0.29 Repeating As A Fraction

Conversion Method:

\[ x = 0.\overline{29} \] \[ 100x = 29.\overline{29} \] \[ 100x - x = 29.\overline{29} - 0.\overline{29} \] \[ 99x = 29 \] \[ x = \frac{29}{99} \]

Fraction Result:

Unit Converter ▲

Unit Converter ▼

From:	To:

1. What Is 0.29 Repeating?

0.29 repeating (written as 0.\(\overline{29}\)) is a decimal number where the digits "29" repeat infinitely. It represents a rational number that can be expressed as a fraction of two integers.

2. Conversion Method

The standard algebraic method to convert repeating decimals to fractions:

\[ \begin{align*} 1. &\text{Let } x = 0.\overline{29} \\ 2. &\text{Multiply by 100: } 100x = 29.\overline{29} \\ 3. &\text{Subtract original: } 100x - x = 29.\overline{29} - 0.\overline{29} \\ 4. &\text{Solve: } 99x = 29 \\ 5. &\text{Result: } x = \frac{29}{99} \end{align*} \]

3. Mathematical Proof

The fraction \(\frac{29}{99}\) exactly equals 0.\(\overline{29}\) because:

29 ÷ 99 = 0.292929...
The repeating block matches the numerator
The denominator is 10^n - 1 (where n is digits in repeating block)

4. Applications

Details: Repeating decimal conversions are essential in:

Exact arithmetic calculations
Computer algebra systems
Number theory proofs
Fractional approximations

5. Frequently Asked Questions (FAQ)

Q1: Why does this method work?
A: It eliminates the infinite repeating portion through subtraction, leaving an exact fraction.

Q2: Can this fraction be simplified?
A: No, 29/99 is already in simplest form (29 is prime and doesn't divide 99).

Q3: How does this compare to terminating decimals?
A: Terminating decimals have denominators with only 2 and 5 as prime factors.

Q4: What about other repeating patterns?
A: The method works for any repeating decimal, adjusting the multiplier by 10^n.

Q5: Is 29/99 an exact representation?
A: Yes, unlike decimal approximations, the fraction is mathematically exact.