# Roots of Unity

The nth Roots of Unity appear when we consider the complex roots of an equation of the form:

$z^n = 1.$

### Solving the Equation

As we have an nth degree polynomial, we will have n complex roots. By converting this to the polar form (by letting   and noting that $1 = e^{2\pi ik}$ for $k\in \mathbb{Z}$), we get the expression:

$r^ne^{ni\theta} = e^{2\pi ik}$

As the magnitude of the right hand side is 1, we can deduce that r = 1, leaving us with $e^{ni\theta} = e^{2\pi ik}$. Quick algebraic manipulation gives us:

$\theta=\frac{2\pi k}n$

Hence, we can conclude that the solutions the polynomial are given by  which can be converted to the trigonometric equation using Euler’s formula:

### Geometry

All roots of unity lie on the unit circle in the complex plane, as all roots have a magnitude of 1.

Additionally, the nth roots of unity are connected in order, they form a regular n sided polygon. This can easily be seen by analysing the arguments of the roots.

### Properties

• The sum of the nth roots of unity is 0.
• If $\zeta$ is a primitive nth root of unity, then the roots of unity can be expressed as $1, \zeta, \zeta^2,\ldots,\zeta^{n-1}$.
• A primitive nth root of unity is such that $\zeta^m\neq 1$ for $1\le m\le n-1$.
• This sequence of powers is n periodic because z j + n = z jz n = z j⋅1 = z j for all values of j.
• For each nth root of unity, $\zeta$, we have that $\zeta^n=1$. Although obvious, this property should not be forgotten as, for example, it can aid with algebraic manipulation.

Pretty huh? M x