Repeating Decimal to Fraction
The calculator converts a repeating decimal to a fraction using the rule of nines and shows a step-by-step solution.
The repeating decimal calculator converts a repeating decimal to a fraction: 0.(3) = 1/3, 0.1(6) = 1/6. Enter the repeating part in brackets.
What is a repeating decimal
A repeating decimal is an infinite decimal in which the same group of digits repeats forever. This group is called the period (repetend) and is written in brackets (or with a bar over it):
0.333… = 0.(3), 0.1666… = 0.1(6), 2.142857142857… = 2.(142857)
- A purely repeating decimal starts repeating right after the point: 0.(3).
- A mixed repeating decimal has other digits between the point and the period: 0.1(6).
Rule for converting to a fraction
- Numerator: the number formed by all digits up to the end of the first period, minus the number formed by the digits before the period.
- Denominator: as many nines as there are digits in the period, followed by as many zeros as there are digits between the point and the period.
Example 1. 0.(3) = 3 − 0/9 = 3/9 = 1/3
Example 2. 0.1(6) = 16 − 1/90 = 15/90 = 1/6
Example 3. 1.25(07) = 12507 − 125/9900 = 12382/9900 = 6191/4950
Why the rule works
Let x = 0.(3). Multiply by 10: 10x = 3.(3). Subtract the original number: 10x − x = 3, so 9x = 3 and x = 3/9 = 1/3. Multiplying by 10, 100, 1000… shifts the period, and the infinite tails cancel when subtracting — that is where the nines in the denominator come from.
Fun fact: 0.(9) = 9/9 = 1 — they are the same number.