Answer :
To determine whether each number is rational or irrational in the given table, follow the definitions of rational and irrational numbers:
Rational Numbers: Numbers that can be expressed as a fraction [tex]\(\frac{a}{b}\)[/tex], where [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are integers, and [tex]\(b \neq 0\)[/tex]. This includes integers, terminating decimals, and repeating decimals.
Irrational Numbers: Numbers that cannot be expressed as a simple fraction. These numbers have non-terminating, non-repeating decimal expansions.
Let's go through each number step-by-step:
1. 5000: Rational - It is an integer.
2. [tex]\(\sqrt{9} = 3\)[/tex]: Rational - [tex]\(3\)[/tex] is an integer.
3. -4: Rational - It is an integer.
4. 3.2 (Repeats): Rational - A repeating decimal can be expressed as a fraction.
5. 3.421553...: Irrational - A decimal that does not terminate or repeat.
6. [tex]\(\sqrt{32}\)[/tex]: Irrational - Cannot be simplified to an integer.
7. [tex]\(\sqrt{81} = 9\)[/tex]: Rational - [tex]\(9\)[/tex] is an integer.
8. 0: Rational - It is an integer.
9. [tex]\(\pi\)[/tex]: Irrational - [tex]\(\pi\)[/tex] has a non-terminating, non-repeating decimal expansion.
10. 12.2: Rational - A terminating decimal.
11. [tex]\(\frac{1}{8}\)[/tex]: Rational - It is a fraction.
12. -22.3: Rational - A terminating decimal.
13. [tex]\(\sqrt{144} = 12\)[/tex]: Rational - [tex]\(12\)[/tex] is an integer.
14. [tex]\(2^5 = 32\)[/tex]: Rational - [tex]\(32\)[/tex] is an integer.
15. [tex]\(\sqrt{21}\)[/tex]: Irrational - Cannot be simplified to an integer.
16. 4.5: Rational - A terminating decimal.
17. 1: Rational - It is an integer.
18. [tex]\(\sqrt{36} = 6\)[/tex]: Rational - [tex]\(6\)[/tex] is an integer.
19. 2 (Terminates): Rational - A terminating number.
20. 5.320..: Irrational - A non-terminating, non-repeating decimal.
21. [tex]\(\frac{\sqrt{100}}{2} = 5\)[/tex]: Rational - [tex]\(5\)[/tex] is an integer.
22. [tex]\(\sqrt{40}\)[/tex]: Irrational - Cannot be simplified to an integer.
23. 5 [tex]\(\frac{2}{5}\)[/tex]: Rational - It can be expressed as an improper fraction.
24. 0.132 (Repeats): Rational - A repeating decimal can be expressed as a fraction.
According to the criteria:
- Rational Numbers:
1. 5000
2. [tex]\(\sqrt{9} = 3\)[/tex]
3. -4
4. 3.2 (Repeats)
5. [tex]\(\sqrt{81} = 9\)[/tex]
6. 0
7. 12.2
8. [tex]\(\frac{1}{8}\)[/tex]
9. -22.3
10. [tex]\(\sqrt{144} = 12\)[/tex]
11. [tex]\(2^5 = 32\)[/tex]
12. 4.5
13. 1
14. [tex]\(\sqrt{36} = 6\)[/tex]
15. 2 (Terminates)
16. [tex]\(\frac{\sqrt{100}}{2} = 5\)[/tex]
17. 5 [tex]\(\frac{2}{5}\)[/tex]
18. 0.132 (Repeats)
- Irrational Numbers:
1. 3.421553...
2. [tex]\(\sqrt{32}\)[/tex]
3. [tex]\(\pi\)[/tex]
4. [tex]\(\sqrt{21}\)[/tex]
5. 5.320..
6. [tex]\(\sqrt{40}\)[/tex]
Rational Numbers: Numbers that can be expressed as a fraction [tex]\(\frac{a}{b}\)[/tex], where [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are integers, and [tex]\(b \neq 0\)[/tex]. This includes integers, terminating decimals, and repeating decimals.
Irrational Numbers: Numbers that cannot be expressed as a simple fraction. These numbers have non-terminating, non-repeating decimal expansions.
Let's go through each number step-by-step:
1. 5000: Rational - It is an integer.
2. [tex]\(\sqrt{9} = 3\)[/tex]: Rational - [tex]\(3\)[/tex] is an integer.
3. -4: Rational - It is an integer.
4. 3.2 (Repeats): Rational - A repeating decimal can be expressed as a fraction.
5. 3.421553...: Irrational - A decimal that does not terminate or repeat.
6. [tex]\(\sqrt{32}\)[/tex]: Irrational - Cannot be simplified to an integer.
7. [tex]\(\sqrt{81} = 9\)[/tex]: Rational - [tex]\(9\)[/tex] is an integer.
8. 0: Rational - It is an integer.
9. [tex]\(\pi\)[/tex]: Irrational - [tex]\(\pi\)[/tex] has a non-terminating, non-repeating decimal expansion.
10. 12.2: Rational - A terminating decimal.
11. [tex]\(\frac{1}{8}\)[/tex]: Rational - It is a fraction.
12. -22.3: Rational - A terminating decimal.
13. [tex]\(\sqrt{144} = 12\)[/tex]: Rational - [tex]\(12\)[/tex] is an integer.
14. [tex]\(2^5 = 32\)[/tex]: Rational - [tex]\(32\)[/tex] is an integer.
15. [tex]\(\sqrt{21}\)[/tex]: Irrational - Cannot be simplified to an integer.
16. 4.5: Rational - A terminating decimal.
17. 1: Rational - It is an integer.
18. [tex]\(\sqrt{36} = 6\)[/tex]: Rational - [tex]\(6\)[/tex] is an integer.
19. 2 (Terminates): Rational - A terminating number.
20. 5.320..: Irrational - A non-terminating, non-repeating decimal.
21. [tex]\(\frac{\sqrt{100}}{2} = 5\)[/tex]: Rational - [tex]\(5\)[/tex] is an integer.
22. [tex]\(\sqrt{40}\)[/tex]: Irrational - Cannot be simplified to an integer.
23. 5 [tex]\(\frac{2}{5}\)[/tex]: Rational - It can be expressed as an improper fraction.
24. 0.132 (Repeats): Rational - A repeating decimal can be expressed as a fraction.
According to the criteria:
- Rational Numbers:
1. 5000
2. [tex]\(\sqrt{9} = 3\)[/tex]
3. -4
4. 3.2 (Repeats)
5. [tex]\(\sqrt{81} = 9\)[/tex]
6. 0
7. 12.2
8. [tex]\(\frac{1}{8}\)[/tex]
9. -22.3
10. [tex]\(\sqrt{144} = 12\)[/tex]
11. [tex]\(2^5 = 32\)[/tex]
12. 4.5
13. 1
14. [tex]\(\sqrt{36} = 6\)[/tex]
15. 2 (Terminates)
16. [tex]\(\frac{\sqrt{100}}{2} = 5\)[/tex]
17. 5 [tex]\(\frac{2}{5}\)[/tex]
18. 0.132 (Repeats)
- Irrational Numbers:
1. 3.421553...
2. [tex]\(\sqrt{32}\)[/tex]
3. [tex]\(\pi\)[/tex]
4. [tex]\(\sqrt{21}\)[/tex]
5. 5.320..
6. [tex]\(\sqrt{40}\)[/tex]
