I need to prove that $\mathbb{C}/\mathbb{R}\cong \mathbb{R}$

Question

I need to prove that $\mathbb{C}/\mathbb{R}\cong \mathbb{R}$, by the this theorem:
$G/\ker(\varphi)\cong Im(\varphi)$.

I tried to find $\varphi$ that will give me this but I didn't succeed.

Maybe it's a mistake at my H.W. But if not, I'll be glad if you will help me...

Thank you!

Answer 1

Hint: Assuming you mean $\langle\mathbb{C},+\rangle$ and $\langle\mathbb{R},+\rangle$: consider $\varphi:\mathbb{C}\to\mathbb{R}$ by $\varphi(z)=\operatorname{im}(z)$, i.e. $\varphi(x+iy)=y$.

Answer 2

I think the map $(a+ib)\mapsto a$ would be helpful.