Neighborhood of Infinity is a cool math blog. From it I learned about Stephen Schanuel (a frequent collaborator of category theory superstar Bill Lawvere) and his extremely cute construction of the real numbers directly from the integers. The intuition is to represent a real number r as a function Z → Z whose slope is r.
It goes like this:
A quasihomomorphism on the integers Z is a map f : Z → Z with the property that the quantity |f(x+y)-f(x)-(y)| is bounded by a constant as x,y vary over Z. That is, it's almost a homomorphism from the additive group of integers to itself: the amount by which it fails to be is just a constant.
(in big-O notation: f is a qh if f(x+y) = f(x) + f(y) + O(1))
Two quasihomomorphisms f, g are considered equivalent if their difference |f(x)-g(x)| is bounded as x varies over Z.
(in big-O notation, f ~ g if f(x) = g(x) + O(1))
The set of real numbers is just the quasihomomorphisms quotiented out by this equivalence. That's it!
Well, except for the arithmetic stuff on top; two add two of these real numbers, add them pointwise as functions, and to multiply them, just compose them. Say a quasihomomorphism is ≥ 0 if it's bounded below on inputs ≥ 0, and that lets you define ordering.
Here's a paper by Ross Street, one by Arthan, and A'Campo's independent discovery of all this.