Natural representation for calculating exponentials?

Question

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?

Answer 1

This may or may not directly answer your question, but I think you would be interested in the factoradic system.

In base 10, the first digit from the right indicates multiples of $10^0$, the second of $10^1$, the third of $10^2$, etc. and because we want each number to have a unique base 10 representation, we restrict our digits to the range $0$ to $9$.

Similarly, in the factoradic system, the first digit from the right indicates multiples of $0!$, the second of $1!$, the third of $2!$, etc. and the digit indicating multiples of $k!$ is restricted to the range $0$ to $k$. You can easily prove that each number has a unique factoradic representation because you can generate the factoradic representation just like you would generate e.g. the binary representation, and vice versa. Indeed, it is convenient that the largest $k$-digit factoradic number is 1 less than the smallest $(k+1)$-digit factoradic number: $$k\cdot k! + (k-1)(k-1)! + \dotsb + 1\cdot 1! + 0\cdot 0! = (k+1)! - 1$$

So what does this system optimize? For one, it is very convenient to number permutations. Suppose you permute the set $\{A, B, C, D, E\}$ to get $\{E, A, C, D, B\}$. To get the factoradic number of this permutation you do the following:

    select from        index (from 0)
 original sequence ---  of selection
 A  B  C  D |E|    --- 4
|A| B  C  D        --- 0
    B |C| D        --- 1
    B    |D|       --- 1
   |B|             --- 0

$40110$ is the factoradic number for the permutation $\{E,A,C,D,B\}$, or $99$ in decimal. Converting the other way is likewise. This means you now have an excellent and precise way to represent permutations (and even combinations) of sets in a computer's numerical memory! And further, if the original sequence makes lexicographic sense (like $\{A,B,C,D,E\}$), the factoradic ordering is the same as the lexicographic ordering!

Another 'optimization' occurs if we extend the factoradic system to fractions. The first digit after the decimal point (or rather, the factoradic point) indicates multiples of $\frac{1}{1!}$, the second of $\frac{1}{2!}$, etc. and the $k$th digit after the factoradic point must be in the range $0$ to $(k-1)$. Now every rational number has a unique terminating representation! (In any fixed-base system, a rational number with denominator coprime to the base has a recurring representation.) Some interesting irrationals are given at the bottom of this page.