When fractions with the denominators of 2 4 8 16 32 produce terminating decimals fractions with the denominators 6 12 18 24 produce repeating decimals why?

Answer 1

Decimal numbers are based on powers of 10. The only factors of 10 are 2 and 5. Any denominator of the form 2^x*5^y will divide 10^x or 10^y - whichever is larger.

In the above example, the first set of denominators are all of the form 2^x and so will divide 10^x. So all fractions will have decimal expressions with x decimal digits (assuming you start with the fraction in simplified form).

The second set of denominators include 3 as a factor and so none of these denominators can be expressed as 2^x*5^y and so none of them are a factor of 10^z for any z.

All that may sound complicated. At its simplest, the rule is that if the only prime factors of the denominator are 2 and 5, then you will get a terminating decimal. Anything else and you won't.