Repeating decimals in rational numbers

When is a rational number going to have a repeating decimal?

It turns out there’s a really easy way to find out.

For example, given:

 3
---
250

I can tell you from now (without touching a calculator) that will have a TERMINATING DECIMAL.

Given

8
-
9

I can tell you from now (without touching a calculator) that will have a REPEATING DECIMAL.

How do I know?

All you have to do is factor the denominator into its prime factors.

 3         3           3
--- = ----------- = --------
250   (2)(5)(5)(5)  (2)(53)

If the denominator of any fraction has the form 2^x5^y with x, y positive integers, then that fraction represents a terminating decimal value. Otherwise, the fraction represents a repeating decimal value.

Said another way, if you can break down the denominator to being just a bunch of 2’s times a bunch of 5’s, then the fraction represents a repeating decimal.

More examples:

 3          3
--- = -------------
900   (22) (32) (52)

       1
= --------------
  (22) (3) (52)

I say the above example will have a repeating decimal because the denominator cannot be totally written as the product of 2’s and 5’s.

Checking with Mr. Calculator, I’m right. 3 / 900 = 0.0033333333 (repeating forever).

Now let us try another example:

 9          9
--- = -------------
900   (22) (32) (52)

      1
= ---------
  (22) (52)

So I say this WILL BE a terminating decimal, because the denominator just becomes 2 x 2 x 5 x 5, which is purely the product of 2’s and 5’s.

Indeed, 1/100 = 0.01.

And that’s the way it is.

Advertisements