Prime Factors in Unit Fractions

Date: 2/29/96 at 2:31:48
From: gary roberts
Subject: terminating decimals

Why does the denominator of a unit fraction have to have prime 
factors of 2 and/or 5 in order to terminate?  :-)

Date: 3/1/96 at 13:3:42
From: Doctor Ethan
Subject: Re: terminating decimals

Well, the easy answer is that those are the prime factors of 
ten, but let's see why that makes sense.

When you write numbers out as decimals you are essentially 
looking at them as a series of fractions.

For instance:

.13278  is

1/10  + 3/100 + 2/1000 + 7/10000 + 8/100000

Or 

13278
------
100000

Does that makes sense?

So when you have a fraction, it can be written as a finite 
decimal if and only if it can be written as a fraction in the 
form of 

n/ 10^k

that is some integer n divided by 10 to some power.

So for instance  3/25  = 12/100 = .12

or 7/16 = 35/80 = 175/400 = 875/2000 = 4375/10000 = .4375

However, if we had 3/7 then no multiplying by stuff on the top 
and bottom will ever make the bottom equal to a power of ten.

Does this make any sense?

If you want more help then write back to us.

-Doctor Ethan,  The Math Forum