Complex Numbers

$z_1$ and $z_2$ are both $n$th roots of unity. This means each one can be written in the form $e^{\frac{i 2\pi r}{n}}$, where $r$ is an integer.Let $z_1=e^{\frac{i 2\pi r_1}{n}}$ and $z_2=e^{\frac{i 2\pi r_2}{n}}$ for some integers $r_1$ and $r_2$.If the line segment joining $z_1$ and $z_2$ makes a right angle at the origin, this means the points $z_1, z_2$, and the origin form a right-angled triangle with the right angle at the origin.This can only happen if $\frac{z_1}{z_2}$ is a purely imaginary number. A complex number is purely imaginary when its argument (angle) is $\frac{\pi}{2}$ or $\frac{3\pi}{2}$, or in general, $\frac{\pi}{2}+m\pi$ where $m$ is an integer.We find: $\frac{z_1}{z_2}=e^{\frac{i 2\pi}{n}(r_1-r_2)}$For this to be purely imaginary, we need: $\frac{2\pi}{n}(r_1-r_2)=\frac{\pi}{2}+m\pi$Divide both sides by $\pi: \frac{2}{n}(r_1-r_2)=\frac{2m+1}{2}$Now solve for $n: n=\frac{4(r_1-r_2)}{2m+1}$Since $n$ must be a positive integer, $2m+1$ must divide evenly into $4(r_1-r_2)$. The simplest way for this to always work for integer values is when $n$ is a multiple of 4 . Therefore, $n=4k$ for some positive integer $k$.