# Modulus of Complex Root of Unity equals 1

## Theorem

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

Let $U_n = \set {z \in \C: z^n = 1}$ be the set of complex $n$th roots of unity.

Let $z \in U_n$.

Then:

$\cmod z = 1$

where $\cmod z$ denotes the modulus of $z$.

## Proof

 $\displaystyle z^n$ $=$ $\displaystyle 1$ $\displaystyle \leadsto \ \$ $\displaystyle \cmod {z^n}$ $=$ $\displaystyle \cmod 1$ $\displaystyle$ $=$ $\displaystyle 1$ $\displaystyle \leadsto \ \$ $\displaystyle \cmod z^n$ $=$ $\displaystyle 1$ Power of Complex Modulus equals Complex Modulus of Power $\displaystyle \leadsto \ \$ $\displaystyle \cmod z$ $=$ $\displaystyle 1$ Positive Real Complex Root of Unity

$\blacksquare$