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.

Remarks

The negative of $[(a, b)]$ is $[(b, a)]$. The map $n \mapsto [(n, 0)]$ embeds $\mathbb{N}$ in $\mathbb{Z}$, and it respects addition and multiplication. Each class contains a pair of the form $(n, 0)$ or $(0, n)$. These classes are the integers $n$ and $-n$.

Explanations

L1 human

The pairs $(5, 2)$, $(4, 1)$ and $(3, 0)$ all stand for the integer $3$, because each first number is $3$ more than the second. The pairs $(2, 5)$ and $(0, 3)$ stand for $-3$. The construction makes new numbers from the natural numbers only, without the assumption that negative numbers already exist.