Question
which set rational numbers is always rational negative numbers no terminating decimal deciamials whole numbered explain why
Answers
GPT-4o mini
To clarify your question, you're asking about which sets of rational numbers are always rational, specifically focusing on negative numbers, non-terminating decimals, and whole numbers. Let's break that down:
1. **Rational Numbers**: By definition, a rational number is any number that can be expressed as the quotient of two integers (i.e., a fraction \( \frac{p}{q} \) where \( p \) and \( q \) are integers and \( q \neq 0 \)).
2. **Negative Numbers**: Negative rational numbers are also rational. For example, \(-\frac{1}{2}\) is a rational number because it can be expressed as the ratio of two integers (-1 and 2).
3. **Non-terminating Decimals**: Non-terminating decimals can be either rational or irrational. A non-terminating decimal that repeats (e.g., 0.333...) is rational because it can be expressed as a fraction \(\frac{1}{3}\). However, a non-terminating decimal that does not repeat (e.g., π or the square root of 2) is irrational and cannot be expressed as a fraction.
4. **Whole Numbers**: Whole numbers (0, 1, 2, 3, ...) are also rational because any whole number \( n \) can be expressed as \( \frac{n}{1} \), which fits the definition of a rational number.
### Summary:
- **Negative Numbers**: Always rational (if expressed as fractions).
- **Non-terminating Decimals**: Not always rational; only those that repeat are rational.
- **Whole Numbers**: Always rational (expressible as fractions).
Therefore, the sets that are always rational are negative numbers (as long as they are in fractional form), and whole numbers. Non-terminating decimals can be a mix of rational or irrational depending on whether they are repeating or not.
1. **Rational Numbers**: By definition, a rational number is any number that can be expressed as the quotient of two integers (i.e., a fraction \( \frac{p}{q} \) where \( p \) and \( q \) are integers and \( q \neq 0 \)).
2. **Negative Numbers**: Negative rational numbers are also rational. For example, \(-\frac{1}{2}\) is a rational number because it can be expressed as the ratio of two integers (-1 and 2).
3. **Non-terminating Decimals**: Non-terminating decimals can be either rational or irrational. A non-terminating decimal that repeats (e.g., 0.333...) is rational because it can be expressed as a fraction \(\frac{1}{3}\). However, a non-terminating decimal that does not repeat (e.g., π or the square root of 2) is irrational and cannot be expressed as a fraction.
4. **Whole Numbers**: Whole numbers (0, 1, 2, 3, ...) are also rational because any whole number \( n \) can be expressed as \( \frac{n}{1} \), which fits the definition of a rational number.
### Summary:
- **Negative Numbers**: Always rational (if expressed as fractions).
- **Non-terminating Decimals**: Not always rational; only those that repeat are rational.
- **Whole Numbers**: Always rational (expressible as fractions).
Therefore, the sets that are always rational are negative numbers (as long as they are in fractional form), and whole numbers. Non-terminating decimals can be a mix of rational or irrational depending on whether they are repeating or not.