To categorize the given numbers 500, 0.11, -9, [tex]\(\sqrt{5}\)[/tex], [tex]\(\frac{22}{7}\)[/tex], [tex]\(\frac{1}{10}\)[/tex], and 0 into different sets, let's go through each set step-by-step:
### a) Natural Numbers
Natural numbers are positive integers starting from 1 and going upwards (1, 2, 3, ...). Among the given numbers, only 500 is a positive integer fitting this category.
- Natural Numbers: [tex]\(\{500\}\)[/tex]
### b) Whole Numbers
Whole numbers include all natural numbers along with zero (0, 1, 2, 3, ...).
- Whole Numbers: [tex]\(\{0, 500\}\)[/tex]
### c) Integers
Integers include all positive and negative whole numbers along with zero (... -3, -2, -1, 0, 1, 2, 3 ...). Among the given numbers, 500, -9, and 0 fit this category.
- Integers: [tex]\(\{0, -9, 500\}\)[/tex]
### d) Rational Numbers
Rational numbers are numbers that can be expressed as a fraction or ratio of two integers (including terminating and repeating decimals). Among the given numbers, the following fit this category: 0, 0.11, -9, 500, [tex]\(\frac{22}{7}\)[/tex], [tex]\(\frac{1}{10}\)[/tex].
- Rational Numbers: [tex]\(\{0, 0.11, -9, 500, \frac{22}{7}, \frac{1}{10}\}\)[/tex]
### e) Irrational Numbers
Irrational numbers cannot be expressed as a simple fraction; their decimal expansions are non-repeating and non-terminating. Among the given numbers, [tex]\(\sqrt{5}\)[/tex] fits this category.
- Irrational Numbers: [tex]\(\{\sqrt{5}\}\)[/tex]
### f) Real Numbers
Real numbers include all rational and irrational numbers. In this case, all the given numbers are real numbers.
- Real Numbers: [tex]\(\{0, 0.11, -9, 500, \sqrt{5}, \frac{22}{7}, \frac{1}{10}\}\)[/tex]
To summarize, the categorizations are:
- a) Natural Numbers: [tex]\(\{500\}\)[/tex]
- b) Whole Numbers: [tex]\(\{0, 500\}\)[/tex]
- c) Integers: [tex]\(\{0, -9, 500\}\)[/tex]
- d) Rational Numbers: [tex]\(\{0, 0.11, -9, 500, 3.142857142857143, 0.1\}\)[/tex]
- e) Irrational Numbers: [tex]\(\{2.23606797749979\}\)[/tex]
- f) Real Numbers: [tex]\(\{0, 0.11, -9, 500, 2.23606797749979, 3.142857142857143, 0.1\}\)[/tex]
