Fraction to decimal converter
Convert a fraction to its decimal equivalent or a decimal back to a simplified fraction, with repeating decimals correctly notated.
How it works
A fraction is a division waiting to happen, and dividing it out by long division is what reveals whether it ends. At each step you carry a remainder; if a remainder ever comes back around to one you have already seen, the digits from that point on must repeat forever, because the same remainder can only ever produce the same next digit. That is how the repeating block above is found — not by dividing and guessing from a rounded 0.3333333, but by watching for the remainder to come back. Which fractions repeat is entirely decided by the denominator once the fraction is in lowest terms: if its only prime factors are 2 and 5 — the factors of ten — it terminates, and otherwise it repeats. So 1/4 and 1/8 stop neatly while 1/3, 1/6, and 1/7 never do. Going the other way, a terminating decimal is just its digits over a power of ten, reduced by the greatest common divisor, which turns 75/100 into 3/4.