I'd have thought that (1,0) represents the real number 1, rather than zero.

You are correct that we are on the unit circle, but we won't ever get back to 1 or -1, because $\pi$ is irrational and thus for $n,m \in \mathbb{Z}$, $n\pi = m \; \Rightarrow \; n = m = 0$, there being no other solutions. In fact, we never get to -1 at all!

So what we have is two infinite sets consisting of points on the unit circle, both of which fill in more and more of the circle as $|n| \to \infty$. But the two sets are disjoint (they share no members). There's perhaps nothing too surprising about that. In the number $e^{i\theta}$, $\theta$ is a real number and we are comfortable with the odd integers and the even integers being a pair of disjoint sets within the reals. But I fine it challenging to think of these two sets on the unit circle being disjoint, because my mind wants to identify them, as $|n| \to \infty$, with the unit circle. Of course, this is a bad idea, because that would suggest that in some sense $1 = -1$ in the complex plane.

Anyway, over to you Hoempa.

You are correct that we are on the unit circle, but we won't ever get back to 1 or -1, because $\pi$ is irrational and thus for $n,m \in \mathbb{Z}$, $n\pi = m \; \Rightarrow \; n = m = 0$, there being no other solutions. In fact, we never get to -1 at all!

So what we have is two infinite sets consisting of points on the unit circle, both of which fill in more and more of the circle as $|n| \to \infty$. But the two sets are disjoint (they share no members). There's perhaps nothing too surprising about that. In the number $e^{i\theta}$, $\theta$ is a real number and we are comfortable with the odd integers and the even integers being a pair of disjoint sets within the reals. But I fine it challenging to think of these two sets on the unit circle being disjoint, because my mind wants to identify them, as $|n| \to \infty$, with the unit circle. Of course, this is a bad idea, because that would suggest that in some sense $1 = -1$ in the complex plane.

Anyway, over to you Hoempa.

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