Whole Numbers – Properties

Whole numbers are the set of numbers that include all the natural numbers and zero (0, 1, 2, 3, …). They do not include any fractions, decimals, or negative numbers. Understanding the properties of whole numbers is fundamental in arithmetic and number theory.

Here are the key properties of whole numbers:

1. Closure Property

• Addition: The sum of any two whole numbers is always a whole number.
• a + b = c where a, b, c ∈ W
• For example, 2+3=5.
• Multiplication: The product of any two whole numbers is always a whole number.
• a × b = c where a, b, c ∈ W
• For example, 2×3=6.

2. Commutative Property

• Addition: The order in which two whole numbers are added does not affect the sum.
• a + b=b + a
• For example, 3+4=4+3.
• Multiplication: The order in which two whole numbers are multiplied does not affect the product.
• a × b = b × a
• For example, 3×4=4×3.

3. Associative Property

• Addition: The way in which whole numbers are grouped in addition does not affect the sum.
• (a + b) + c = a + (b + c)
• For example, (2 + 3) + 4 = 2 + (3 + 4).
• Multiplication: The way in which whole numbers are grouped in multiplication does not affect the product.
• (a × b) + c = a × (b + c)
• For example, (2×3)×4=2×(3×4).

4. Distributive Property

• This property connects addition and multiplication. It states that multiplying a number by a sum is the same as multiplying each addend individually and then adding the products.
• a × (b + c) = (a × b)+(a × c)
• For example, 2×(3+4)=(2×3)+(2×4).

5. Identity Property

• Addition (Additive Identity): The sum of any whole number and zero is the number itself.
• a + 0=a
• For example, 5+0=5.
• Multiplication (Multiplicative Identity): The product of any whole number and one is the number itself. a×1=a
• For example, 5×1=5.

6. Zero Property of Multiplication

• The product of any whole number and zero is zero. a×0=0
• For example, 5×0=0.

Summary

These properties help in simplifying calculations and understanding the structure of whole numbers. Recognizing and applying these properties allows for efficient problem-solving in arithmetic and algebra.