# What are roots of unity?

Nov 20, 2014

A root of unity is a complex number that when raised to some positive integer will return 1.

It is any complex number $z$ which satisfies the following equation:

${z}^{n} = 1$

where $n \in \mathbb{N}$, which is to say that n is a natural number. A natural number is any positive integer: (n = 1, 2, 3, ...). This is sometimes referred to as a counting number and the notation for it is $\mathbb{N}$.

For any $n$, there may be multiple $z$ values that satisfy that equation, and those values comprise the roots of unity for that n.

When $n = 1$
Roots of unity: $1$

When $n = 2$
Roots of unity: $- 1 , 1$

When $n = 3$
Roots of unity = $1 , \frac{1 + \sqrt{3} i}{2} , \frac{1 - \sqrt{3} i}{2}$

When $n = 4$
Roots of unity = $- 1 , i , 1 , - i$