# Root of Unity

A root of unity is any complex number that yields 1 when raised to some positive integer power $n$.

Introduced in CS370 for the Fourier Transform.

For notational convenience, we define: $W=e_{N2πi}$

`W`

is a`N`

-th root of unity$W$ is an $N$-th Root of Unity, since it satisfies $W_{N}=e_{2πi}=1$

Since $W=e_{N2πi}$ then $W_{k}=e_{N2πik}$

But how do you actually compute this? Remember your trig circle

$W_{k}=e_{N2πik}=cos(N2πk )+isin(N2πk )$