Question
Which of the following rational numbers can be expressed as a repeating decimal
Answers
Answered by
GPT-4o mini
To determine which rational numbers can be expressed as repeating decimals, it’s important to understand that all rational numbers can be represented as either terminating or repeating decimals. A rational number is defined as the quotient of two integers, \( a/b \), where \( a \) and \( b \) are integers and \( b \neq 0 \).
The following characteristics apply:
- **Terminating Decimals**: A decimal is terminating if the prime factorization of the denominator (when the rational number is in its simplest form) contains only the primes 2 and/or 5. For example, \( \frac{1}{2} = 0.5 \) and \( \frac{1}{8} = 0.125 \) are terminating decimals.
- **Repeating Decimals**: A decimal is repeating if the prime factorization of the denominator (again, in simplest form) contains other primes besides 2 and 5. For instance, \( \frac{1}{3} = 0.333...\) is a repeating decimal because the denominator 3 is neither 2 nor 5.
If you have a specific list of rational numbers you'd like to analyze to see whether they can be expressed as repeating decimals, please provide that list, and I can help you determine which ones are repeating decimals!
The following characteristics apply:
- **Terminating Decimals**: A decimal is terminating if the prime factorization of the denominator (when the rational number is in its simplest form) contains only the primes 2 and/or 5. For example, \( \frac{1}{2} = 0.5 \) and \( \frac{1}{8} = 0.125 \) are terminating decimals.
- **Repeating Decimals**: A decimal is repeating if the prime factorization of the denominator (again, in simplest form) contains other primes besides 2 and 5. For instance, \( \frac{1}{3} = 0.333...\) is a repeating decimal because the denominator 3 is neither 2 nor 5.
If you have a specific list of rational numbers you'd like to analyze to see whether they can be expressed as repeating decimals, please provide that list, and I can help you determine which ones are repeating decimals!