# How to show the circle group $\mathbb{T}$ is isomorphic to $\mathbb{C}^*/\mathbb{R}^+$?

How to show the circle group $$\mathbb{T}=\{z\in\mathbb{C}:|z|=1\}$$ is isomorphic to $\mathbb{C}^*/\mathbb{R}^+$?

-
What is $C^*/R^+$ ? – Amr Jan 7 '13 at 2:10
Have you learned any theorems with quotients and isomorphisms in them? – Jonas Meyer Jan 7 '13 at 2:11
From context it appears that $\mathbb C^*=\mathbb C\setminus \{0\}$ with multiplication, and $\mathbb R^+$ is the subgroup of positive real numbers with multiplication, but I agree with @Amr that it wouldn't hurt to make this explicit. – Jonas Meyer Jan 7 '13 at 2:13
@JonasMeyer I see. Just to make sure what is the circle group ? – Amr Jan 7 '13 at 2:15
@Amr: Many of these things can be looked up in Wikipedia: circle group. – MJD Jan 7 '13 at 2:39

$$f:C^*\to \Bbb T\,\,\,,\,\,f(z):=\frac{z}{||z||}\;\;?....$$
@hxhxhx88: You can also show that $\Bbb T\cong\frac{\Bbb R}{\Bbb Z}$ via $\phi:\Bbb R\to \Bbb T, \phi(x)=\text{e}^{2\pi x i}$. – Babak S. Jan 7 '13 at 6:56