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Physics

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  • $\begingroup$ With this structure is there any way to get units with fractional exponents? Eg $12 \ \mathrm{m^{3/2}}$ $\endgroup$ Commented Sep 6 at 21:02
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    $\begingroup$ @Dale: It would take some work, but I'm guessing you could do it, by introducing a one-dimensional vector space $M_2$ that you think of as "the space of square roots of masses" and adding some structure in the form of an isomorphism $M_2\otimes_{\mathbb R}M_2\rightarrow M$ (where $M$ is the space of masses). This sort of thing adds fractional exponents ``one denominator at a time'', and you might prefer a structure that incorporates all of them at once. I can imagine what that would look like, but would have to think about the details. $\endgroup$ Commented Sep 6 at 23:37
  • $\begingroup$ The space of velocities is not a linear vector space, as I realized while I was writing my answer to complement yours. The space of four-velocities is linear, though. $\endgroup$ Commented Sep 7 at 3:45
  • $\begingroup$ What is $\otimes_{\mathbb R}$? $\endgroup$ Commented Sep 8 at 3:28
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    $\begingroup$ @TheRizzler: If $V$ and $W$ are real vector spaces, then $V\otimes_{\mathbb R} W$ (which is called the tensor product of $V$ and $W$ over ${\mathbb R})$ is the universal recipient of a bilinear map from the product $V\times W$. It has a basis the various $v_i\otimes w_j$ where the $v_i$ form a basis of $V$ and the $W_j$ form a basis of $W$. $\endgroup$ Commented Sep 8 at 3:47