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.

Distributivity of Multiplication over Addition:

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.

Representation of Whole Numbers on Number Line

Division as The Inverse of Multiplication

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