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  "body": "\n## Statement\n\nLet $x \\ge 0$ be a real number with a [[decimal expansion|06XY4X9430]] that is eventually periodic.\nThat is, there are $N \\ge 0$ and $p \\ge 1$ with $d_{k + p} = d_k$ for each $k \u003e N$.\nThen $x$ is a [[rational number|0NCJ5WXN6Z]].\n\n## Remarks\n\nProof: $10^{N+p} x - 10^N x$ is an integer, because the periodic tails cancel. Thus $x$ is a quotient of an integer by $10^N (10^p - 1)$.\nFor example, $x = 0.\\overline{36}$ gives $100 x - x = 36$, thus $x = 36/99 = 4/11$.\nThe converse is also true: the decimal expansion of each rational number is eventually periodic.\n\n## Explanations\n\n### L1 human\n\nA decimal that repeats the same block of digits forever is always a fraction.\nShift the decimal point by the length of the block and subtract: the endless tail disappears.\nFor $0.777\\dots$, this gives $10 x - x = 7$, thus $x = 7/9$.\n",
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