I think this may be a very stupid question but here goes: I have a basic understanding of complex numbers and know that you can raise a number to a complex power, etc. But it seems to me that regular integers remain in a privileged position in mathematics in the following case: If we have an equation, we talk about the number of roots. The roots might be complex, the number of roots must always be an integer from zero to n. Is there any kind of math where having a complex or even simply a fractional number of roots is possible?
This idea of the number roots of an equation being complex is just one, possibly very stupid example, but I think it illustrates what I am thinking of. I would be totally happy with someone telling me this is a stupid question not worth pursuing.
I think a good example of what you are searching for is Fractional Calculus. You can find more complete info for this on Wikipedia, but the gist of it is this:
Differentiation is treated as an operator, denoted $D$; i.e $D(f)$ is the first derivative of $f$. The second derivative, $D(D(f))$, yields the composed operator denoted by $D^2$; similarly the $n$-th derivative is denoted $D^n$, which covers the positive integers. We stipulate $D^0$ to be the identity function. The application of the definite integral of $f$ from $0$ to $x$ yields the operator $D^{-1}$; iterating this yields all operators $D^{-n}$.
Now for the 'W.T.F?' part: There is a general expression for $D^{-n}$ as a single integral combining the function $f$ with other terms, one of which is the factorial of $n$. This factorial is the ONLY part of the expression that actually requires $n$ to be an integer. The move here is replacement of the factorial with an equivalent call to the Gamma function! This yields an expression for $D^{-r}$ for noninteger values of $r$! One then extends this to positive fractions $r$ by applying $D^m$ to $D^{r-m}$ for some positive integer $m$.
Mind blown? Mine was!