1/3 is 0.(3), not 0.333333
Divides numerator by denominator and gives the exact answer: a terminating decimal, or a recurring one with the period in brackets rather than a truncated tail.
A fraction is a division written down, so converting means dividing the top by the bottom. The interesting part is why some divisions stop and others never do.
The rule is short. If the simplified denominator contains only twos and fives, the decimal terminates. Any other prime factor — three, seven, eleven — produces a repeating block. That is why 1/8 is exact and 1/3 and 1/7 are not.
The exact decimal with the repeating block marked when there is one The simplified fraction and an approximate value for calculations The same quantity as a percentage The period is found by honest long division tracking remainders, so even long ones show up: 1/7 repeats on 142857. Very long periods are cut at forty digits, beyond which the notation stops being readable. Can I tell in advance whether it repeats? Simplify the fraction and factorise the denominator. Only twos and fives means it terminates. Any other prime means it repeats.
What do the brackets mean? The repeating group. 0.41(6) means 0.41666… forever. It is standard mathematical notation, not an abbreviation.
How long can a period be? At most the denominator minus one. 1/7 repeats over six digits — 142857 — which is the maximum for seven. Such denominators are called full reptend primes.
How do I convert a repeating decimal back to a fraction? A period of n digits sits over n nines: 0.(3) = 3/9 = 1/3, and 0.(27) = 27/99 = 3/11. Digits before the period add zeros after the nines.
Worked example Five eighths
Input: 5/8
Output: 0.625 — it terminates
Note: The denominator 8 is 2³, twos only, so no period appears. 5/12 gives 0.41(6) instead: twelve carries a factor of three.
