Why is the isomorphism $\phi(x + \mathbb{Z}) = e^{2\pi xi}$ natural?

Question

Let $G = \mathbb{R}$, $H = \mathbb{Z}$. The quotient group $G/H$ is isomorphic to the circle subgroup $S^1$ of the multiplicative group $\mathbb{C}^\times$. Why is the isomorphism $\phi : G/H \rightarrow S^1$, $\phi(x + \mathbb{Z}) = e^{2\pi xi}$ a "natural" choice in the sense that is the "obvious" one to take? For instance, if we are told that $G/H$ is isomorphic to $S^1$, what intuition would lead us to the $\phi$ described above?

Answer 1

Think about how addition works in $G/H$. If we let $[x]$ be shorthand for $x + \mathbb Z$, then for example $[1/2] + [2/3] = [7/6] = [1/6]$. Once we get past $1$, we "wrap around" starting at $0$ again as if we were on a circle.

This is similar to clock arithmetic on an analog clock, which works modulo $12$, for example $[11] + [2] = [13] = [1]$.

So in that sense it's natural to identify $G/H$ with arithmetic on a circle, and the map $[x] \mapsto e^{2\pi i x}$ is a convenient way to do that. It simply maps $[x]$ to the point whose angle is $2\pi [x]$ on the unit circle.