# Is R under addition isomorphic to R\{0} under multiplication?

1. Dec 27, 2009

### 3029298

1. The problem statement, all variables and given/known data
Is $$\mathbb{R}$$ under addition isomorphic to $$\mathbb{R}\backslash{0}$$ under multiplication?

3. The attempt at a solution
It is true that $$\mathbb{R}$$ under addition is isomorphic to $$\mathbb{R}_{>0}$$ under multiplication, by using the bijection $$\phi : \mathbb{R}\rightarrow\mathbb{R}_{>0}$$ with $$\phi\left(x\right)=e^x$$. But we cannot use this isomorphism for $$\mathbb{R}\backslash{0}$$ since the exponential is positive everywhere, and if we define it to be negative for $$x<0$$ then we cannot map to the interval $$(0,1)$$. Intuitively this gives a hint that there is no isomorphism possible... Can I do something with the fact that we know that if there is an isomorphism, it must map inverses to inverses, therefore $$\phi(0)=1$$? I really need a hint to be able to find the answer... Thanks for any help!

Last edited: Dec 27, 2009
2. Dec 27, 2009

### rasmhop

Let $\varphi : (\mathbb{R},+) \to (\mathbb{R}\setminus \{0\},\times)$ be such an isomorphism. For an arbitrary element $x \in \mathbb{R}$ we have:
$$\varphi(x) = \varphi(x/2 + x/2) = \varphi(x/2)^2 \geq 0$$

3. Dec 27, 2009

### 3029298

Beautiful argument! Didn't think of that... thanks!