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!