Recurring decimals (also called repeating decimals) are decimal numbers that contain a repeating sequence of digits after the decimal point. This sequence of digits continues infinitely. For example:

- \( \frac{1}{3} = 0.333\ldots \) where the digit "3" repeats indefinitely, denoted as \( 0.\overline{3} \).
- \( \frac{1}{6} = 0.1666\ldots \), where the digit "6" repeats, denoted as \( 0.1\overline{6} \).

### Types of Recurring Decimals
1. **Pure Recurring Decimals**: The repeating sequence starts immediately after the decimal point. Example: \( 0.\overline{7} = 0.7777\ldots \).
2. **Mixed Recurring Decimals**: The repeating sequence does not start right after the decimal point but after a few non-repeating digits. Example: \( 0.16\overline{7} = 0.167777\ldots \).

### Converting Recurring Decimals to Fractions
Recurring decimals can often be converted into fractions using algebraic methods. Here's a typical approach for converting a simple recurring decimal like \( 0.\overline{3} \) into a fraction:

Let \( x = 0.\overline{3} \).
- Multiply both sides by 10: \( 10x = 3.\overline{3} \).
- Subtract the original equation \( x = 0.\overline{3} \) from \( 10x = 3.\overline{3} \):
  
  \[
  10x - x = 3.\overline{3} - 0.\overline{3}
  \]
  \[
  9x = 3
  \]
- Divide by 9: \( x = \frac{3}{9} = \frac{1}{3} \).

This process can be applied to more complex repeating decimals as well.

### Properties of Recurring Decimals
1. **Rational Numbers**: Any recurring decimal can be expressed as a fraction, meaning all repeating decimals are rational numbers.
2. **Period**: The number of digits in the repeating sequence is called the period of the recurring decimal. For instance, in \( 0.\overline{142857} \), the period is 6.

### Examples:
1. \( 0.\overline{6} = \frac{2}{3} \)
2. \( 0.\overline{9} = 1 \)
3. \( 0.12\overline{3} = \frac{37}{300} \)

Would you like more details or examples on recurring decimals?

### Related Questions:
1. How can mixed recurring decimals be converted to fractions?
2. Can every rational number be expressed as a recurring decimal?
3. What is the period of the recurring decimal for \( \frac{1}{7} \)?
4. How do recurring decimals differ from terminating decimals?
5. Are there any special cases when converting recurring decimals to fractions?

**Tip**: A decimal that terminates (ends after a certain number of digits) can be converted to a fraction by expressing it as a ratio of powers of 10.

Explore recurring decimals and learn how to convert them into fractions using algebraic methods. This page covers pure and mixed recurring decimals, along with examples like 0.\overline{3} and 0.16\overline{7}. Suitable for students in Grades 6-8.

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