Why is $\mathbb{R}/\mathbb{Z}$ isomorphic to the complex numbers of module one?

$\mathbb R/\mathbb Z$ identifies each point on the real line with a unique point in the interval $[0,1)$. We can stretch this interval by a factor of $2\pi$ and wrap it around the circumference of the unit circle in the complex plane.

Because the real number $1$ maps to $0$ in the interval $[0,1)$ we find that joining the ends together works, and each real number can be identified with a number of circuits of the unit circle.

It is like treating the whole real line as a piece of elastic and wrapping it around the unit circle so that all the integers end up in the same place.

Because the homomorphism $$ \varphi\colon\mathbb{R}\to \mathbb{C}\setminus\{0\} $$ (the domain is a group with respect to addition, the codomain with respect to multiplication) defined by $$ \varphi(t)=\cos(2\pi t)+i\sin(2\pi t) $$ has the unit circle as its image and $\mathbb{Z}$ as its kernel.

Hint : $x \rightarrow e^{ix}$