# The mysterious fractions

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

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

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

These decimals list every $n$ 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? $81$ is $9^2$, $9801$ is $99^2$, $998001$ is $999^2$.

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

Can you find a fraction that, when converted to decimal, lists every $n$ 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 :http://www.youtube.com/watch?v=daro6K6mym8

- 6 years, 7 months ago

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

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

- 6 years, 7 months ago

Though, for the Fibonacci sequence, note that with $x = \frac{1}{10}$, you 'add' the tens digit to the preceding units digit, so you don't get the sequence of $0.112358132134\ldots$, but instead $\frac{10}{89} = 0.1123595\ldots$. Ah, if only patterns were verified by checking the first 5 terms.

Here's a spinoff question.

Is $0.112358132134 \ldots$ rational or irrational?

Staff - 6 years, 7 months ago

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

- 6 years, 7 months ago

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

- 6 years, 7 months ago