Properties of Whole Numbers

The reading material provided on this page for Properties of Whole Numbers is specifically designed for students in grades 5 and 6. So, let's begin!

Whole Numbers

Whole numbers are the set of numbers that includes all positive integers (1, 2, 3, ...) as well as 0. Whole numbers do not include negative numbers or fractions.

Properties of Whole Numbers

Closure Property

Whole numbers are closed under addition and multiplication. This means that if you add or multiply two whole numbers, the result will always be another whole number.
If a and b are two whole numbers, then,

• a + b = c
• a x b = c
• c is a whole number

Example: 7 + 2 = 9 is a whole number
7 x 2 = 14 is a whole number

The resulting values in the above examples are whole numbers.

Commutative Property

Whole numbers have the commutative property of addition and multiplication. This means that the order of the numbers does not matter when adding or multiplying them.
If a and b are two whole numbers, then.

• a + b = b + a
• a x b = b x a

Example: 3 + 7 = 7 + 3 = 10
2 x 5 = 5 x 2 = 10

From the above examples we can observe that the result is not affected by the change in the order of numbers.

Associative Property

Whole numbers have the associative property of addition and multiplication. This means that changing the grouping of the numbers when adding or multiplying them will not affect the result.
Let a, b, and c are three whole numbers, then.

• a + (b + c) = (a + b) + c
• a x (b x c) = (a x b) x c

Example: 3 + (4 + 5) = (3 + 4) + 5
3 + 9 = 7 + 5
12 = 12

L.H.S = R.H.S
5 x (3 x 4) = (5 x 3) x 4
5 x 12 = 15 x 4
60 = 60
L.H.S = R.H.S

From the above examples, we can observe that the result is not affected by the change in the order of numbers.

Distributive Property

Multiplication is distributive over addition and subtraction. It means that if a, b and c are whole numbers then.

• a x (b + c) = (a x b) + (a x c)
• a × (b - c) = (a × b) - (a × c )

Example: 2 x (3 + 4) = (2 x 3) + (2 x 4)
2 x 7 = 6 + 8
14 = 14
L.H.S. = R.H.S.

Identity Property

Whole numbers have an identity element. The identity element for addition is 0 and the identity element for multiplication is 1.

Example: 29 + 0 = 29
29 x 1 = 29
By the above examples, we can say that if zero is added to any whole number, then the value of the original number does not change.

Similarly, when we multiply any whole number by 1, then the value of the actual number remains unchanged.

NOTE:

1. Whole numbers are not closed under division: When you divide a whole number by another whole number, the result may not always be a whole number.

For example, 7 ÷ 2 is not a whole number, but 8 ÷ 4 is a whole number.

2. Whole numbers are non-negative: Whole numbers are always positive or zero. They do not include negative numbers, such as -1, -2, -3, etc.