# Properties of Multiplication

The properties of multiplication on whole numbers are discussed below; these properties will help us in finding the product of even very large numbers conveniently.

Closure Property:

If a and b are two numbers, then their product a × b is also a whole number.

In other words, if we multiply two whole numbers, we get a whole number.

Verification:

In order to verify this property, let us take a few pairs of whole numbers and multiply them;

For example:

(i) 8 × 9 = 72

(ii) 0 × 16 = 0

(iii) 11 × 15 = 165

(iv) 20 × 1 = 20

We find that the product is always a whole numbers.

Commutativity:

The multiplication of whole numbers is commutative.

In other words, if a and b are any two whole numbers, then a × b = b × a.

Verification:

In order to verify this property, let us take a few pairs of whole numbers and multiply these numbers in different orders as shown below;

For example:

(i) 7 × 6 = 42 and 6 × 7 = 42

Therefore, 7 × 6 = 6 × 7

(ii) 20 × 10 = 200 and 10 × 20 = 200

Therefore, 20 × 10 = 10 × 20

(iii) 15 × 12 = 180 and 12 × 15 = 180

Therefore, 15 × 12 = 12 × 15

(iv) 12 × 13 = 156 and 13 × 12

Therefore, 12 × 13 = 13 × 12

We find that in whatever order we multiply two whole numbers, the product remains the same.

`

Multiplication By Zero:

If a is any whole number, then a × 0 = 0 × a = 0.

In other words, the product of any whole number and zero is always zero.

Verification:

In order to verify this property, we take some whole numbers and multiply them by zero as shown below;

For example:

(i) 20 × 0 = 0 × 20 = 0

(ii) 1 × 0 = 0 × 1 = 0

(iii) 115 × 0 = 0 × 115 = 0

(iv) 0 × 0 = 0 × 0 = 0

(v) 136 × 0 = 0 × 136 = 0

We observe that the product of any whole number and zero is zero.

Multiplicative Identity:

If a is any whole number, then a × 1 = a = 1 × a.

In other words, the product of any whole number and 1 is the number itself.

Verification:

In order to verify this property, we find the product of different whole numbers with 1 as shown below:

For example:

(i) 13 × 1 = 13 = 1 × 13

(ii) 1 × 1 = 1 = 1 × 1

(iii) 25 × 1 = 25 = 1 × 25

(iv) 117 × 1 = 117 = 1 × 117

We see that in each case a × 1 = a = 1 × a.

The number 1 is called the multiplication identity or the identity element for multiplication of whole numbers because it does not change the identity (value) of the numbers during the operation of multiplication.

Associativity:

If a, b, c are any whole numbers, then

(a × b) × c = a × (b × c)

In other words, the multiplication of whole numbers is associative, that is, the product of three whole numbers does not change by changing their arrangements.

Verification:

In order to verify this property, we take three whole numbers say a, b, c and find the values of the expression (a × b) × c and a × (b × c) as shown below :

For example:

(i) (2 × 3) × 5 = 6 × 5 = 30 and 2 × (3 × 5) = 2 × 15 = 30

Therefore, (2 × 3) × 5 = 2 × (3 × 5)

(ii) (1 × 5) × 2 = 5 × 2 = 10 and 1 × (5 × 2) = 1 × 10 = 10

Therefore, (1 × 5) × 2 = 1 × (5 × 2)

(iii) (2 × 11) × 3 = 22 × 3 = 66 and 2 × (11 × 3) = 2 × 33 = 66

Therefore, (2 × 11) × 3 = 2 × (11 × 3).

(iv) (4 × 1) × 3 = 4 × 3 = 12 and 4 × (1 × 3) = 4 × 3 = 12

Therefore, (4 × 1) × 3 = 4 × (1 × 3).

We find that in each case (a × b) × c = a × (b × c).

Thus, the multiplication of whole numbers is associative.

If a, b, c are any three whole numbers, then

(i) a × (b + c) = a × b + a × c

(ii) (b + c) × a = b × a + c × a

In other words, the multiplication of whole numbers distributes over their addition.

Verification:

In order to verify this property, we take any three whole numbers a, b, c and find the values of the expressions a × (b + c) and a × b + a × c as shown below :

For example:

(i) 3 × (2 + 5) = 3 × 7 = 21 and 3 × 2 + 3 × 5 = 6 + 15 =21

Therefore, 3 × (2 + 5) = 3 × 2 + 3 × 5

(ii) 1 × (5 + 9) = 1 × 14 = 15 and 1 × 5 + 1 × 9 = 5 + 9 = 14

Therefore, 1 × (5 + 9) = 1 × 5 + 1 × 9.

(iii) 2 × (7 + 15) = 2 × 22 = 44 and 2 × 7 + 2 × 15 = 14 + 30 = 44.

Therefore, 2 × (7 + 15) = 2 × 7 + 2 × 15.

The Number Zero

Properties of Whole Numbers

Successor and Predecessor

Representation of Whole Numbers on Number Line

Properties of Subtraction

Properties of Multiplication

Properties of Division

Division as The Inverse of Multiplication