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  "body": "\n## Statement\n\nLet $\\alpha$ be a real number, and let $p$ and $q$ be positive integers such that\n$$\\left| \\alpha - \\frac{p}{q} \\right| \u003c \\frac{1}{2 q^2}.$$\nThen $p / q$ is a [[convergent|0SRWNR42BQ]] of the [[continued fraction expansion|0SJ7Z6VKWK]] of $\\alpha$.\n\n## Remarks\n\nSo the very good rational approximations of $\\alpha$ are all convergents.\nThe constant $\\frac{1}{2}$ cannot be replaced by $1$: not every fraction with $\\left| \\alpha - \\frac{p}{q} \\right| \u003c \\frac{1}{q^2}$ is a convergent.\nA use: it shows that the solutions of [[Pell's equation|0AF3DANJ76]] come from the convergents of $\\sqrt{d}$.\n\n## History\n\nAdrien-Marie Legendre proved the theorem in his Essai sur la théorie des nombres of 1798.\n\n## Explanations\n\n### L1 human\n\nIf a fraction is extremely close to $\\alpha$, then the continued fraction algorithm finds it.\nSo to look for the best approximations of a number, you only need to compute its convergents.\n",
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