# The Eudoxus reals

Let’s call a function $f : {\mathbb Z} \to {\mathbb Z}$ a near-endomorphism of $\mathbb Z$ if there is a constant $C>0$ such that $|f(a+b)-f(a)-f(b)| \leq C$ for all $a,b \in \mathbb Z$. The set of near-endomorphisms of $\mathbb Z$ will be denoted by $N$. We put an equivalence relation $\sim$ on $N$ by declaring that $f \sim g$ iff the function $f-g$ is bounded, and let ${\mathbb E}$ denote the set of equivalence classes.

It’s not difficult to show that defining $f+g$ in terms of pointwise addition and $f \cdot g$ in terms of composition of functions turns ${\mathbb E}$ into a commutative ring. And it turns out that this ring has a more familiar name… Before reading further, can you guess what it is?