Fractions with Repeating Decimals

Date: 02/24/97 at 14:08:30
From: Anonymous
Subject: Fractions with Repeating Decimals

How do you know if a fraction will have a repeating decimal or if it 
will terminate?

Thanks!

Date: 03/14/97 at 11:45:59
From: Doctor Bill
Subject: Re: Fractions with Repeating Decimals

Dear Cameron,

The answer is not too obvious.  

First, put the fraction in lowest terms.  Now look at the denominator.  
If the only prime factors of the denominator are 2's and 5's, then the 
decimal expansion of the fraction will terminate.  Otherwise, it will 
repeat.  Another way to say this condition is that the denominator 
(when the fraction is in lowest terms) can be written of the form 
(2^n)*(5^m), where n and m are integers greater than or equal to zero 
(the ^ means exponent).

Why is this true?  It has to do with the fact that our number system 
is in base 10 and the only factors to 10 are 5 and 2.  If you take any 
terminating fraction, like 4.25, you can multiply it by a power of 10 
to get an integer.  This is illustrated in the following equation (p/q 
is the fraction representation for 4.25 in lowest terms):

4.25 = 100 * p/q  =  an integer.

Thus the 100 must have cancelled out q, and therefore the only factors 
of q were 2's and 5's.

We hope this helps,
                   
-Doctors Bill and Ken,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/