## multiplicative reals as a real vector space [closed]

Can the multiplicative group $\mathbb{R}^*$ of nonzero real numbers be given the structure of a real vector space? I know that the positive reals can, namely by defining scalar multiplication by $a$ to be exponentiation by $a$ (i.e., $a\cdot x = x^a$), but this obviously doesn't work when we allow negative reals.

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Do you want addition in the vector space to correspond to multiplication of real numbers? In this case the formula $(-1)(-1)= 1\cdot 1$ shows that this is impossible. – Richard Stanley Dec 25 2010 at 3:30
The comments on this question mathoverflow.net/questions/15214/… also apply here. – Zev Chonoles Dec 25 2010 at 4:21
Richard, I can't believe I didn't notice that! Thanks. – Avi Steiner Dec 25 2010 at 6:50