Whole numbers are a fundamental concept in mathematics. They are defined as the set of non-negative integers, which includes zero and all positive integers but excludes fractions, decimals, and negative numbers. The set of whole numbers is represented as:
W={0,1,2,3,4,5,…}
Key Properties of Whole Numbers:
Closure Property: Whole numbers are closed under addition and multiplication. This means that when you add or multiply two whole numbers, the result is always a whole number. For example:
3+5 = 8 (a whole number)
4×2 = 8 (a whole number)
Commutative Property: Addition and multiplication of whole numbers are commutative, meaning the order of the numbers doesn’t affect the result. For example:
2+3 = 3+2 = 5
4×6 = 6×4 = 24
Associative Property: Whole numbers follow the associative property under addition and multiplication, meaning the way numbers are grouped doesn’t affect the result. For example:
(2+3)+4 = 2+(3+4) = 9
(2×3)×4 = 2×(3×4) = 24
Distributive Property: Multiplication of whole numbers distributes over addition and subtraction. For example:
3×(2+5) = (3×2)+(3×5) = 6+15 = 21
Identity Properties:
Additive Identity: 0 is the additive identity because adding 0 to any whole number does not change its value (e.g., 5+0=55 + 0 = 55+0=5).
Multiplicative Identity: 1 is the multiplicative identity because multiplying any whole number by 1 gives the same number (e.g., 7×1=77 \times 1 = 77×1=7).
Examples:
Identify whole numbers: From the set −2,0,3.5,5,1, the whole numbers are 0,5,10, 5, 10,5,1.
Using the distributive property: 4×(3+2) = 4×3+4×2 = 12+8 = 20.
Difference Between Natural and Whole Numbers:
Natural numbers are used for counting and start from 1, while whole numbers include all natural numbers plus zero. This means that zero is a whole number but not a natural number.
Whole numbers play a critical role in various mathematical operations and are essential for understanding basic arithmetic and number theory.
