Sure, let's analyze the sets and see if -1.576 belongs to any of them.
1. Natural Numbers:
Natural numbers (often denoted by [tex]\(\mathbb{N}\)[/tex]) are the set of positive integers starting from 1, 2, 3, and so on. Negative numbers, decimals, and fractions are not included in natural numbers.
- Set: {1, 2, 3, ...}
- [tex]\(-1.576\)[/tex] does not belong to the set of natural numbers.
2. Whole Numbers:
Whole numbers (often denoted by [tex]\(\mathbb{W}\)[/tex]) are the set of natural numbers including zero. They do not include negative numbers or decimals.
- Set: {0, 1, 2, 3, ...}
- [tex]\(-1.576\)[/tex] does not belong to the set of whole numbers.
3. Integers:
Integers (often denoted by [tex]\(\mathbb{Z}\)[/tex]) include all whole numbers and their negative counterparts. However, it does not include fractions or decimal numbers.
- Set: {..., -3, -2, -1, 0, 1, 2, 3, ...}
- [tex]\(-1.576\)[/tex] does not belong to the set of integers because it is a decimal number.
4. Rational Numbers:
Rational numbers (often denoted by [tex]\(\mathbb{Q}\)[/tex]) are numbers that can be expressed as the quotient of two integers (i.e., in the form [tex]\(\frac{a}{b}\)[/tex], where [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are integers and [tex]\(b \neq 0\)[/tex]). Decimal numbers that terminate or repeat are also rational.
- Example: [tex]\(\frac{-1576}{1000} = -1.576\)[/tex]
- [tex]\(-1.576\)[/tex] belongs to the set of rational numbers because it can be expressed as a fraction.
5. Irrational Numbers:
Irrational numbers (often denoted by [tex]\(\mathbb{I}\)[/tex]) cannot be expressed as a simple fraction; their decimal expansions are non-terminating and non-repeating.
- Examples: [tex]\(\pi, \sqrt{2}\)[/tex]
- [tex]\(-1.576\)[/tex] does not belong to the set of irrational numbers because it can be expressed as a fraction.
6. Real Numbers:
Real numbers (often denoted by [tex]\(\mathbb{R}\)[/tex]) are all the numbers that can be found on the number line. This set includes all rational and irrational numbers.
- [tex]\(\mathbb{R} = \mathbb{Q} \cup \mathbb{I}\)[/tex]
- [tex]\(-1.576\)[/tex] belongs to the set of real numbers because it is a rational number.
Summarizing, the number [tex]\(-1.576\)[/tex] belongs to the following sets:
- Rational numbers
- Real numbers
