Perhaps because of my computer science background, I tend to see math as finding "optimal" representations for computing different things. For example, in the complex plane, whereas the usual x,y-coordinate representation is ideal for performing additions, the polar representation is better for computing multiplications (maybe this has more to do with the exponential "transforming" multiplications to additions...?). Of course also the radix-representation of natural numbers (compared to say, latin numbers) is the most "efficient" for performing arithmetic.
I am wondering if there are other such "natural" representations, for example for calculating exponentials, or factorials, etc. Please do not shy away from pointing me towards higher-level math theories or ideas.
I also had the Riesz Representation theorem (linear functions can be represented by inner product with a single element) in mind when asking this question, and then found a whole bunch of "representation" theorems: https://en.wikipedia.org/wiki/Representation_theorem. Is there a general framework for thinking about these ideas (perhaps category theory?) or are they just spread all over different mathematical fields without links?