1. What is 0.23 Recurring?

0.23 recurring (written as 0.\(\overline{23}\)) is a decimal number where the digits "23" repeat infinitely. It represents the fraction 23/99.

2. Conversion Method

The standard method to convert repeating decimals to fractions:

Where:

• \( ab \) — The repeating digits (23 in this case)
• The denominator has as many 9's as there are repeating digits

Example: For 0.\(\overline{23}\), there are 2 repeating digits, so we use 99 as denominator.

3. Mathematical Proof

Let \( x = 0.\overline{23} \)
Then \( 100x = 23.\overline{23} \)
Subtract: \( 100x - x = 23.\overline{23} - 0.\overline{23} \)
\( 99x = 23 \)
\( x = \frac{23}{99} \)

4. Using the Calculator

Instructions: Enter the repeating decimal in the format "0.23..." (including the dots). The calculator will automatically convert it to its fractional form.

5. Frequently Asked Questions (FAQ)

Q1: Why does this method work?
A: The method uses algebra to eliminate the infinite repeating portion by shifting and subtracting.

Q2: What if the decimal doesn't start repeating immediately?
A: For patterns like 0.2333..., use different methods accounting for non-repeating digits.

Q3: Can all repeating decimals be converted to fractions?
A: Yes, all repeating decimals represent exact fractions (rational numbers).

Q4: How to simplify the resulting fraction?
A: Find the greatest common divisor (GCD) of numerator and denominator.

Q5: Is 0.999... really equal to 1?
A: Yes, by this same method: 0.\(\overline{9}\) = 9/9 = 1 exactly.