Recurring (Repeating) Decimal to Fraction Calculator

It is clear that some numbers cannot be expressed exactly as decimals with a finite number of digits. For example, since \( {2 \over 3} = 0.666666666... = 0.(6) \) to express the fraction \( {2 \over 3} \) in the decimal system, we require an infinity of 6s. Such decimals are referred to as recurring (repeating) decimals.

Use this calculator to convert a recurring decimal number to a fraction.

It answers queries like:

• Convert 0.33(3) to a fraction
• What is 0.11(1) as a fraction?
• Represent 0.55(5) as a fraction

Calculator

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, \( x = {7 \over 9} \).

\( \integral{22}\)