In the sciences, we can do dimensional analysis and unit checks to verify whether or not the LHS and the RHS have the same units. If we have the following function:$$y=f(x)=x^{2}$$ what ensures the preserving of units? I have a feeling it is the exponent of 2 which is not dimensionless, but if we write it as:$$y=x^{2}=x\times x$$ where can I see balancing out of the dimensions?
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$\begingroup$ What context is this arising in? The square of a scalar is still a scalar, but you can certainly square things that aren't scalars. (And, in purely mathematical settings, one doesn't typically do dimensional analysis. Similar ideas with homogeneity exist, and just generally respecting symmetries) $\endgroup$– Milo BrandtFeb 19, 2016 at 1:26
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If you're doing mathematics, usually you work with dimensionless quantities, so it makes no sense to try to do dimensional analysis.
answered Feb 18, 2016 at 23:24