Natural Numbers: Natural numbers are the counting numbers starting from 1 and extending infinitely. They are denoted by the symbol "N." Examples of natural numbers include 1, 2, 3, 4, 5, and so on.
Whole Numbers: Whole numbers are the set of natural numbers along with zero. They are denoted by the symbol "W." Examples of whole numbers include 0, 1, 2, 3, 4, and so on.
Integers: Integers are the set of whole numbers along with their negative counterparts. They are denoted by the symbol "Z." Examples of integers include ..., -3, -2, -1, 0, 1, 2, 3, ... The ellipsis represents that integers extend infinitely in both the positive and negative directions.
Rational Numbers: Rational numbers are numbers that can be expressed as the quotient or fraction of two integers, where the denominator is not zero. They are denoted by the symbol "Q." Examples of rational numbers include 1/2, 3/4, -5/7, 2, and -3.25.
Properties of Rational Numbers:
1. Closure: The sum, difference, product, or quotient of any two rational numbers is always a rational number. For example, if we add two rational numbers 1/3 and 2/5, the result is 11/15, which is also a rational number.
2. Commutativity: The order of addition and multiplication does not affect the result for rational numbers. For example, if we add 1/2 and 3/4, the result is the same as adding 3/4 and 1/2. Similarly, multiplication of rational numbers follows the same rule.
3. Associativity: The associative property holds for addition and multiplication of rational numbers. For example, if we add three rational numbers (1/2 + 3/4) + 2/3, it is equal to 1/2 + (3/4 + 2/3).
4. Distributivity: Multiplication is distributive over addition for rational numbers. For example, if we multiply (1/2) by the sum of (3/4 + 2/3), it is equal to (1/2 * 3/4) + (1/2 * 2/3).
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