Function $z(\theta) = e^{i \theta} + e^{i \theta \pi}$ is not periodic because $\pi$ is irrational.

Question

There is this popular video going around that shows $\pi$ is irrational using visualization of the function, $z(\theta) = e^{i \theta} + e^{i \theta \pi}$. I understand the reason intuitively, both the functions have different periodicities so the sum is not going to be periodic unless one of the periodicities is an integral multiple of the other.

Here is my attempt outline at proving it mathematically. First find the roots of the function, my idea is if the function is going to repeat itself than it should be some integer multiple of the roots. If I can prove there is no integer multiple of the roots that has same values for the function $z(\theta)$ (other than 0) then it can't be periodic. Does this make sense?

Answer 1

Hint Suppose the that $z(\theta)$ has period $T > 0$. Then, $$2 = z(0) = z(T) = e^{iT} + e^{\pi i T} .$$ Thus, $e^{i T} = e^{i \pi T} = 1$. (Why?)

In particular there are integers $p, q$ such that $T = 2 \pi q$ and $\pi T = 2 \pi p$. Dividing gives $\pi = \frac{p}{q} ,$ which contradicts the irrationality of $\pi$.