Mathematics
Repeating decimal to fraction converter
Type a recurring decimal with the repeating block in brackets, such as 0.1(6), and get its fraction in lowest terms. Or type a fraction and see the decimal with the repeating block marked.
Recurring decimals and their fractions
Every fraction, divided out, gives a terminating or a recurring decimal. It terminates when the denominator, in lowest terms, has only the factors 2 and 5; otherwise the digits repeat forever, because a division has only finitely many possible remainders and sooner or later one comes back. 1/7 = 0.(142857) repeats every six digits, 1/97 every 96.
Going back is finding the fraction. The school rule: the numerator is the number without its point minus the part before the repeating block; the denominator is as many 9s as the block has digits, followed by as many 0s as there are digits before it. For 0.1(6): (16 − 1) / 90 = 15/90 = 1/6.
The rule works because multiplying by a power of 10 shifts the block, and subtracting two copies of the number cancels the endless tails. It is also why 0.(9) is exactly 1: the difference between the two numbers would be smaller than any positive amount.
Common mistakes
- Using only 9s in the denominator when there are digits before the block: 0.1(6) needs 90, not 9.
- Forgetting to subtract the non-repeating part in the numerator.
- Not reducing the fraction: 15/90 is right, but the expected answer is 1/6.
Frequently asked questions
Why is 0.(9) equal to 1?
Because 1/3 = 0.(3) and three times 1/3 is 1, while three times 0.(3) is 0.(9). By the rule: 9/9 = 1. Two different numbers would have a positive gap between them, and here there is none.
How can I tell whether a fraction gives a recurring decimal?
Reduce it and factor the denominator: with only 2s and 5s the decimal terminates, otherwise it recurs. With neither 2 nor 5 it is purely recurring; with them alongside other factors it is mixed.
How long can the repeating block be?
At most the denominator minus one: 1/7 has 6 digits, 1/17 has 16, 1/97 has 96.
How this calculation works
Decimal a.b(c) with k digits b before the block and a block c of p digits: fraction = (abc − ab) / (99…9 00…0), with p nines and k zeros. Terminating a.b: ab / 10^k. Fraction to decimal: long division; the block starts when a remainder repeats, and its length is the stretch between the two repeats.
Related calculators
Prime factorisation
Prime factors of a number up to 30 digits, with the division ladder, divisors and a primality test.
GCD and LCM
Greatest common divisor and least common multiple with prime factorization.
Fractions
Addition, subtraction, multiplication and division of fractions, reduced to lowest terms.
Scientific notation
Scientific and engineering notation, significant figures and rounding of any number.