How to Convert an Infinite Decimal to a Fraction

By Michael Keenan

Infinite decimals can be tricky to convert to fractions because you cannot simply put the decimal over the appropriate multiple of 10. Converting an infinite decimal to a fraction can better help you to represent the number. For example, 0.3636... may be harder to grasp than 36/99. You can only convert repeating infinite decimals to fractions. For example, pi does not terminate or repeat so while it is commonly approximated as 22/7, it is not exact.

Set the repeating fraction equal to x. For example, if your infinite decimal is 0.18232323... you would write x=0.182323...

Determine the repeating length of the decimal. The repeating length is the number of digits in the repeating pattern. For example, 0.182323... has a repeating length of 2 because the pattern is "23." If your decimal was 0.485485485.... the repeating length would be 3.

Multiply each side of the equation from step 1 by 10^R, where R is the repeating length. For example, since 0.182323... has a repeating length of 2, and 10^2 is 100, you would get 100x=18.2323...

Subtract the equation in Step 1 from the equation in Step 3. For example, you would subtract x=0.182323... from 100x=18.2323... and you would get 99x=18.05.

Solve the equation in Step 4 for x. For example, with 99x=18.05 you would divide by 99 on both sides so you would have x=18.05/99, or 1805/9900.

Simplify the fraction found in Step 4. For example, 1805/9900 simplifies to 361/1980.

About the Author

Mark Kennan is a writer based in the Kansas City area, specializing in personal finance and business topics. He has been writing since 2009 and has been published by "Quicken," "TurboTax," and "The Motley Fool."