Terminating or repeating?

Divide the numerator by the denominator and one of two things happens:

• it terminates: the remainder becomes 0 and the decimal stops, as in 3/8 = 0.375;
• it repeats for ever: 5/11 = 0.454545…, with 45 repeating.

"Why must it repeat?" Dividing by 7, the only possible remainders are 1 to 6 (a remainder of 0 would end the division). So within at most six steps some remainder must come back, and from then on the division goes round the same loop.

You can tell which, without dividing. Write p/q in lowest terms, then break q into primes.

Only 2s and/or 5s in q → the decimal terminates. You can then make the denominator a power of 10, because 10 = 2 × 5: 3/20 = 3/(2² × 5) = 15/100 = 0.15. If q has any other prime (3, 7, 11, …), no power of 10 can be reached, and the decimal repeats. So 2/3 = 0.666… does not terminate.

Remember 'lowest terms' first. 3/15 looks as if it has a 3 in the denominator, but 3/15 = 1/5 = 0.2, which terminates.

How many decimal places? The larger of the two powers of 2 and 5: 18/125 = 18/5³ = 144/1000 = 0.144 (3 places). A denominator 2³ × 5 also gives 3 places.