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