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  "body": "\n## Statement\n\nDefine an order on the [[rational numbers|0NCJ5WXN6Z]] as follows. For integers $a, c$ and positive integers $b, d$, let $\\frac{a}{b} \\le \\frac{c}{d}$ when $a d \\le b c$ in $\\mathbb{Z}$.\nThis relation is well defined, and with it $\\mathbb{Q}$ is an [[ordered field|010JCV9J4E]].\n\n## Remarks\n\nThe order on $\\mathbb{Q}$ extends the [[order on the integers|0F6FM9Y4Y6]].\nThis is the only order on $\\mathbb{Q}$ that makes it an ordered field.\nThe ordered field $\\mathbb{Q}$ is Archimedean, but it is not complete: the set $\\{q \\in \\mathbb{Q} : q^2 \u003c 2\\}$ has no least upper bound in $\\mathbb{Q}$.\n\n## Explanations\n\n### L1 human\n\nTo compare two fractions, write them with positive denominators and cross-multiply.\nFor example, $2/3 \u003c 3/4$ because $2 \\cdot 4 = 8 \u003c 9 = 3 \\cdot 3$.\n",
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