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?

#Fractions #CosinesGroup #RecurringDecimals

Easy Math Editor

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
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Comments

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

Anish Puthuraya - 7 years, 5 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 $\frac{10^{-n}}{(1-10^{-n})^2}$ .

Cody Johnson - 7 years, 5 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?

Calvin Lin Staff - 7 years, 5 months ago

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

Cody Johnson - 7 years, 5 months ago

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

Rahul Saha - 7 years, 5 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$