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?

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