Repeating Decimal to Fraction Calculator
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Enter Your Decimal
Type any decimal number into the calculator. Supports standard decimals, repeating patterns, and measurement values.
Instant Conversion
Our algorithm converts your decimal to a fraction instantly, finding the greatest common divisor to simplify it to the lowest terms.
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See your fraction in simplified form, as a mixed number, and with a complete step-by-step breakdown of the conversion process.
What is a Repeating Decimal?
A repeating decimal (or recurring decimal) has a sequence of one or more digits that repeats infinitely. Examples include 0.333... and 0.1666...
How to Convert Repeating Decimals to Fractions
We use algebra to eliminate the repeating part:
- Let x equal the repeating decimal.
- Multiply x by a power of 10 to shift the decimal point past the first repeating block.
- Multiply x by a power of 10 to shift the decimal point just before the repeating block.
- Subtract the second equation from the first to eliminate the repeating part.
- Solve for x to find the fraction, then simplify using the GCD.
Common Repeating Decimals
| Decimal | Fraction |
|---|---|
| 0.111... | 1/9 |
| 0.1666... | 1/6 |
| 0.333... | 1/3 |
| 0.666... | 2/3 |
| 0.8333... | 5/6 |
| 0.0909... | 1/11 |
Standard Calculator
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16ths Calculator
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Frequently Asked Questions
Common questions about repeating decimals
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01 What is a repeating decimal?
A repeating decimal is a decimal number that has a sequence of one or more digits that repeat infinitely, such as 0.333... or 0.142857...
02 How do you convert a repeating decimal to a fraction?
We use an algebraic method. Let x equal the repeating decimal, multiply by a power of 10 to shift the decimal, subtract to cancel out the repeating part, and solve for the fraction.
03 What does the bar notation mean?
The bar notation (vinculum) placed over a sequence of digits indicates that those digits repeat forever. For example, 0.16 with a bar over the 6 means 0.16666...
04 Are all repeating decimals rational numbers?
Yes, all repeating decimals are rational numbers, meaning they can always be expressed exactly as a fraction containing integers.
05 What is the fraction for 0.999...?
The repeating decimal 0.999... is exactly equal to 1. Using the algebraic method, if x = 0.999..., then 10x = 9.999... Subtracting x gives 9x = 9, so x = 1.
06 How to convert recurring decimal to fraction calculator?
'Recurring' and 'repeating' decimals mean the same thing — digits that repeat infinitely (e.g., 0.666... or 0.272727...). Our repeating decimal to fraction calculator handles both pure repeating decimals (like 0.333...) and mixed repeating decimals (like 0.1666...) with full step-by-step algebraic solutions.
07 How do I convert a repeating decimal to a fraction?
Use the algebraic method: Let x equal the repeating decimal (e.g., x = 0.333...). Multiply both sides by 10 (or 100, etc.) to shift the decimal (10x = 3.333...). Subtract the original equation from the new one (10x - x = 3), which gives 9x = 3, so x = 3/9 = 1/3. For mixed repeating decimals, adjust the multiplier to align the repeating blocks.