# Argument of x to the n Equals n Times The Argument

## Theorem

Let $z$ be a complex number.

Then:

- $\forall n \in \N_{>0}: \map \arg {z^n} = n \map \arg z$

## Proof

For $n = 1$

- $\map \arg {z^1} = 1 \cdot \map \arg z$

Assuming the result is true for $n = k$, we have:

\(\displaystyle \map \arg {z^{k + 1} }\) | \(=\) | \(\displaystyle \map \arg {z z^k}\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \map \arg z + \map \arg {z^k}\) | Argument of Product equals Sum of Arguments | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \map \arg z + k \map \arg z\) | by our induction hypothesis | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \paren {k + 1} \map \arg z\) |

$\blacksquare$