# 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.

#### Calculator

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} \\).