Show that:

1. The additive group $\displaystyle \mathbb {R} / \mathbb {Z} $ is isomorphic to the multiplicative group of all complex numbers of modulus 1.

2. The additive group $\displaystyle \mathbb {Q} / \mathbb {Z} $ is isomorphic to the multiplicative group of all complex roots of unity.

Okay, so I understand that I need find a mapping between these groups, prove they are well-defined, linear, and one to one and onto, but would anyone please explain to me how the complex numbers of mod 1 and roots of unity going to behave?

Complex numbers of mod 1, wouldn't that be any combination of a(i)?

Roots of unity, well, I know how it would work in integers, but complex, I'm a bit lost on that.

Thanks for all your help!!!