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  "body": "\n## Definition\n\nLet $x$ be a real number.\nPut $x_0 = x$ and $a_0 = \\lfloor x_0 \\rfloor$.\nFor $n \\ge 0$, if $x_n \\neq a_n$, put $x_{n+1} = \\dfrac{1}{x_n - a_n}$ and $a_{n+1} = \\lfloor x_{n+1} \\rfloor$. If $x_n = a_n$, the process stops.\nThe continued fraction expansion of $x$ is the [[simple continued fraction|06XF7BCFFJ]] $[a_0; a_1, a_2, \\dots]$, finite or infinite.\n\n## Remarks\n\nHere $\\lfloor y \\rfloor$ is the largest integer that is at most $y$. Since $0 \u003c x_n - a_n \u003c 1$, each $a_{n+1}$ with $n \\ge 0$ is a positive integer.\nFor example, $\\frac{43}{19} = [2; 3, 1, 4]$, and $\\sqrt{2} = [1; 2, 2, 2, \\dots]$.\nThe process stops if and only if $x$ is rational, by the [[termination theorem|0PMY0YQYXX]].\nFor a rational $x$, the steps are the steps of the [[Euclidean algorithm|0TRNT4X7J0]] on its numerator and denominator.\n\n## Explanations\n\n### L1 human\n\nTake the integer part of $x$, then flip the remaining fractional part, and repeat.\nThe integer parts that you collect are the terms of the continued fraction of $x$.\n",
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