TITLE: Things that should be positive integers...really?
QUESTION [19 upvotes]: Kronecker. Nuff said. Even the numbers themselves historically started 
as positive integers and were subsequently generalized to hell and back.
Here are some other well known concepts that "should" involve $\mathbb{N}$ 
but were generalized to $\mathbb{Q}$, $\mathbb{R}$ or even $\mathbb{C}$:

Dimension $\rightarrow$ Hausdorff dimension.
Factorial $\rightarrow$ gamma function.
Differentation $\rightarrow$ half-differentation (etc.)

So, can you extend this small to a big list? 
(Motivation: Some hypothetic knot polynomial I calculated with
demanded a dimension of its associated group representation 
- thus the "rt" tag - of 60/11. That is noooooot boding well 
for its existence. :-)

REPLY [8 votes]: Let $f:\mathbb{Z}_p\to\mathbb{Z}_p$ be a "nice" map on the $p$-adic integers (or a map on some more general space with a $p$-adic topology). People who study $p$-adic dynamcis investigate what the iterates of $f$ do to points of the space. So if we fix a point $\alpha\in\mathbb{Z}_p$, we can define an iteration map
$$
  I  : \mathbb{N} \longrightarrow \mathbb{Z}_p,\qquad
    I(n) = f^n(\alpha).
$$
The map $I$ is naturally defined on $\mathbb{N}$, and if $f$ is invertible, then it clearly extends to $\mathbb{Z}$. But for various applications, one would like to evaluate $I(n)$ for $n\in\mathbb{Z}_p$. So the example is 

iteration an integral number of times $\to$ iteration a $p$-adic number of times. 

A very pretty application of this idea is in the paper:
Bell, J. P. ;  Ghioca, D. ;  Tucker, T. J.  The dynamical Mordell-Lang problem for étale maps.
 Amer. J. Math. 132  (2010),  no. 6, 1655--1675.