Writing Repeating Decimals as Fractions

Date: 11/8/95 at 19:12:32
From: Anonymous
Subject: Mathematics Grade 8

When you are expressing a repeating decimal as a fraction, why is it
you use 9, 99, 999, or 9999 etc. in the denominator?  

For example, 0.5555...(5 repeating) equals 5/9.

Please help and thanks in advance.

Jadon

Date: 11/12/95 at 14:37:56
From: Doctor Jeremy
Subject: Re: Mathematics Grade 8

It comes from a well-known result in series.  If you have the series

      1    1    1          1
x  =  - + -- + -- + ... + --
 n    a    2    3          n
          a    a          a

as n increases, the sum gets closer to 1/(a-1).  So as you 
keep going through the digits of .111111..., which is

 1     1     1     1
--- + --- + --- + --- + ...
10      2     3     4
      10    10    10

your sum gets closer and closer to 1/(10-1) = 1/9.  Since .5555 is 5 
times that, the sum is 5 times (1/9) or 5/9.

Similarly, you get a denominator of 99 because it is 100-1.

-Doctor Jeremy,  The Geometry Forum