The *n*th Roots of Unity appear when we consider the complex roots of an equation of the form:

**Solving the Equation**

As we have an *n*th degree polynomial, we will have **n** complex roots. By converting this to the polar form (by letting and noting that for ), we get the expression:

As the magnitude of the right hand side is 1, we can deduce that r = 1, leaving us with . Quick algebraic manipulation gives us:

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 *n*th 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
*n*th roots of unity is**0**. - If is a primitive
*n*th root of unity, then the roots of unity can be expressed as .- A primitive
*n*th root of unity is such that for . - This sequence of powers is
*n*periodic because*z*^{j + n}=*z*^{j}⋅*z*^{n}=*z*^{j}⋅1 =*z*^{j}for all values of*j*.

- A primitive
- For each
*n*th root of unity, , we have that . Although obvious, this property should not be forgotten as, for example, it can aid with algebraic manipulation.

### Representations of Roots of Unity

Pretty huh? M x