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  "body": "\n## Definition\n\nA real number $x$ is a Liouville number if for each positive integer $n$ there are integers $p$ and $q$ with $q \u003e 1$ and\n$$0 \u003c \\left| x - \\frac{p}{q} \\right| \u003c \\frac{1}{q^n}.$$\n\n## Remarks\n\nThe standard example is Liouville's constant $\\sum_{k=1}^{\\infty} 10^{-k!} = 0.110001000\\ldots$.\nEach Liouville number is irrational, and in fact [[transcendental|079EGSD32Y]].\nThe numbers $e$ and $\\pi$ are transcendental, but they are not Liouville numbers.\nThe Liouville numbers are uncountable, but they form a set of Lebesgue measure zero.\n\n## Explanations\n\n### L1 human\n\nA Liouville number has rational approximations that are extremely good, better than any fixed power of $1/q$.\nAlgebraic numbers cannot be approximated this well, so a Liouville number is not algebraic.\n",
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