In solving $n^{\text{th}}$ roots of unity, when using the expression $e ^{2ki\pi}$ , $\;k = 0,\pm1, \pm2, \ldots$ , why must "$k$" must be an integer?
I understand there's a substitution going on, but why must "$k$" be an integer? Wouldn't that limit the range of the angle? I suppose in the original expression $z = e ^ {in\theta}$ where $\theta$ is an arbitrary angle. Why would a substitution using $2k\pi$ where k is an integer be equivalent as $n\theta$? As $n$ is also an integer, wouldn't that suggest that the original $\theta$ must also be a multiple of $2\pi$?
If I just ignore the "why" and keep going on, I can very well understand the process of solving roots of unity, but I am just a bit curious about this "why". I've searched for Wikipedia and textbooks and stuff but they all seem to not have a explanation...
Thanks a lot!