The mysterious fractions

What value do you get when you convert 181\frac {1}{81}to decimal? You get 0.0123456790123456790...0.0123456790123456790....

What value do you get when you convert 19801\frac {1}{9801} to decimal? You get 0.000102030405060708091011...96979900010.000102030405060708091011...9697990001.

What value do you get when you convert 1998001\frac {1}{998001} to decimal? You get 0.000001002003...100101102...996997999000...0.000001002003...100101102...996997999000....

These decimals list every nn digit numbers (81 is 1, 9801 is 2, 998001 is 3, etc.) apart from the second last number. There is a pattern to find one of these fractions.

Can you see something special about the denominators? 8181 is 929^2, 98019801 is 99299^2, 998001998001 is 9992999^2.

This means that if you did 199980001\frac {1}{99980001} you would get 0.0000000100020003...9996999799990000...0.0000000100020003...9996999799990000....

Can you find a fraction that, when converted to decimal, lists every nn digit number?

Note by Sharky Kesa
6 years, 7 months ago

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Watch the numberphile video which explains this very well at :

Anish Puthuraya - 6 years, 7 months ago

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This is a result of generating functions, and plugging x=110x=\frac{1}{10} into the function. For example, the generating function for the Fibonacci sequence is x1xx2\frac{x}{1-x-x^2}. If we plug in x=110x=\frac{1}{10}, we get 1/101(1/10)(1/10)2=10100101=1089=0.11235\frac{1/10}{1-(1/10)-(1/10)^2}=\frac{10}{100-10-1}=\frac{10}{89}=0.11235\dots.

To answer your question, sure you can. If you want to list every nn-digit number, you'll want to have the sum i=1i×10inn\sum_{i=1}^\infty i\times10^{-in-n}. Recall that the generating function for ii is 1(1x)2\frac{1}{(1-x)^2}, so we'll have 10n(110n)2\frac{10^{-n}}{(1-10^{-n})^2}.

Cody Johnson - 6 years, 7 months ago

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Though, for the Fibonacci sequence, note that with x=110 x = \frac{1}{10} , you 'add' the tens digit to the preceding units digit, so you don't get the sequence of 0.1123581321340.112358132134\ldots , but instead 1089=0.1123595 \frac{10}{89} = 0.1123595\ldots . Ah, if only patterns were verified by checking the first 5 terms.

Here's a spinoff question.

Is 0.112358132134 0.112358132134 \ldots rational or irrational?

Calvin Lin Staff - 6 years, 7 months ago

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You see the effects of carrying the digit, yes, but it seemed more magical to post the first 5 digits.

Cody Johnson - 6 years, 7 months ago

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Irrational. I would provide a proof,but then I would be guilty of stealing your answer from MSE.

Rahul Saha - 6 years, 7 months ago

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