0.1231313 Recurring As A Fraction

Recurring Decimal to Fraction Conversion:

\[ 0.12\overline{313} = \frac{122}{989} \]

The recurring decimal 0.1231313... is equal to:

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1. Understanding Recurring Decimals

A recurring decimal is a decimal number that has digits that repeat infinitely. In 0.1231313..., the "313" sequence repeats indefinitely after the initial "12".

2. Conversion Process

To convert 0.12\overline{313} to a fraction:

\[ \begin{align*} 1. \text{Let } x &= 0.123131313... \\ 2. \text{Multiply by 100: } 100x &= 12.3131313... \\ 3. \text{Multiply by 100000: } 100000x &= 12313.131313... \\ 4. \text{Subtract: } 100000x - 100x &= 12313.131313... - 12.3131313... \\ 5. \text{Result: } 99900x &= 12200.818181... \\ 6. \text{Solve for x: } x &= \frac{12200}{99900} = \frac{122}{989} \end{align*} \]

3. Mathematical Proof

Verification: Dividing 122 by 989 gives 0.1231313..., confirming our conversion is correct.

4. Practical Applications

Details: Converting recurring decimals to fractions is essential in exact calculations, algebra, and when precise representations are needed in mathematics and engineering.

5. Frequently Asked Questions (FAQ)

Q1: Why convert recurring decimals to fractions?
A: Fractions provide exact representations while decimals are often approximations, especially important in precise calculations.

Q2: How do you identify the repeating pattern?
A: Look for the sequence of digits that repeats indefinitely after the decimal point.

Q3: Can all recurring decimals be converted to fractions?
A: Yes, all repeating decimals represent rational numbers and can be expressed as fractions.

Q4: What's the difference between terminating and recurring decimals?
A: Terminating decimals have finite digits, while recurring decimals have an infinite repeating pattern.

Q5: How do you simplify the resulting fraction?
A: Find the greatest common divisor (GCD) of numerator and denominator and divide both by it.