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 2x5y 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.