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text: '[summary(Technical): A rational number is a number which may be written as $\\frac{a}{b}$, where $a$ is an [-48l] and $b$ is a nonzero integer.]\n\nThe rational numbers are either whole numbers or fractions of whole numbers, like $0,$ $1$, $2$, $\\frac{1}{2}$, $\\frac{97}{3}$, $-17$, $\\frac{-85}{1993},$ and so on. The [3jz set] of rational numbers is written $\\mathbb Q.$\n\n[54z Irrational numbers] like [49r $\\pi$] and [e $e$] are _not_ included in $\\mathbb Q;$ the rational numbers are only those numbers which can be written as $\\frac{a}{b}$ for integers $a$ and $b$ (where $b \\neq 0$).\n\nFormally, $\\mathbb{Q}$ is the [-3gz] of the [-field_of_fractions] of $\\mathbb{Z}$ (the [3gq ring] of [48l integers]).\nThat is, each $q \\in \\mathbb Q$ is an expression $\\frac{a}{b}$, where $b$ is a nonzero integer and $a$ is an integer, together with certain rules for addition and multiplication.\nThe rational numbers are the last intermediate stage on the way to constructing the [4bc real numbers], but they are also very interesting and important in their own right.\n\nOne intuition about the rational numbers is that once we've created the [4bc real numbers], then a real number $x$ is a rational number if and only if it may be written as $\\frac{a}{b}$, where $a, b$ are integers and $b$ is not $0$.\n[todo: add division as a requisite]\n\n# Examples\n\n- The integer $1$ is a rational number, because it may be written as $\\frac{1}{1}$ (or, indeed, as $\\frac{2}{2}$ or $\\frac{-1}{-1}$, or $\\frac{a}{a}$ for any nonzero integer $a$).\n- The number [49r $\\pi$] is not rational ([513 proof]).\n- The number $\\sqrt{2}$ (being the unique positive real which, when multiplied by itself, yields $2$) is not rational ([548 proof]).\n\n# Properties\n\n- There are infinitely many rationals. (Indeed, every integer is rational, because the integer $n$ may be written as $\\frac{n}{1}$, and there are infinitely many integers.)\n- There are countably many rationals ([511 proof]). Therefore, because there are [2w0 uncountably] many real numbers, [-almost_all] real numbers are not rational.\n- The rationals are [dense_metric_space dense] in the reals.\n- The rationals form a [481 field] ([4zr proof]). Indeed, they are a subfield of the real numbers.\n\n# Construction\n\nInstead of taking the reals and selecting a certain collection which we label the "rationals", [construction_of_complexes_from_naturals it is possible to construct the rationals] given access only to the natural numbers; and from the rationals we may construct the reals.\nIn some sense, this approach is cleaner than starting with the reals and producing the rationals, because the natural numbers are very intuitive objects but the real numbers are less so.\nWe can be closer to satisfying some deep existential unease if we can build the reals out of the much-simpler naturals.\n\nAs an analogy, being able to produce a block of wood given access to a wooden table is much less satisfying than the other way round, and we run into blocks of wood "in the wild" so we are pretty convinced that there actually are such things as blocks of wood.\nOn the other hand, we almost never see wooden tables in nature, so we can't be quite as sure that they're real until we've built one ourselves.\n\nSimilarly, everyone recognises broadly what a counting number is, and they're out there in the wild, but the rational numbers are somewhat less "natural" and their existence is less intuitive.\n\n[todo: add simpler lenses]',
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