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# 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
3 years, 9 months ago

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Watch the numberphile video which explains this very well at :http://www.youtube.com/watch?v=daro6K6mym8

- 3 years, 9 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 $$\frac{10^{-n}}{(1-10^{-n})^2}$$.

- 3 years, 9 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 - 3 years, 9 months ago

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

- 3 years, 9 months ago

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

- 3 years, 9 months ago