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### Repeating Decimals

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Date: 01/24/2001 at 12:11:39
From: LoraAnn Barclift
Subject: Repeating decimals

By looking at the denominator, how can you determine if the decimal
representation of a fraction will terminate or repeat?

By observation I have concluded that the fraction will repeat if the
denominator is prime or a multiple of 3. Is there a theory in number
theory that justifies this conclusion?
```

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Date: 01/24/2001 at 16:19:27
From: Doctor Greenie
Subject: Re: Repeating decimals

Hi, LoraAnn -

You have done some nice work with your observations; however, your
conclusion is not correct (as is often the case when you try to draw a
conclusion from observation....)

(1) It is ALMOST always the case that the decimal representation is
repeating if the denominator is a prime.  The exceptions are 2 and 5.

(2) It IS always true that the decimal representation is repeating if
the denominator is a multiple of 3.

(3) There are many repeating decimals in which the denominator is
neither prime nor a multiple of 3 - such as 14 and 35.

To characterize the denominators of common fractions that will have
repeating decimal representations, it is easier to characterize the
denominators that will produce terminating decimal fractions. Then any
denominator that doesn't produce a terminating decimal will produce a
repeating decimal.

The key to identifying the denominators that produce terminating
decimals lies in the names for the decimal numbers.

Here are some fractions that have terminating decimal representations:

.5 = "five tenths" = 5/10  ( = 1/2 in reduced form)
.35 = "thirty-five hundredths" = 35/100  ( = 7/20)
.123 = "one hundred twenty-three thousandths" = 123/1000

And here are a few that have repeating decimal representations:

.33333... = "point 3 repeating" (?)  ( = 1/3)
.027027... = "point 027 repeating" (?)  ( = 1/37, believe it or
not...)
.416666... = ???  ( = 5/12; I don't know how you would say it in
words)

Look at the denominators in these examples and see what distinguishes
the denominators in the first group from those in the second group.

Write back if you want more help with this. You can also undoubtedly
find more information on this subject in the Dr. Math archives.  Click
on the "Search the Archives" link on the main Dr. Math page and try
"terminating decimal" or "repeating decimal" as the phrase to search
for.

- Doctor Greenie, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
Middle School Fractions

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