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$$ \frac{1}{121} = 0.00\ \overbrace{8264}\ \overbrace{4628}\ 09\ \overbrace{91735}\ \overbrace{53719} \ldots $$ The entire $22$-digit repetend appears here. It begins with the first digit after the decimal point. The sequence $8264$ gets reversed and appears as $4628$, and then the same happens with $91735$. But the $09$ between those two doesn't fit any such pattern that I've noticed.

Can anything intelligent be said about this? Is it an instance of some phenomenon that has other instances? Where is it mentioned in the literature?

Michael Hardy
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2 Answers2

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Try looking at it like this: $$ \frac{1}{121} = 0.0\ \overbrace{08264}^a\ \overbrace{46280}^\bar a\ 9\ \overbrace{91735}^b\ \overbrace{53719}^\bar b\ 0\ 08264 \ldots $$ It becomes much more obvious what's going on if you look at it with this patterning. Notice that $a+b=99999$ and $\bar a+\bar b=99999$, while $a$ and $\bar a$, and $b$ and $\bar b$, are digit-reversals of each other. The remaining two digits sum to 9, naturally.

Glen O
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    Well, I see that your "reputation" is 1961, a number that looks the same if you rotate it $180^\circ$. And now I've ruined that by up-voting this answer. – Michael Hardy May 05 '13 at 03:01
  • But I cannot doubt that there is more to be said. – Michael Hardy May 05 '13 at 03:02
  • Repeat length is set by finding the minimum $n$ for which $10^n \equiv 1 \pmod {121}$ In this case it is $22$. If it is even, you have $10^{\frac n2}\equiv -1 \pmod {121}.$ Then each digit in the second half of the repeat is 9 minus the corresponding digit in the first repeat. This explains Glen O's point that $a+b=99999$. The same behavior is seen in $\frac 17=0.\overline {142857}$ where the sum of the first three digits and the last three digits is $999.$ Another way to see it is $\frac 1n+\frac {n-1}n=0.\overline 9$ I still haven't explained the relation between $a$ and $\overline a$ – Ross Millikan May 05 '13 at 04:19
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Interesting, http://www.wolframalpha.com/input/?i=1%2F121. From the "more digits" tab, it looks as though the pattern begins with 00, then a 4 digit sequence and the reverse concatenation, and alternates to 09, followed by a 5 digit sequence and the reverse concatenation.

So the pattern suggests a sequence of the form 00(++++)(----)09(+++++)(-----)00... where the plus signs are the digit sequence and the minus signs are the reverse. The sequence (without it's reverse) after the 00 have an even numbered sequence of even digits while the sequence after 09 have an odd numbered sequence of odd digits.

very lovely, no clue as to why though.

marchimedes
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