Decimals: Terminating or Repeating?

Date: 01/26/2001 at 15:10:31
From: Seegee
Subject: How to tell if decimals are terminating or repeating?

How can you tell just by looking at a fraction whether, in decimal 
form, it will terminate or repeat? My math teacher said there 
was a way, but I don't see how. Please help.

Date: 01/26/2001 at 15:49:23
From: Doctor Greenie
Subject: Re: How to tell if decimals are terminating or repeating?

Hi, Seegee -

If a decimal fraction terminates, then it has a name like one of the 
following:

    " ____ tenths"
    " ____ hundredths"
    " ____ thousandths"
    " ____ ten-thousandths"
    ...
    " ____ millionths"
    ...
    " ____ ten-billionths"
    ...

etc., etc.

When you write these numbers as common fractions, what is special 
about the denominators?

The answer to that question should be a big hint toward the answer to 
your question, but it won't give the complete answer. For example, 
here are a couple of fractions whose decimal representations terminate 
but that don't have names from the "infinite" list above: 3/4 (= .75) 
and 5/8 (= .625).

So why do these two have terminating decimals, while a fraction like 
1/3 does not? It is because the first two can be written as equivalent 
fractions with names from the list above, while the fraction 1/3 
cannot:

    3/4 = 75/100 = seventy-five hundredths
    5/8 = 625/1000 = six hundred twenty-five thousandths

but you can't write 

    1/3 = a/10
or      = b/100
or      = c/1000
or  ....

where a, b, c, or any other of the numerators are integers.

I have still only hinted at the precise answer to your question. If 
you can't quite figure out the whole answer after studying what I've 
written, you can find the complete answer in the Dr. Math archives.  
Click on the "Search the Archives" link on the main Dr. Math page and 
use "repeating decimal" or "terminating decimal" as the phrase to 
search for (do not use quotation marks, but be sure to click on the 
button that makes the search engine look for the entire phrase 
instead of the individual words).  The search will provide you with 
links to several pages where this question is discussed.

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

Date: 01/26/2001 at 15:32:14
From: Doctor Rob
Subject: Re: How to tell if decimals are terminating or repeating?

Thanks for writing to Ask Dr. Math, Seegee.

The fraction will terminate if and only if the denominator has for
prime divisors only 2 and 5, that is, if and only if the denominator
has the form 2^a * 5^b for some exponents a >= 0 and b >= 0. The
number of decimal places until it terminates is the larger of a 
and b.

The proof of this lies in the fact that every terminating decimal
has the form n/10^e, for some e >= 0 (e is the number of places to
the right that the decimal point must be moved to give you an integer,
and n is that integer), and every fraction of that form has a
terminating decimal found by writing down n and moving the decimal
point e places to the left. Now when you cancel common factors from 
n/10^e = n/(2*5)^e = n/(2^e*5^e), it may reduce the exponents
in the denominator, but that is all that can happen.

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