Answer :
To determine which set of numbers are rational numbers but not integers, whole numbers, or natural numbers, we need to analyze each set individually.
1. Set [tex]\(\{\sqrt{2}, \pi, \sqrt{3}\}\)[/tex]
- [tex]\(\sqrt{2}\)[/tex] and [tex]\(\sqrt{3}\)[/tex] are square roots of non-perfect squares, making them irrational numbers.
- [tex]\(\pi\)[/tex] is a well-known irrational number.
- Therefore, [tex]\(\{\sqrt{2}, \pi, \sqrt{3}\}\)[/tex] consists entirely of irrational numbers.
2. Set [tex]\(\{0,1,2,3\}\)[/tex]
- These numbers are all integers. Since:
- Integers include whole numbers and natural numbers.
- Whole numbers include 0 and positive numbers.
- Natural numbers are positive integers starting from 1.
- Therefore, [tex]\(\{0,1,2,3\}\)[/tex] are integers, whole numbers, and natural numbers.
3. Set [tex]\(\{\sqrt{-1}, \sqrt{-4}, \sqrt{-5}\}\)[/tex]
- [tex]\(\sqrt{-1}\)[/tex] is the imaginary unit [tex]\(i\)[/tex], and [tex]\(\sqrt{-4}\)[/tex] can be written as [tex]\(2i\)[/tex], while [tex]\(\sqrt{-5}\)[/tex] is [tex]\(\sqrt{5}i\)[/tex].
- All these numbers are imaginary, not rational.
4. Set [tex]\(\{-2,-3,-4\}\)[/tex]
- These numbers are all negative integers.
- Negative integers are not whole numbers or natural numbers.
- Therefore, [tex]\(\{-2,-3,-4\}\)[/tex] consists of integers but not whole or natural numbers.
5. Set [tex]\(\left\{\frac{1}{2}, \frac{1}{4}, \frac{1}{8}\right\}\)[/tex]
- Each number is a simple fraction.
- Fractions represent rational numbers but not integers, whole numbers, or natural numbers.
Based on this analysis, the set of numbers that are rational numbers but not integers, whole numbers, or natural numbers is:
[tex]\[ \left\{\frac{1}{2}, \frac{1}{4}, \frac{1}{8}\right\} \][/tex]
Thus, the answer is:
[tex]\[ \boxed{5} \][/tex]
1. Set [tex]\(\{\sqrt{2}, \pi, \sqrt{3}\}\)[/tex]
- [tex]\(\sqrt{2}\)[/tex] and [tex]\(\sqrt{3}\)[/tex] are square roots of non-perfect squares, making them irrational numbers.
- [tex]\(\pi\)[/tex] is a well-known irrational number.
- Therefore, [tex]\(\{\sqrt{2}, \pi, \sqrt{3}\}\)[/tex] consists entirely of irrational numbers.
2. Set [tex]\(\{0,1,2,3\}\)[/tex]
- These numbers are all integers. Since:
- Integers include whole numbers and natural numbers.
- Whole numbers include 0 and positive numbers.
- Natural numbers are positive integers starting from 1.
- Therefore, [tex]\(\{0,1,2,3\}\)[/tex] are integers, whole numbers, and natural numbers.
3. Set [tex]\(\{\sqrt{-1}, \sqrt{-4}, \sqrt{-5}\}\)[/tex]
- [tex]\(\sqrt{-1}\)[/tex] is the imaginary unit [tex]\(i\)[/tex], and [tex]\(\sqrt{-4}\)[/tex] can be written as [tex]\(2i\)[/tex], while [tex]\(\sqrt{-5}\)[/tex] is [tex]\(\sqrt{5}i\)[/tex].
- All these numbers are imaginary, not rational.
4. Set [tex]\(\{-2,-3,-4\}\)[/tex]
- These numbers are all negative integers.
- Negative integers are not whole numbers or natural numbers.
- Therefore, [tex]\(\{-2,-3,-4\}\)[/tex] consists of integers but not whole or natural numbers.
5. Set [tex]\(\left\{\frac{1}{2}, \frac{1}{4}, \frac{1}{8}\right\}\)[/tex]
- Each number is a simple fraction.
- Fractions represent rational numbers but not integers, whole numbers, or natural numbers.
Based on this analysis, the set of numbers that are rational numbers but not integers, whole numbers, or natural numbers is:
[tex]\[ \left\{\frac{1}{2}, \frac{1}{4}, \frac{1}{8}\right\} \][/tex]
Thus, the answer is:
[tex]\[ \boxed{5} \][/tex]
