=$ $\displaystyle 1$ $\displaystyle e^{i \pi / 2}$ $=$ $\displaystyle i$ $\displaystyle e^{i \pi}$ $=$ $\displaystyle -1$ $\displaystyle e^{3 i \pi / 2}$ $=$ $\displaystyle -i$

### Complex $5$th Roots of Unity

The complex $5$th roots of unity are the elements of the set:

$U_n = \set {z \in \C: z^5 = 1}$

They are:

 $\displaystyle e^{0 \pi / 5}$ $=$ $\displaystyle 1$ $\displaystyle e^{2 \pi / 5}$ $=$ $\displaystyle \dfrac {\sqrt 5 - 1} 4 + i \dfrac {\sqrt {10 + 2 \sqrt 5} } 4$ $\displaystyle e^{4 \pi / 5}$ $=$ $\displaystyle -\dfrac {1 + \sqrt 5} 4 + i \sqrt {\dfrac 5 8 - \dfrac {\sqrt 5} 8}$ $\displaystyle e^{6 \pi / 5}$ $=$ $\displaystyle -\dfrac {1 + \sqrt 5} 4 - i \sqrt {\dfrac 5 8 - \dfrac {\sqrt 5} 8}$ $\displaystyle e^{8 \pi / 5}$ $=$ $\displaystyle \dfrac {\sqrt 5 - 1} 4 - i \dfrac {\sqrt {10 + 2 \sqrt 5} } 4$

### Complex $6$th Roots of Unity

The complex $6$th roots of unity are the elements of the set:

$U_n = \set {z \in \C: z^6 = 1}$

They are:

 $\displaystyle e^{0 i \pi / 6}$ $=$ $\displaystyle 1$ $\displaystyle e^{i \pi / 3}$ $=$ $\displaystyle \frac 1 2 + \frac {i \sqrt 3} 2$ $\displaystyle e^{2 i \pi / 3}$ $=$ $\displaystyle -\frac 1 2 + \frac {i \sqrt 3} 2$ $\displaystyle e^{i \pi}$ $=$ $\displaystyle -1$ $\displaystyle e^{4 i \pi / 3}$ $=$ $\displaystyle -\frac 1 2 - \frac {i \sqrt 3} 2$ $\displaystyle e^{5 i \pi / 3}$ $=$ $\displaystyle \frac 1 2 - \frac {i \sqrt 3} 2$

## Also see

• Results about the complex roots of unity can be found here.