(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Is [tex]\mathbb{R}[/tex] under addition isomorphic to [tex]\mathbb{R}\backslash{0}[/tex] under multiplication?

3. The attempt at a solution

It is true that [tex]\mathbb{R}[/tex] under addition is isomorphic to [tex]\mathbb{R}_{>0}[/tex] under multiplication, by using the bijection [tex]\phi : \mathbb{R}\rightarrow\mathbb{R}_{>0}[/tex] with [tex]\phi\left(x\right)=e^x[/tex]. But we cannot use this isomorphism for [tex]\mathbb{R}\backslash{0}[/tex] since the exponential is positive everywhere, and if we define it to be negative for [tex]x<0[/tex] then we cannot map to the interval [tex](0,1)[/tex]. 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 [tex]\phi(0)=1[/tex]? I really need a hint to be able to find the answer... Thanks for any help!

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

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