Decimal to Fraction and Repeating Decimal Converter

Every terminating or repeating decimal is a fraction, and the recipe never changes.

A terminating decimal like 0.75 has 2 places, so it equals 75/100, which reduces by 25 to 3/4. A repeating decimal like 0.(3) never ends, but it is a fraction too. The nines-and-zeros rule says the denominator has one 9 for each digit of the repeating block and one 0 for each digit between the decimal point and the block, and the numerator is the number without the point minus the number without the point and without the repeating block. For 0.1(6) the numerator is 16 − 1 = 15, the denominator is 90 and 15/90 reduces to 1/6. For 2.4(54) the numerator is 2454 − 24 = 2430, the denominator is 990 and the result is 27/11, or 2 5/11 as a mixed number. The algebra method lands in the same place: call the number x, multiply by 10 and by 100 until the repeating blocks line up, then subtract (100x − 10x = 16 − 1). The reverse trip works too: a fraction in lowest terms gives a terminating decimal exactly when its denominator has only the prime factors 2 and 5; otherwise it repeats, and the period has at most (denominator − 1) digits (1/7 repeats 142857, 1/17 has 16 digits and 1/97 has 96). That is why 0.999... equals 1: 9/9 = 1. Irrational numbers like π or √2 have no such fraction; Approximate mode uses continued fractions to find the best fraction with a limited denominator (22/7 and 355/113 for π).

Type the number and the solution shows up right away.

1. Type a decimal (0.75), a repeating decimal with the repeating part in parentheses (0.1(6)), with an ellipsis (0.333...) or with a bar over the repeating digit. The buttons insert ( ), an ellipsis, a bar and a sign.
2. Read the simplified fraction, the mixed number and the decimal check, and switch between the nines-and-zeros rule and the algebra method to follow every step.
3. Use Fraction to decimal to find the repeating and non-repeating parts of any fraction, and Approximate to replace a long decimal with the best fraction under a denominator limit.

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Sources and references for this tool

These references help contextualize formulas, standards, APIs and limitations used on this page. They do not replace professional validation when a result has legal, financial, medical or operational impact.

• Repeating decimalWikipedia, Repeating decimals: the fraction rule, 0.999... = 1 and when a decimal terminates.

Frequently asked questions

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