Show that $\frac{3}{5} + i\frac{4}{5}$ isn't a root of unity Intuitively, I see why $\frac{3}{5} + i\frac{4}{5}$ is not a root of unity because $\frac{2\pi}{\arctan(4/3)}$ appears to be irrational when I plug into my calculator. But how to I show this rigorously?I think contradiction should work, but still I wasnt able to show this rigorously. Any help will be appreciated. (Also: I first tried using the idea that roots of unity had angle $\frac{2\pi}{n}$ for n in the natural numbers but then I realized that n should be replaced by rational number because $e^{\frac{i4\pi}{7}}$ is a root of unity too.) 

Hint: An $n$th root of unity has the form $e^{2\pi ik/n}$ with $0\leq k\leq n-1$.
By Euler's formula, $r(\cos \phi + i\sin\phi) = re^{i\phi}$.
