Question
repeating decimal fractions.
Answers
GPT-4o mini
Repeating decimal fractions (or recurring decimals) are decimal numbers that have a sequence of digits that repeat infinitely. For example, the fraction \( \frac{1}{3} \) is represented as the decimal \( 0.333... \), which can be written as \( 0.\overline{3} \), indicating that the digit 3 repeats indefinitely.
### How to Identify Repeating Decimals
1. **Recognition**: If you find a decimal number where a specific sequence of digits repeats, it is a repeating decimal. For instance:
- \( 0.142857142857... \) can be written as \( 0.\overline{142857} \) because the sequence "142857" repeats.
2. **Fractions**: Many rational numbers (fractions where both the numerator and the denominator are integers) can be expressed as either terminating decimals or repeating decimals. A fraction will result in a repeating decimal if the denominator (in simplest form) contains prime factors other than 2 and 5.
### Examples of Converting to Repeating Decimals
1. **Example 1**: \( \frac{1}{7} \)
- Performing long division gives \( 0.142857142857... \)
- This can be expressed as \( 0.\overline{142857} \).
2. **Example 2**: \( \frac{1}{6} \)
- Long division gives \( 0.166666... \)
- This can be expressed as \( 0.1\overline{6} \).
3. **Example 3**: \( \frac{4}{11} \)
- Long division results in \( 0.363636... \)
- It can be expressed as \( 0.\overline{36} \).
### Converting Repeating Decimals to Fractions
To convert a repeating decimal back into a fraction, you can follow these steps:
1. **Let \( x \) equal the repeating decimal**. For example, let \( x = 0.\overline{36} \).
2. **Multiply \( x \) by a power of 10 that shifts the decimal point to the right to match the start of the repeat**. In this case:
- Multiply by 100 (because the repeat is 2 digits long): \( 100x = 36.\overline{36} \).
3. **Set up an equation** by subtracting the original \( x \) from the new equation:
\[
100x - x = 36.\overline{36} - 0.\overline{36}
\]
\[
99x = 36
\]
4. **Solve for \( x \)**:
\[
x = \frac{36}{99} = \frac{4}{11} \text{ (after simplifying)}
\]
### Summary
- Repeating decimals are decimal numbers that contain a digit (or group of digits) that repeat indefinitely.
- A fraction results in a repeating decimal if its reduced form has prime factors other than 2 or 5.
- You can convert a repeating decimal back to a fraction using algebraic techniques.
If you have specific questions or need more detailed explanations or examples related to repeating decimals, feel free to ask!
### How to Identify Repeating Decimals
1. **Recognition**: If you find a decimal number where a specific sequence of digits repeats, it is a repeating decimal. For instance:
- \( 0.142857142857... \) can be written as \( 0.\overline{142857} \) because the sequence "142857" repeats.
2. **Fractions**: Many rational numbers (fractions where both the numerator and the denominator are integers) can be expressed as either terminating decimals or repeating decimals. A fraction will result in a repeating decimal if the denominator (in simplest form) contains prime factors other than 2 and 5.
### Examples of Converting to Repeating Decimals
1. **Example 1**: \( \frac{1}{7} \)
- Performing long division gives \( 0.142857142857... \)
- This can be expressed as \( 0.\overline{142857} \).
2. **Example 2**: \( \frac{1}{6} \)
- Long division gives \( 0.166666... \)
- This can be expressed as \( 0.1\overline{6} \).
3. **Example 3**: \( \frac{4}{11} \)
- Long division results in \( 0.363636... \)
- It can be expressed as \( 0.\overline{36} \).
### Converting Repeating Decimals to Fractions
To convert a repeating decimal back into a fraction, you can follow these steps:
1. **Let \( x \) equal the repeating decimal**. For example, let \( x = 0.\overline{36} \).
2. **Multiply \( x \) by a power of 10 that shifts the decimal point to the right to match the start of the repeat**. In this case:
- Multiply by 100 (because the repeat is 2 digits long): \( 100x = 36.\overline{36} \).
3. **Set up an equation** by subtracting the original \( x \) from the new equation:
\[
100x - x = 36.\overline{36} - 0.\overline{36}
\]
\[
99x = 36
\]
4. **Solve for \( x \)**:
\[
x = \frac{36}{99} = \frac{4}{11} \text{ (after simplifying)}
\]
### Summary
- Repeating decimals are decimal numbers that contain a digit (or group of digits) that repeat indefinitely.
- A fraction results in a repeating decimal if its reduced form has prime factors other than 2 or 5.
- You can convert a repeating decimal back to a fraction using algebraic techniques.
If you have specific questions or need more detailed explanations or examples related to repeating decimals, feel free to ask!