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)(5^{3})

If the denominator of any fraction has the form 2^{x}5^{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 (2^{2}) (3^{2}) (5^{2}) 1 = -------------- (2^{2}) (3) (5^{2})I say the above example

because the denominatorwill have a repeating decimalbe totally written as the product of 2’s and 5’s.cannotChecking with Mr. Calculator, I’m right. 3 / 900 = 0.0033333333 (repeating forever).

Now let us try another example:

9 9 --- = ------------- 900 (2^{2}) (3^{2}) (5^{2}) 1 = --------- (2^{2}) (5^{2})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.