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  "body": "\n## Definition\n\nLet $b \\ge 2$ be an integer, and let $x$ be a real number.\nWrite the fractional part $x - \\floor{x}$ in base $b$ as $0.d_1 d_2 d_3 \\ldots$ with digits $d_i \\in \\set{0, 1, \\dots, b-1}$.\nWhen there are two such expansions, take the one that does not end in an infinite run of the digit $b - 1$.\nFor a finite string $w$ of $k$ digits and an integer $n \\ge 1$, let $N(w, n)$ be the number of indices $i$ with $1 \\le i \\le n - k + 1$ such that $d_i d_{i+1} \\cdots d_{i+k-1} = w$.\nThe number $x$ is *normal in base $b$* if\n$$\\lim_{n \\to \\infty} \\frac{N(w, n)}{n} = b^{-k}$$\nfor every $k \\ge 1$ and every string $w$ of $k$ digits.\nThe number $x$ is *absolutely normal* if it is normal in base $b$ for every integer $b \\ge 2$.\n\n## Remarks\n\nExample: Champernowne's number $0.1234567891011121314\\ldots$, formed by the decimal digits of the positive integers in order, is normal in base $10$.\nNon-example: $x = 1/3 = 0.333\\ldots$ in base $10$ is not normal, since for $w = 1$ one has $N(w, n) = 0$ for all $n$, so the limit is $0 \\ne 1/10$.\nIn the same way, no [[rational number|0NCJ5WXN6Z]] is normal in any base, because its expansion is eventually periodic.\nA number is *simply normal* in base $b$ if the condition holds for strings of length $k = 1$.\nNormality in base $b$ is equivalent to the [[equidistribution modulo one|XM6DH57BWY]] of the sequence $(b^n x)_{n \\ge 0}$.\nHere $\\floor{x}$ is the [[floor|08GR54CKEA]] of $x$.\n",
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