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  "body": "\n## Statement\n\nThe number [[$\\pi$|0C1VYVNV2N]] is irrational.\n\n## Remarks\n\nIvan Niven gave a short proof by calculus. Assume $\\pi = a/b$, and let $f(x) = \\frac{x^n (a - b x)^n}{n!}$.\nThen $\\int_0^\\pi f(x) \\sin x \\, dx$ is a positive integer for every $n$. But it tends to $0$ as $n \\to \\infty$, a contradiction.\nIn 1882, Ferdinand von Lindemann proved the stronger fact that $\\pi$ is transcendental. Mathlib states irrationality as `irrational_pi`.\n\n## History\n\nJohann Heinrich Lambert gave the first proof in the 1760s. He used a continued fraction for $\\tan x$.\n\n## Explanations\n\n### L1 human\n\nThe number $\\pi$ is not a fraction of two integers.\nSo its decimal expansion never ends and never repeats.\n",
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