# Recurring Decimal To Fraction Calculator

Use this calculator to convert a repeating decimal to a fraction. [Fraction to recurring decimal calculator] ( /show/calculator/fraction-to-recurring-decimal) is also available.

It answers queries like: * Convert 0.(3) to a fraction * Convert 0.33333... to a fraction * What is 0.(1) as a fraction? * Represent 0.(5) as a fraction Some numbers cannot be expressed exactly as decimals with a finite number of digits. For example, since 2/3 = 0.666666666..., to express the fraction 2/3 in the decimal system, we require an infinity of 6s. Such decimals are referred to as __recurring (or repeating) decimals__. ##Recurring decimal to fraction## Every recurring decimal has a representation as a fraction. To see that, consider a recurring fraction of the form: \$$2.5(34) = 2.534343434343434...\$$ Let's convert the recurring part of the decimal to an infinite geometric series: \$$2.5 + 0.0(34) = 2.5 + 0.034 \cdot 10^{0} + 0.034 \cdot 10^{-2} + 0.034 \cdot 10^{-4}... = 2.5 + 0.034 \cdot {\sum^{\infty}_{i=0} (10^{-2i)})} \$$ And from the formula for the sum of a geometric series we get: \$$2.5 + { {34\over 1000} \over 1 - 10^{-2} } = {25 \over 10} + {34 \over 990} \$$ which means the whole expression is a fraction. ##General Formula## We can rewrite the formula above with variables to get something more general: \$${n + r \cdot 10^{-p} \cdot \sum^{\infty}_{i=0} (10^{-i \cdot j})} = n + {r \cdot 10^{-p} \over 1 - 10^{-j} }\$$ where: \$$n \$$ is the non-recurring part \$$r \$$ r is the recurring part \$$j \$$ is the length of \$$r \$$ \$$p \$$ is the number digits preceding the recurring part and the decimal point \$$+ 1 \$$ ##Method for Human-beings## There are better methods of finding the desired fractions than using the above formula. Let's use it on an example. What fraction is \$$0.(7) \$$ equal to? Let \$$x = 0.(7) \$$. Then \$$10x = 7.777777... = 7 + 0.(7) = 7 + x \$$. So, \$$9x = 7 \$$ and lastly, \$$x = {7 \over 9} \$$.