Repeating Decimal to Fraction Calculator

A repeating decimal has one or more digits that cycle infinitely after the decimal point. It is written with an ellipsis (0.333…) or with a bar (vinculum) over the repeating block. Examples: 0.333… (single digit 3 repeats), 0.272727… (two-digit block 27 repeats), and 0.8333… (8 does not repeat, only 3 repeats). Every repeating decimal is a rational number and can be expressed as an exact fraction.

Use the algebraic method: (1) Let x equal the repeating decimal. (2) Multiply x by 10^n where n is the length of the repeating block, to shift the decimal past one full cycle. (3) If there is a non-repeating prefix, create a second equation that shifts the decimal to just before the repeating part. (4) Subtract the two equations to cancel the infinite tail. (5) Solve for x and simplify with the GCD. Example: x = 0.333…; 10x = 3.333…; 10x − x = 3; 9x = 3; x = 1/3.

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...

Yes, all repeating decimals are rational numbers, meaning they can always be expressed exactly as a fraction containing integers.

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.

Enter the non-repeating part of the decimal (everything before the repeating block) in the first field, and the repeating digits in the second field. For example, for 0.1666…, enter 0.1 and 6. The calculator applies the algebraic method and shows the simplified fraction with a complete step-by-step proof.

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.