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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 $\mathbb{C}$:

  1. Dimension $\rightarrow$ Hausdorff dimension.
  2. Factorial $\rightarrow$ gamma function.
  3. 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. :-)

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$S_n$ $\to$ $S_t$ (Deligne). – darij grinberg Sep 1 '11 at 12:54
Also, the classical: $\binom{n}{k}$ $\to$ $\binom{x}{k}=\frac{x\left(x-1\right)...\left(x-k+1\right)}{k!}$. – darij grinberg Sep 1 '11 at 12:56
I would rather be interested in the complementary question: what entities, defined on the positive integers, cannot be extended in any sensible way to a larger set? – Federico Poloni Sep 1 '11 at 13:38
Oh, and this should definitely be Community Wiki – Federico Poloni Sep 1 '11 at 13:39
The order of a numerical method in solving $f(x)=0$. At the beginning, it appears to be an integer: $1$ for many methods based upon the Picard contraction principle, $2$ for the Newton--Raphson method. But the order of the secant method is $\phi$, the Gloden ratio! – Denis Serre Sep 1 '11 at 14:38
up vote 16 down vote accepted

The writhe is the fundamental differential geometric invariant of a closed space curve. I think it is the most useful topological invariant outside mathematics- biologists use it to study circular DNA molecules, and chemists use it in the study of long polymers. For space curve $C(t)$ it's defined as the double integral
$\frac{1}{4\pi}\int_{C\times C}\frac{C^\prime(s)\times C^\prime(t)\cdot (C(s)-C(t))}{|C(s)-C(t)|^3}ds dt.$

but most people think of it as the number of positive crossings minus the number of negative crossings. This quantity is naturally an integer. The integral formula is based on the Gauss integral for the linking number, but has a complicated history, with a lot of contribution from non-mathematicians.

But, what to do, most real-life long molecules aren't closed space curves. And so biologists, chemists, and physicists, followed by mathematicians, generalized the writhe to open space curves. The idea is that writhe makes sense for a tangle diagram, so they integrated over all projection angles of the open space curve. The result is a definition for the writhe of an open space curve, which is a real number (which can be efficiently estimated). I think it's differential geometry's most useful real numbers for studying open space curves where they occur in biology, chemistry, and physics.

A nice survey of writhe in various contexts is Berger and Prior's The writhe of open and closed space curves.

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The natural extension of Euler characteristic to orbifolds is valued in Q.

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There are several natural extensions of Euler characteristic to orbifolds. Some with values in $\mathbb{Q}$, others with vaues in $\mathbb{Z}$. – euklid345 Sep 1 '11 at 15:34

Motivic integration, where the underlying measures are valued in rings of motives.

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Which is a simultaneous generalization of the integer-valued functions "Euler characteristic" and "count points over finite fields"! – JSE Sep 1 '11 at 16:14
Amen! Also, I guess I bent the rules a little, since measures aren't usually $\mathbf{Z}$-valued, but I think the discrepancy between real numbers and rings of motives is big enough that this still counts. :) – David Hansen Sep 1 '11 at 20:25

A natural generalization of cardinality of sets is groupoid cardinality, which is a real number.

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nice to know this term (totally new to me)! – Suvrit Sep 1 '11 at 22:10
Be warned that number theorists call it "mass" instead of groupoid cardinality. – Omar Antolín-Camarena Sep 3 '11 at 15:02

It's interesting to note that this has happened with several notions of "dimension". Krull dimension of rings has been extended to notions as GK-dimension, for example.

As a complementary answer... what would be a ring of characteristic $-\pi$?

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A ring of characteristic $\pi$ should at least be perfectly round! – Roland Bacher Sep 2 '11 at 9:16

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.

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(Probably should count towards your Dimension example)

Sobolev Spaces of Integer Dimension $\rightarrow$ Sobolev(–Slobodeckij) Spaces of fractional Dimension

Has Important applications for numerically solving boundary integrals

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