WolframAlpha shows for 1/53
$0.0\overline{1886792452830}$
as the repeating decimal. Why is it not
$0.\overline{0188679245283}$ instead?
For example, Wikipedia shows for 1/81
$0.\overline{012345679}$
as the repeating decimal, not
$0.0\overline{123456790}$
Here are some more examples of fractions and the repeating-decimal representations of them in Wolfram Alpha: \begin{align} \frac{1}{1665} = \frac{6}{9990} &= 0.000\overline{600}, \\[.7ex] \frac{617}{49999950} = \frac{1234}{99999900} &= 0.0000\overline{123400}, \end{align} but \begin{align} \frac{67}{666} = \frac{1}{10} + \frac{6}{9990} &= 0.1\overline{006}, \\[.7ex] \frac{2500306}{24999975} = \frac{1}{10} + \frac{1234}{99999900} &= 0.10\overline{001234}. \end{align}
The question is, why are so many decimal places displayed in the first two examples, where the "repetition" bar could just as well have been placed over earlier digits to write these numbers as $0.0\overline{006}$ and $0.00\overline{001234}$? After all, the second two examples show that Wolfram Alpha will sometimes show a repunit that begins with a zero.
I think a clue may be found in the "plaintext" representation of repeating decimals in the Wolfram language. For example, $1/1665$ is represented by
{{{6, 0, 0}}, -3}. That is, for a decimal fraction with some leading zeros, a kind of scientific notation is used. The-3signifies an exponent of $-3$, that is, the entire number is shifted three places to the right. That is, $$ \frac{1}{1665} = 0.\overline{600} \times 10^{-3}. $$ This is much like scientific notation, except that the decimal point is placed to the left of the first non-zero digit instead of to the right of it.The Wolfram representation of $67/6660$, on the other hand, is
{{1, {0, 0, 6}}, -1}, that is, $$ \frac{67}{6660} = 0.1\overline{006} \times 10^{-1}. $$The pattern seems to be to avoid a leading zero inside the curly brackets of the "plaintext" notation, preferring to use a negative exponent instead. When the only digits are repeating digits, this means the first digit of the repunit (the repeating block of digits) is non-zero. But if there are non-repeating non-zero digits, the number of digits before the repeating block is minimized, even if this means that the repeating block starts with a zero.
But Wolfram Alpha still has some surprises in its representation of repeating decimals. The equations below show some other repeating decimals as displayed by Wolfram Alpha: \begin{align} \frac{2000}{333} = \frac{6}{999} \times 1000 &= 6.00\overline{600}, \\[.7ex] \frac{1667}{333} = 5 + \frac{6}{999} &= 5.00\overline{600}, \\[.7ex] \frac{17003}{3330} = \frac{51}{10} + \frac{6}{999} &= 5.1\overline{060}. \end{align}
Interestingly, the first repeating decimal, $6.00\overline{600}$, has the "plaintext" representation
{{{6, 0, 0}}, 1}, that is, $$ 0.\overline{600} \times 10^1. $$ People are not accustomed to repunits that span the decimal point, so it makes sense that Wolfram Alpha would not display something like $\overline{6.00}$. Instead, the second repunit is displayed under the "repeating" bar. The "plaintext" representation of $5.1\overline{060}$ is{{5, 1, {0, 6, 0}}, 1}, which is exactly what is displayed. But the "plaintext" Wolfram representation of $5.00\overline{600}$ is{{5, {0, 0, 6}}, 1}, so why isn't it displayed as $5.\overline{006}$? I can only speculate about this: perhaps the rule for displaying numbers with repunits is to display the fractional part using the "plaintext" representation of the fractional part alone. For example, Wolfram Alpha tells us that $$ \frac{168500}{333} = 506.00\overline{600}, $$ not $506.\overline{006}$, and not $506.0\overline{060}$ despite the fact that the "plaintext" representation of this fraction is{{5, {0, 6, 0}}, 3}. That is, instead of putting the "repeating" bar over any of the repunits from the "plaintext" representation of the entire number, Wolfram Alpha uses the repunit from its "plaintext" representation of the fractional part.