Question
Which set of RATIONAL numbers below is ALWAYS rational? Circle one and explain why.
Negative numbers?
Non Terminating decimals?
Decimals?
Whole Numbers?
Negative numbers?
Non Terminating decimals?
Decimals?
Whole Numbers?
Answers
Answered by
GPT-4o mini
The set that is ALWAYS rational is **Whole Numbers**.
### Explanation:
- **Whole Numbers**: This set includes 0, 1, 2, 3, and so on. Whole numbers can be expressed as fractions where the denominator is 1 (e.g., 3 can be written as \( \frac{3}{1} \)), which makes them rational numbers.
- **Negative Numbers**: While many negative numbers are rational (like -1, -2, etc.), not all negative numbers are rational. For example, negative square roots of non-perfect squares (like \(-\sqrt{2}\)) are irrational.
- **Non Terminating Decimals**: Non-terminating decimals can be either rational or irrational. For example, the decimal representation of \( \frac{1}{3} = 0.333...\) is non-terminating and rational, while \(\pi\) (approximately 3.14159...) is an example of a non-terminating decimal that is irrational.
- **Decimals**: This category includes both terminating (like 0.5) and non-terminating decimals (like 0.333...). Terminating decimals are rational, but non-terminating decimals can be either rational or irrational.
Thus, whole numbers are the only set listed that is always rational.
### Explanation:
- **Whole Numbers**: This set includes 0, 1, 2, 3, and so on. Whole numbers can be expressed as fractions where the denominator is 1 (e.g., 3 can be written as \( \frac{3}{1} \)), which makes them rational numbers.
- **Negative Numbers**: While many negative numbers are rational (like -1, -2, etc.), not all negative numbers are rational. For example, negative square roots of non-perfect squares (like \(-\sqrt{2}\)) are irrational.
- **Non Terminating Decimals**: Non-terminating decimals can be either rational or irrational. For example, the decimal representation of \( \frac{1}{3} = 0.333...\) is non-terminating and rational, while \(\pi\) (approximately 3.14159...) is an example of a non-terminating decimal that is irrational.
- **Decimals**: This category includes both terminating (like 0.5) and non-terminating decimals (like 0.333...). Terminating decimals are rational, but non-terminating decimals can be either rational or irrational.
Thus, whole numbers are the only set listed that is always rational.