SEARCH HOME
Math CentralQuandaries & Queries

search

Question from Emma, a student:

This week, my Algebra teacher told us about the pattern between infinitely repeating
decimals and their corresponding fractions.
(ex. .2222222...= 2/9, .151515...=15/99, 456456456...=456/999, etc.)
I was just wondering the reason why this pattern occurs.
Is there a certain element that causes this pattern to occur?\
Thanks
-Emma

Hi Emma,

I am going to try to explain using the repeating decimal 0.15151515...

Write 0.15151515... as

15/100 + 15/10000 + 15/1000000 + 15/100000000 + ...
=15/100 + 15/1002 + 15/1003 + 15/1004 + ...

This is the geometric series

a + ar + ar2 + ar3 + ar4 + ...

with a = 15/100 and r = 1/100.

I don't know if you have seen geometric series in your algebra classes yet but if you haven't you will see them soon. In a response to a previous question Claude and I showed that if |r| < 1 then the series

a + ar + ar2 + ar3 + ar4 + ...

approaches a/(1 - r) as the number of terms approaches infinity. In the case of 0.15151515.. this means that

15/100 + 15/10000 + 15/1000000 + 15/100000000 + ...
=15/100 + 15/1002 + 15/1003 + 15/1004 + ...
=(15/100)/(1 -1/100) = 0.15/99
=(15/100)/(99/100)
=15/99

A similar argument works for all the examples you sent.

Penny

About Math Central
 

 


Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.
Quandaries & Queries page Home page University of Regina PIMS