About the Repeating Decimal to Fraction Calculator
This repeating decimal to fraction calculator turns a recurring decimal into the exact fraction it represents. Put the repeating digits in brackets — 0.(36) means 0.363636…, and 2.1(45) means 2.1454545… — or type a few repeats followed by "..." (0.8333...) and the calculator detects the repeating block for you. You get the simplified fraction, the mixed number, the numerator and denominator, and the multiply-and-subtract steps used to find it.
It is meant for students learning why every repeating decimal is rational, for checking homework on recurring decimals, and for anyone who has a calculator result like 0.142857142857 and wants to know it is 1/7. Mixed decimals with a non-repeating part before the cycle (like 0.58333…) are handled the same way.
When you use "..." the repeating block is a best guess: the shortest pattern that repeats at least twice at the end. For an unambiguous answer, mark the block with brackets or an overline.
How to use the repeating decimal to fraction calculator
- 1Type the decimal, putting the repeating digits in brackets: 0.(36) or 2.1(45).
- 2Or type a few repeats and end with "..." (for example 0.8333...).
- 3Read the simplified fraction and the mixed number.
- 4Follow the working to see the multiply-and-subtract method.
Formula and method
Write the decimal as an integer part I, a non-repeating block A with a digits and a repeating block R with r digits. Multiplying x by 10ᵃ⁺ʳ and by 10ᵃ produces two numbers with identical repeating tails; subtracting them cancels the infinite tail and leaves (10ᵃ⁺ʳ − 10ᵃ)·x = AR − A, where AR and A are read as whole numbers. Dividing and reducing by the greatest common divisor gives the fraction in lowest terms.
For a purely repeating decimal (a = 0) the rule is simply "repeating digits over that many 9s": 0.(36) = 36/99 = 4/11 and 0.(142857) = 142857/999999 = 1/7. Every repeating decimal is rational, and a fraction in lowest terms repeats exactly when its denominator has a prime factor other than 2 or 5.
- I
- Integer part
- A, a
- Non-repeating digits after the point and how many there are
- R, r
- Repeating block and its length
Worked examples
0.(36) = 0.363636…
Let x = 0.3636…. Then 100x = 36.3636…, and subtracting gives 99x = 36, so x = 36/99 = 4/11 after dividing by 9.
0.1(6) = 0.1666…
100x = 16.666… and 10x = 1.666…; subtracting gives 90x = 15, so x = 15/90 = 1/6.
2.1(45) = 2.1454545…
The non-repeating part is "1" and the repeating block is "45". 1000x − 10x = 2145.45… − 21.45… = 2124, so x = 2124/990 = 118/55 = 2 8/55.
0.142857142857... auto-detected
The shortest block that repeats is 142857 (six digits), so x = 142857/999999 = 1/7.
0.(9) equals 1
10x = 9.999… and x = 0.999…, so 9x = 9 and x = 1. A recurring 9 is just another way of writing the next number up.
Frequently asked questions
How do you convert a repeating decimal to a fraction?+
Set x equal to the decimal, multiply by a power of 10 that shifts one full repeating block left of the point, and subtract the original so the repeating tails cancel. Solve for x and simplify. For 0.(7): 10x − x = 7, so x = 7/9.
What is 0.333… as a fraction?+
0.333… = 3/9 = 1/3. Similarly 0.666… = 2/3, 0.111… = 1/9, 0.0909… = 1/11 and 0.8333… = 5/6.
Is 0.999… really equal to 1?+
Yes. Using the same method, 10x − x = 9.999… − 0.999… = 9, so x = 1. There is no number between 0.999… and 1, so they are the same real number.
Are all repeating decimals rational?+
Yes — every repeating or terminating decimal can be written as a fraction of two integers, and conversely every fraction gives a decimal that eventually repeats or terminates. Irrational numbers like π never settle into a repeating block.
How do I enter a decimal where only some digits repeat?+
Put only the repeating digits in brackets. For 0.58333… type 0.58(3); for 1.2454545… type 1.2(45). The digits before the bracket are the non-repeating part.