Repeating Decimal to Fraction Calculator
Convert repeating decimals into simplified fractions, mixed numbers, and clear step-by-step algebra you can check by division.
Repeating decimals have exact fraction values
A repeating decimal to fraction calculator rewrites a decimal with a repeating block as an exact rational number. Enter the non-repeating digits and the repeating pattern using parentheses, such as 0.(3) or 1.2(34).
A repeating decimal to fraction calculator converts repeating decimals into exact fractions. Enter the decimal, mark the repeating digits, then simplify the result. For example, 0.(3) equals 1/3, and 0.(27) equals 27/99, which simplifies to 3/11.
Supported notation: Use parentheses for the repeating part, brackets like 0.[6], or ellipses like 0.333... when the pattern is obvious.
Exact arithmetic: The calculator uses integer arithmetic, reduces by the greatest common divisor, and avoids decimal rounding.
Quick check: Divide the final numerator by the final denominator. The decimal should repeat with the same block.
Simplified Fraction
For --
Mixed Number
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Shown when the absolute value is greater than 1.
Repeating Block
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Unreduced Setup
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This is reduced by the greatest common divisor.
Check
Divide Back
Numerator divided by denominator returns the decimal pattern.
Step-by-step notes
How to Use This Calculator
- Enter the decimal: Put the repeating digits inside parentheses, such as 0.(6) or 2.41(6).
- Use ellipses only for obvious repeats: The calculator can read examples like 0.333... or 1.1666....
- Convert the value: The calculator builds the fraction from the non-repeating and repeating digit counts.
- Review the simplified result: The fraction is reduced by the greatest common divisor.
- Check by division: Divide the numerator by the denominator to confirm the repeating decimal pattern.
Repeating Decimal Rules of Thumb
A repeating decimal is a decimal number where one digit or a block of digits continues forever. Because the repetition is predictable, every repeating decimal can be written as a fraction.
The denominator pattern depends on how many digits repeat and how many decimal places come before the repeat. A one-digit repeat uses a 9, a two-digit repeat uses 99, and a three-digit repeat uses 999 before place-value shifts are applied.
- Pure repeating decimal: 0.(7) becomes 7/9.
- Mixed repeating decimal: 0.1(6) has one non-repeating digit before the repeat.
- Terminating decimal: 2.125 converts with a power of 10 in the denominator, then reduces.
- Negative decimal: Convert the absolute value first, then apply the negative sign to the fraction.
The same method works for long repeating blocks like 0.(142857), which is the decimal form of 1/7.
Common Decimal Forms
| Decimal Type | Example | Fraction Setup | Simplified Result | Notes |
|---|
Tip: Parentheses are the clearest way to mark exactly which digits repeat.
Repeating Decimal to Fraction Formula
Use this formula when a decimal has a nonrepeating part followed by a repeating cycle. Let A be the integer made from the whole number, nonrepeating digits, and one repeat block. Let B be the integer made from the whole number and nonrepeating digits only.
Fraction = (A - B) / (10^n x (10^r - 1))
Here, n is the number of nonrepeating decimal digits and r is the number of repeating digits. Reduce the fraction after applying the formula.
Example: 0.1(6)
A = 16, B = 1, n = 1, and r = 1, so (16 - 1)/(10 x 9) = 15/90 = 1/6.
Terminating Decimal Formula
For a terminating decimal with d decimal places, use digits / 10^d, then reduce. For example, 2.125 = 2125/1000 = 17/8.
Step-by-Step Method
The cleanest conversion method is to count the non-repeating digits, count the repeating digits, then subtract the shifted integer strings.
1. Split the Digits
Identify the whole number, the non-repeating decimal digits, and the repeating block.
2. Build the Fraction
Subtract the non-repeat integer string from the repeat-included string, then divide by the matching 9s and 0s denominator.
3. Reduce
Divide the numerator and denominator by their greatest common divisor to get the simplest fraction.
Further reading: Khan Academy: Writing Repeating Decimals as Fractions
Where Fraction Conversion Is Useful
Exact fractions are easier to use in algebra, ratios, measurement conversions, and classroom math because they avoid rounded decimal approximations.
Homework and tests: Many math problems expect an exact fraction instead of a rounded decimal answer.
Measurements: Repeating decimals sometimes appear when converting between metric, imperial, and fractional units.
Algebra: Fractions preserve exact values when solving equations or simplifying expressions.
How to Check the Answer
The easiest check is to divide the final numerator by the final denominator. The quotient should match the original decimal and repeat in the same place.
Fraction check = numerator divided by denominator
For example, 1/6 = 0.1666..., so it can be written as 0.1(6).
- Check the repeat mark: Make sure only the repeating digits are inside parentheses.
- Check the sign: Negative decimals should produce negative fractions.
- Check simplification: The numerator and denominator should not share a common factor greater than 1.
Interesting Fact
Repeating decimals are not approximate values. They are exact rational numbers with a pattern that continues forever. A famous example is 0.999..., which equals 1 because 9/9 reduces to 1.
Frequently Asked Questions
What does a repeating decimal to fraction calculator do?
It converts a decimal with a repeating digit pattern into an exact fraction. Instead of rounding the number, the calculator uses place value to build a rational number and then reduces the result to a clean solution.
How should I mark the repeating digits or bar notation?
Put the repeating digits in parentheses, which is the text-entry version of bar notation. For example, type 0.(3) for a bar over 3 and 1.2(34) when the cycle is 34.
Can the calculator handle a decimal with no repeating cycle?
Yes. If there is no repeating cycle, the calculator treats the input as a terminating decimal. The conversion places the digits over the correct power of 10, then simplifies the fraction.
Why does a repeating decimal always become a rational number?
A repeating decimal is rational because its infinite pattern can be canceled with subtraction after multiplying by powers of 10. In algebra terms, that creates a finite equation, and solving the equation gives an exact fraction.
What if the nonrepeating part or ellipsis notation is ambiguous?
Use parentheses if there is any doubt about the nonrepeating part. For example, 0.123123... is clearer as 0.(123), while 0.12(3) means only the final digit repeats.
How does the formula reduce the numerator and denominator?
The formula first creates a setup fraction, then divides the numerator and denominator by their greatest common divisor. That produces the simplest equivalent fraction, such as reducing 3/9 to 1/3.
Is 0.999... really equal to 1 in math?
Yes. In math, the repeating decimal 0.(9) converts to 9/9, and 9/9 simplifies to 1. The fraction and the decimal name the same value.
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Disclaimer: This repeating decimal to fraction calculator is an educational math tool. Always verify exact values for coursework, exams, engineering, finance, or any setting where precision matters.
Last updated: May 5, 2026