An intersting fomalization the problem of finding $#(n)$, the minimum length description needed to represent a number $n$, can be found in the notion of Kolmogorov complexity. Basically, this asks "What is the shortest description we can give to a computer s.t. when running that description the computer outputs $n$?", denoted $K(n)$.

Luckily, it turns out that our selection of a language to describe this is not particularly important so long as both can be interpreted on a computer. Specifically, $K_1(n)$ and $K_2(n)$ differ by at most a constant indpendent of $n$, where $K_i$ is the complexity function for two different description languages.

Unfortunately, $K$ is not computable, which is to say while $K(n)$ exists we do not really have a way to find it. This of course does not mean that we cannot provide an upper bound by finding some description.

If this sort of stuff is interesting to you, and want a more detailed treatment, I would suggest taking CS475/CS373 or at least check out "Introduction to the Theory of Computation" by Sipser.