To determine which set includes the number [tex]\(\frac{5}{9}\)[/tex], we need to evaluate the properties of this number according to the definitions of different sets of numbers:
1. Natural Numbers:
- Natural numbers are positive integers starting from 1.
- Since [tex]\(\frac{5}{9}\)[/tex] is not a whole number and not an integer, it is not a natural number.
- Result: [tex]\(\frac{5}{9}\)[/tex] is not a natural number. Therefore, it is indicated by 0.
2. Whole Numbers:
- Whole numbers are non-negative integers, starting from 0.
- Since [tex]\(\frac{5}{9}\)[/tex] again is neither whole nor an integer, it does not belong to this set.
- Result: [tex]\(\frac{5}{9}\)[/tex] is not a whole number. Therefore, it is indicated by 0.
3. Integers:
- Integers include all positive and negative whole numbers and zero.
- As [tex]\(\frac{5}{9}\)[/tex] is not a whole number, it is not an integer.
- Result: [tex]\(\frac{5}{9}\)[/tex] is not an integer. Therefore, it is indicated by 0.
4. Rational Numbers:
- Rational numbers are numbers that can be expressed as the ratio of two integers (i.e., [tex]\(\frac{a}{b}\)[/tex] where [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are integers and [tex]\(b \neq 0\)[/tex]).
- [tex]\(\frac{5}{9}\)[/tex] can be written as the ratio of two integers 5 and 9.
- Result: [tex]\(\frac{5}{9}\)[/tex] is a rational number. Therefore, it is indicated by 1.
5. Irrational Numbers:
- Irrational numbers are numbers that cannot be expressed as a simple fraction.
- Since [tex]\(\frac{5}{9}\)[/tex] can be expressed as a fraction, it is not irrational.
- Result: [tex]\(\frac{5}{9}\)[/tex] is not an irrational number. Therefore, it is indicated by 0.
6. Real Numbers:
- Real numbers include all rational and irrational numbers.
- Since [tex]\(\frac{5}{9}\)[/tex] is a rational number, it is also a real number.
- Result: [tex]\(\frac{5}{9}\)[/tex] is a real number. Therefore, it is indicated by 1.
Summarizing all these points, [tex]\(\frac{5}{9}\)[/tex] fits into the following sets:
- Rational numbers: 1
- Real numbers: 1
Thus, when we check each set for the number [tex]\(\frac{5}{9}\)[/tex], the results are:
- Natural numbers: 0
- Whole numbers: 0
- Integers: 0
- Rational numbers: 1
- Irrational numbers: 0
- Real numbers: 1
