Are the groups $\mathbb{C}$ and $\mathbb{R}^2$ isomorphic under addition?

Question

Are the groups $\mathbb{C}$ and $\mathbb{R}^2$ isomorphic under addition?

Answer 1

Yes. Under the usual definition of $(\mathbb{C},+)$, they are isomorphic by definition. That's because the usual definition is:

• $\mathbb{C}=\mathbb{R}^2$;
• the addition in $\mathbb C$ is the same as in $\mathbb{R}^2$.