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  "body": "\n## Statement\n\nLet $\\alpha$ be a real [[algebraic number|0MB4TGMXG6]] of degree $d \\ge 2$.\nThen there is a constant $c \u003e 0$ such that $\\left| \\alpha - \\frac{p}{q} \\right| \u003e \\frac{c}{q^d}$ for all integers $p$ and $q$ with $q \u003e 0$.\nAs a consequence, each [[Liouville number|0J1R5NP36S]] is [[transcendental|079EGSD32Y]].\n\n## Remarks\n\nThe proof uses the minimal polynomial $f$ of $\\alpha$: $q^d f(p/q)$ is a nonzero integer, and the mean value theorem bounds $f(p/q)$ by $|\\alpha - p/q|$.\nFor a Liouville number $x$, the approximations are better than $c / q^d$ for every $d$, so $x$ is not algebraic.\nRoth's theorem improves the exponent $d$ to $2 + \\varepsilon$ for each $\\varepsilon \u003e 0$.\n\n## History\n\nJoseph Liouville proved the bound in 1844. With it, he gave the first proof that transcendental numbers exist.\n\n## Explanations\n\n### L1 human\n\nAlgebraic numbers cannot be approximated by fractions too well: the error is at least a constant times $1/q^d$.\nA number with better approximations of every order cannot be algebraic.\nThis gave the first explicit transcendental numbers.\n",
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