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  "body": "\n## Statement\n\nLet $x$ be a real number.\nThe [[continued fraction expansion|0SJ7Z6VKWK]] of $x$ is finite if and only if $x$ is rational.\n\n## Remarks\n\nIf $x = a/b$ with $b \u003e 0$, the expansion follows the [[Euclidean algorithm|0TRNT4X7J0]] on $a$ and $b$. So it stops after finitely many steps.\nConversely, a finite simple continued fraction is a rational number.\nFor example, $\\frac{43}{19}$: $43 = 2 \\cdot 19 + 5$, $19 = 3 \\cdot 5 + 4$, $5 = 1 \\cdot 4 + 1$, $4 = 4 \\cdot 1$. So $\\frac{43}{19} = [2; 3, 1, 4]$.\nA rational number has exactly two finite expansions, since $[a_0; \\dots, a_n] = [a_0; \\dots, a_n - 1, 1]$ when $a_n \u003e 1$.\n\n## Explanations\n\n### L1 human\n\nFor a fraction, the continued fraction algorithm is the Euclidean algorithm in another form.\nThe remainders get smaller, so the process must end.\nAn irrational number gives an infinite continued fraction.\n",
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