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  "body": "\n## Statement\n\nThe number [[$e$|0MVCFMW7S4]] is irrational.\n\n## Remarks\n\nAssume $e = a/b$ with positive integers $a, b$. Then $x = b! \\big( e - \\sum_{n=0}^{b} \\frac{1}{n!} \\big)$ is an integer.\nBut the [[series for $e$|0G4D3ZKTP8]] gives $0 \u003c x \u003c \\sum_{k=1}^{\\infty} (b + 1)^{-k} = \\frac{1}{b} \\le 1$. No integer lies strictly between $0$ and $1$.\nThis short proof is due to Joseph Fourier.\n\n## History\n\nLeonhard Euler wrote the first proof in 1737. It was published seven years later.\n\n## Explanations\n\n### L1 human\n\nThe series for $e$ converges so fast that $e$ cannot be a fraction.\nIf $e = a/b$, a suitable multiple of the tail of the series would be an integer between $0$ and $1$.\n",
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