Converting Repeating Decimals into Fractions

Date: 10/22/2003 at 00:31:18
From: mike
Subject: rational numbers

I am looking for a rational number with a repeating/non-terminating 
decimal expansion where the repeating pattern is 12 digits long.  How 
can I find such a number?

Date: 10/22/2003 at 02:01:08
From: Doctor Luis
Subject: Re: rational numbers

Hi Mike,

I'll teach you a simple method to convert repeating decimals into a
rational number form.  Let's say I want the five digit pattern 12345
to repeat itself over and over.

         _____
   x = 0.12345 or 0.12345123451234512345....

I can exploit the fact that the decimal repeats itself by multiplying
this decimal by 10^n, where n is the length of the pattern.  In our 
case, we multiply by 10^5 = 100,000.  That will move the decimal point 
5 places to the right:

                     _____          _____
           x =     0.12345 = 0.1234512345
                     _____
  100000 * x = 12345.12345

Note that the decimal part is the same as in the original number.
Hmm..so what would happen if we subtracted these two equations?
                     _____
  100000 * x = 12345.12345
                     _____
 -     1 * x =     0.12345
========================== 
   99999 * x = 12345

All the repeating decimals cancel when we subtract, and we get an 
integer on the right side of the equation! On the left side, 100,000x
- 1x = 99,999x.

Now we divide both sides by 99999 and we find that x, our original 
number, is equal to 12345/99999 or 4115/33333 (expressed as a reduced
fraction).  Thus, we know that:

         _____   12345   4115
  0.1234512345 = ----- = -----
                 99999   33333

You can check this result by dividing either fraction on your
calculator to see if you get the repeating decimal.

For numbers like y = 6.0121212..., where there's a finite pattern
before the repeating one, you have to be a bit more careful.  You need
to bring out the repeating decimal pattern like this

   y = 6 + 0.01212...
     = 6 + (1/10)*(0.121212...)

Now you can work with x = 0.1212... and apply the method you just
learned to express it as a fraction (x = 12/99 = 4/33).  Then
substitute it back into the expression for y:

   y = 6 + (1/10)*(4/33) = 6 + 4/330 = 6 + 2/165 = 992/165

You should now be able to apply this method help you find a rational 
number with a 12 digit long repeating pattern.

I hope this helped!  Let us know if you have any more questions.

- Doctor Luis, The Math Forum
  http://mathforum.org/dr.math/