# An interesting fraction

• Apr 24th 2012, 05:25 PM
Soroban
An interesting fraction

$\displaystyle\frac{1}{998,001} \;=\;0.\overline{000\,001\,002\,003\,004\,005\, \hdots\,996\,997\,999}\, \hdots$

The decimal representation contains all the 3-digit numbers except 998
. . and the 2997-digit cycle repeats forever.

This is just one of a family of such fractions.
Can you determine the underlying characteristic?
• Apr 24th 2012, 09:43 PM
princeps
Re: An interesting fraction
$f(n)=\frac{10^n}{\left(10^n-1\right)^2}$

where $n$ is a number of digits of integer (fractional) part of denominator .
• Apr 25th 2012, 12:44 AM
a tutor
Re: An interesting fraction
More interesting fractions..

$\frac{100}{9899}$

$\frac{10100}{970299}$

$\frac{10000}{970299}$
• Apr 25th 2012, 09:52 AM
Soroban
Re: An interesting fraction
Hello, a tutor!

Very nice fractions!

$\frac{100}{9899} \;=\;0.01\,01\,02\:03\;05\,08\,13\,\hdots$
. . . . . . .Fibonacci sequence

$\frac{10100}{970299} \;=\;0.01\,04\:09\,16\,25\,\hdots$
. . . . . . . . Squares

$\frac{10000}{970299} \;=\;0.01\,03\;06\,10\,15\,21\,\hdots$
. . . . . . . . Triangular numbers

One more . . .

$\frac{1}{89} \;=\;0.0\,1\,1\,2\,3\:5\,9\,5\,5\,\hdots$
. . . . . Fibonacci sequence

Code:

1
--  =  0.01
89      001
0002
00003
000005
0000008
00000013
000000021
.
.
.