I am completely stuck with this exercise.
We have an equivalence relation $\mathcal R$ of $\mathbb{R}$ in $\mathbb{R}$ such as : $\ x \mathcal{R} y \iff x - y \in \mathbb{Z} $
The question I am asked is to prove that the quotient set $E/\mathcal{R}$ is a bijection with the set $\mathbb{U}$ of all complex numbers of modulus 1.
In such exercise, I usually start by writing the relationship as a function, which then helps me to prove the bijection. But here, I can't even write the function, let alone understanding how to represent complex numbers of modulus 1...
Many thanks for any tips!
Function $ f: \mathbb{R} \rightarrow \mathbb{U} : x \mapsto e^{2\pi i x} $ induces the bijection between $\mathbb{R}/\mathcal{R}$ and $\mathbb{U}$.