Convert Recurring Decimals to Fractions
Enter a number using parentheses for the repeating digits.
- Pure recurring:
0.(3),2.(142857) - Mixed recurring:
1.2(34),12.45(9) - Terminating decimal:
0.125
What this recurring decimals as fractions calculator does
This recurring decimals as fractions calculator turns repeating decimal numbers into simplified fractions instantly.
It supports common notation where repeating digits are written in parentheses, such as 0.(6) or 3.41(27).
It also works for terminating decimals like 0.375, so you can use one tool for both decimal types.
Why convert recurring decimals into fractions?
Fractions are exact. Decimals can look exact, but repeating decimals are actually infinite. For example,
0.333... goes on forever and is exactly equal to 1/3. Converting to fractions helps with:
- Algebra and equation solving
- Comparing values precisely
- Working with ratios and proportions
- Avoiding rounding errors in calculations
How the conversion works
1) Pure recurring decimal
Example: x = 0.(7)
Multiply by 10 (one repeating digit): 10x = 7.(7)
Subtract: 10x - x = 7 so 9x = 7
Therefore, x = 7/9.
2) Mixed recurring decimal
Example: x = 1.2(34)
First shift past non-repeating part (1 digit): 10x = 12.(34)
Then shift one full repeat block (2 digits): 1000x = 1234.(34)
Subtract: 1000x - 10x = 1222
So 990x = 1222, and x = 1222/990 = 611/495.
3) Terminating decimal
Example: 0.125 = 125/1000, then reduce by dividing numerator and denominator by 125:
1/8.
Input format guide
- Use at least one digit before the decimal point:
0.(3)instead of.(3) - Use parentheses to mark repeating digits:
4.56(78) - Integers are valid too:
5becomes5/1 - Negative values are supported:
-2.1(6)
Examples you can try
0.(9)=1/10.(12)=4/333.(142857)=22/75.08(3)=61/12
FAQ
Does this calculator simplify fractions?
Yes. Results are reduced to lowest terms using a greatest common divisor (GCD) step.
Can I use large repeating blocks?
Yes. The calculator uses exact integer arithmetic in JavaScript (BigInt) so it can handle large
numerators and denominators without floating-point precision issues.
Is 0.(9) really equal to 1?
Yes. Mathematically, 0.999... is exactly 1. This tool will return 1/1.