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  "body": "\n## Summary\n\nThe integers can be constructed as classes of pairs of [[natural numbers|0MWKJ44V5A]].\nThe idea is that the pair $(a, b)$ stands for the difference $a - b$.\nOn the set $\\mathbb{N} \\times \\mathbb{N}$, define $(a, b) \\sim (c, d)$ when $a + d = b + c$. This relation is an equivalence relation.\nThe set $\\mathbb{Z}$ of [[integers|0C4JS9RF5M]] is the set of the equivalence classes $[(a, b)]$.\nThe operations on classes are $[(a, b)] + [(c, d)] = [(a + c, b + d)]$ and $[(a, b)] \\cdot [(c, d)] = [(a c + b d, a d + b c)]$.\nThese operations do not depend on the choice of the pairs in the classes.\n\n## Remarks\n\nThe negative of $[(a, b)]$ is $[(b, a)]$.\nThe map $n \\mapsto [(n, 0)]$ embeds $\\mathbb{N}$ in $\\mathbb{Z}$, and it respects addition and multiplication.\nEach class contains a pair of the form $(n, 0)$ or $(0, n)$. These classes are the integers $n$ and $-n$.\n\n## Explanations\n\n### L1 human\n\nThe pairs $(5, 2)$, $(4, 1)$ and $(3, 0)$ all stand for the integer $3$, because each first number is $3$ more than the second.\nThe pairs $(2, 5)$ and $(0, 3)$ stand for $-3$.\nThe construction makes new numbers from the natural numbers only, without the assumption that negative numbers already exist.\n",
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