Recurring Decimal to Fraction Converter
Use this calculator to convert repeating (recurring) decimals into a simplified fraction.
Examples: 0.(3) → whole 0, non-repeating empty, repeating 3 | 2.1(6) → whole 2, non-repeating 1, repeating 6
How to use this recurring decimal calculator
A recurring decimal (also called a repeating decimal) is a decimal where one or more digits repeat forever. This tool converts that number into an exact fraction and then simplifies it.
- Enter the whole number part (left of the decimal point).
- Enter any non-repeating digits after the decimal point (if there are any).
- Enter the repeating block of digits.
- Click Convert to Fraction to see the exact and simplified result.
What is a recurring decimal?
A recurring decimal has a repeating cycle. For example: 0.3333..., 1.272727..., and 5.104444.... In shorthand notation, we write these as 0.(3), 1.(27), and 5.10(4).
Every recurring decimal can be written as a rational number, which means it can be expressed as a fraction of integers. That is exactly what this repeating decimal to fraction converter does.
Conversion method (the math behind it)
Case 1: Pure repeating decimal
If the number is 0.(ab) where ab repeats, then: numerator = repeating block, denominator = as many 9s as repeating digits. Example: 0.(45) = 45/99 = 5/11.
Case 2: Mixed decimal (non-repeating + repeating)
If the number looks like W.N(R) where:
- W is the whole number part,
- N is the non-repeating part (length m),
- R is the repeating part (length k),
then the fractional part is:
(N × (10k − 1) + R) / (10m × (10k − 1)).
Add W, then simplify by dividing top and bottom by the greatest common divisor (GCD).
Examples
- 0.(3) → 1/3
- 0.(6) → 2/3
- 1.2(5) → 113/90
- 2.1(6) → 13/6
- 0.08(3) → 1/12
Common mistakes when converting repeating decimals
- Forgetting to separate non-repeating and repeating digits.
- Using the wrong count of 9s and 0s in the denominator setup.
- Not simplifying the final fraction.
- Dropping the negative sign for negative values.
Why this tool is useful
Whether you are in middle school algebra, high school math, exam prep, or tutoring, this recurring decimal as a fraction calculator helps you verify answers instantly. It is especially useful for homework checks, fraction drills, and understanding rational numbers.
FAQ
Can repeating decimals always be fractions?
Yes. Every repeating decimal is rational and can be written exactly as a fraction.
What if I leave repeating digits blank?
The calculator will treat your input as a terminating decimal and still convert it to a fraction. For a true recurring decimal conversion, include the repeating block.
Does this calculator simplify fractions?
Yes. It shows both the unsimplified and simplified forms, plus a mixed number when relevant.