Recurring Decimals To Fractions Calculator

Recurring Decimal Conversion:

\[ x = 0.\overline{dddd} \Rightarrow \text{Fraction} = \frac{\text{Non-repeating part followed by repeating part} - \text{Non-repeating part}}{10^{\text{total digits}} - 10^{\text{non-repeating digits}}} \]

Fraction Result:

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1. What is Recurring Decimal to Fraction Conversion?

Recurring decimals (repeating decimals) are decimal numbers that have digits that repeat infinitely. Converting them to fractions provides an exact representation of the value, which is often more useful in mathematical calculations.

2. How Does the Calculator Work?

The calculator uses the following mathematical principle:

\[ x = 0.\overline{a_1a_2...a_n} \Rightarrow x = \frac{a_1a_2...a_n}{10^n - 1} \]

For mixed recurring decimals (with non-repeating and repeating parts):

\[ x = 0.b_1b_2...b_m\overline{a_1a_2...a_n} \Rightarrow x = \frac{b_1b_2...b_ma_1a_2...a_n - b_1b_2...b_m}{10^{m+n} - 10^m} \]

Explanation: The algorithm subtracts the non-repeating part and divides by a number composed of 9s (for repeating digits) and 0s (for non-repeating digits).

3. Importance of Decimal-Fraction Conversion

Details: Exact fractional representations are crucial in precise mathematical calculations, computer algebra systems, and when working with periodic numbers in physics and engineering.

4. Using the Calculator

Tips: Enter the recurring decimal in either format: 0.333... or 0.(3) for 1/3, or 0.12(34) for 0.12343434... (which equals 611/4950).

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between terminating and recurring decimals?
A: Terminating decimals have finite digits (e.g., 0.5), while recurring decimals have infinite repeating digits (e.g., 0.333...).

Q2: Can all recurring decimals be converted to fractions?
A: Yes, all recurring decimals are rational numbers and can be expressed as fractions of integers.

Q3: How do I represent repeating decimals in input?
A: Use either parentheses (e.g., 0.(6)) or ellipsis (e.g., 0.666...). Both represent 2/3.

Q4: What about decimals with multiple repeating patterns?
A: The calculator handles any single repeating block. For complex patterns, separate calculations may be needed.

Q5: Why does 0.999... equal 1?
A: The infinite series 0.9 + 0.09 + 0.009 + ... converges exactly to 1, which can be proven algebraically or with limits.