Recurring Decimal to Fraction Calculator
Convert any repeating decimal into an exact, fully simplified fraction in seconds. Enter the non-repeating and repeating digits below to see the precise fraction plus the full algebraic working.
Decimal Details
Split your recurring decimal into its parts and convert it to a fraction
Leave as 0 if your decimal is less than 1, e.g. for 0.333…
Digits right after the decimal point that do NOT repeat. Leave blank if the repeating block starts immediately, as in 0.333…
The digit or block of digits that repeats forever, written without the bar or dots.
Your Fraction Result
Simplified fraction, decimal check, and full working
Enter your recurring decimal’s digits above and click Convert to Fraction to see the exact simplified fraction and step-by-step working.
The Algebra Behind Recurring Decimals
Every recurring decimal is a rational number, meaning it can always be written as a fraction of two whole numbers. The trick is multiplying the decimal by powers of ten so the repeating block cancels out when you subtract one equation from another.
π Pure Recurring Decimals
When the repeating block starts immediately after the decimal point, such as 0.3333… or 0.142857142857…, the numerator is simply the repeating block of digits, and the denominator is a string of 9s with one 9 for each repeating digit.
0.3Μ = 3 β 9 = 1 β 3
Once you have the numerator and denominator, divide both by their greatest common divisor to reach the simplified fraction.
π Mixed Recurring Decimals
When some digits don’t repeat before the repeating block begins, such as 0.16666… where the 1 doesn’t repeat but the 6 does, the formula adjusts for both parts: subtract the non-repeating digits from the full digit string, then divide by 9s followed by 0s.
0.16Μ = (16 β 1) β 90 = 15 β 90 = 1 β 6
The number of 9s matches the repeating digits, and the number of trailing 0s matches the non-repeating digits.
Common Recurring Decimals as Fractions
These familiar repeating decimals come up constantly in maths, exams, and everyday calculations. Use this table to check your conversions at a glance.
| Recurring Decimal | Fraction | Simplified | Repeating Block |
|---|---|---|---|
| 0.111… | 1β9 | 1β9 | 1 digit |
| 0.222… | 2β9 | 2β9 | 1 digit |
| 0.333… | 3β9 | 1β3 | 1 digit |
| 0.666… | 6β9 | 2β3 | 1 digit |
| 0.999… | 9β9 | 1 | 1 digit |
| 0.1666… | 15β90 | 1β6 | 1 digit (mixed) |
| 0.8333… | 75β90 | 5β6 | 1 digit (mixed) |
| 0.121212… | 12β99 | 4β33 | 2 digits |
| 0.142857142857… | 142857β999999 | 1β7 | 6 digits |
| 0.090909… | 9β99 | 1β11 | 2 digits |
Recurring Decimals FAQ
Everything you need to know about converting repeating decimals into exact fractions, including pure and mixed recurring decimals.
Let x equal the recurring decimal. Multiply x by a power of 10 so that the repeating block lines up with itself, subtract the original equation from the new one to cancel the repeating part, then solve for x as a ratio of two whole numbers. Finally, simplify the fraction by dividing both the numerator and denominator by their greatest common divisor.
A terminating decimal has a finite number of digits after the decimal point, such as 0.25. A recurring (or repeating) decimal has one or more digits that repeat forever, such as 0.3333… or 0.142857142857…, and is shown with a bar or dots over the repeating digits.
Yes. Any decimal that either terminates or eventually repeats a fixed block of digits forever can always be written as a fraction of two integers, which is the definition of a rational number. Decimals that never settle into a repeating pattern, such as pi, are irrational and cannot be converted to an exact fraction.
A bar or dot placed above one or more digits indicates that those digits repeat infinitely. For example, 0.6 with a bar over the 6 means 0.6666… continuing forever, while 0.142857 with a bar over the whole block means that six-digit sequence repeats endlessly.
A mixed recurring decimal has a non-repeating part right after the decimal point followed by a repeating block, such as 0.1666… where the 1 does not repeat but the 6 does. These are converted by separating the non-repeating and repeating digits and applying a formula that accounts for both parts before simplifying the resulting fraction.
Using the standard algebraic method, if x = 0.999… then 10x = 9.999…, and subtracting the first equation from the second gives 9x = 9, so x = 1. This is a well-known mathematical identity confirming that the recurring decimal 0.999… and the whole number 1 are the same value, not just very close approximations.