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  "body": "\n## Summary\n\nThe rational numbers can be constructed as classes of pairs of [[integers|0C4JS9RF5M]].\nThe idea is that the pair $(a, b)$ with $b \\neq 0$ stands for the fraction $a / b$.\nOn the set of pairs $(a, b)$ with $a, b \\in \\mathbb{Z}$ and $b \\neq 0$, define $(a, b) \\sim (c, d)$ when $a d = b c$. This relation is an equivalence relation.\nThe set $\\mathbb{Q}$ of [[rational numbers|0NCJ5WXN6Z]] is the set of the equivalence classes, and we write $a / b$ for the class of $(a, b)$.\nThe operations are $\\frac{a}{b} + \\frac{c}{d} = \\frac{a d + b c}{b d}$ and $\\frac{a}{b} \\cdot \\frac{c}{d} = \\frac{a c}{b d}$. They do not depend on the choice of the pairs.\n\n## Remarks\n\nThe same construction works for each integral domain $R$. It gives the field of fractions of $R$, the smallest field that contains $R$.\nThe map $a \\mapsto a / 1$ embeds $\\mathbb{Z}$ in $\\mathbb{Q}$.\n\n## Explanations\n\n### L1 human\n\nA fraction is a pair of a top number and a nonzero bottom number.\nThe pairs $(1, 2)$ and $(2, 4)$ are different pairs, but $1 \\cdot 4 = 2 \\cdot 2$, so they give the same rational number.\nThe construction makes this precise: a rational number is the collection of all the pairs that name it.\n",
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