Recurring Decimal to Fraction Calculator

Understanding Recurring Decimals and Fractions

What are Recurring Decimals?

A recurring decimal, also known as a repeating decimal, is a decimal representation of a rational number where a digit or a group of digits repeats indefinitely after the decimal point. For example, 0.333333... (where 3 repeats infinitely) is a recurring decimal.

Formula for Converting Recurring Decimals to Fractions

The general formula for converting a recurring decimal to a fraction is:

\[x = 0.abcdefdefdef...\]

\[1000x = abc.defdefdef...\]

\[999x = abc\]

\[x = \frac{abc}{999}\]

Where:

• \(x\) is the recurring decimal
• \(abc\) is the part before the recurring digits
• \(def\) is the recurring part

Converting Recurring Decimals to Fractions

To convert a recurring decimal to a fraction, follow these steps:

1. Let x be the recurring decimal
2. Multiply both sides by an appropriate power of 10 to shift the decimal point
3. Subtract the original equation from the new equation
4. Solve for x

Calculation Steps

Let's convert 0.3333... to a fraction:

1. Let x = 0.3333...
2. Multiply both sides by 10: 10x = 3.3333...
3. Subtract x from 10x: 9x = 3
4. Solve for x: x = 3/9 = 1/3

Example with Visual Representation

Let's visualize the conversion of 0.3333... to 1/3:

This diagram shows that the recurring decimal 0.3333... is equivalent to the fraction 1/3.