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

[3029298]

Homework Statement

Is $$\mathbb{R}$$ under addition isomorphic to $$\mathbb{R}\backslash{0}$$ under multiplication?

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!

 

[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:
$$\varphi(x) = \varphi(x/2 + x/2) = \varphi(x/2)^2 \geq 0$$

 

[3029298]

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