Construction of the integers from the natural numbers

Summary

The integers can be constructed as classes of pairs of natural numbers. The idea is that the pair $(a, b)$ stands for the difference $a - b$. On the set $\mathbb{N} \times \mathbb{N}$, define $(a, b) \sim (c, d)$ when $a + d = b + c$. This relation is an equivalence relation. The set $\mathbb{Z}$ of integers is the set of the equivalence classes $[(a, b)]$. The operations on classes are $[(a, b)] + [(c, d)] = [(a + c, b + d)]$ and $[(a, b)] \cdot [(c, d)] = [(a c + b d, a d + b c)]$. These operations do not depend on the choice of the pairs in the classes.