Properties of Multiplication

There are six properties of multiplication of whole numbers that will help to solve the problems easily.

The six properties of multiplication are Closure Property, Commutative Property, Zero Property,  Identity Property, Associativity Property and Distributive Property.

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 of  Whole Numbers:

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 of  Whole Numbers / Order Property of  Whole Numbers:

The multiplication of whole numbers is commutative.

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

We can multiply numbers in any order. The product does not change when the order of numbers is changed.

When multiplying any two numbers, the product remains same regardless of the order of multiplicands. We can multiply numbers in any order, the product remains the same.

For Example:

(i) 7 × 4 = 28

(ii) 4 × 7 = 28


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


(V) 1122 × 324 = 324 × 1122

(vi) 21892 × 1582 = 1582 × 21892



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


III. Multiplication By Zero/Zero Property of Multiplication of Whole Numbers:

When a number is multiplied by 0, the product is always 0.

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.

When 0 is multiplied by any number the product is always zero.

For example:

(i) 3 × 0 = 0 + 0 + 0 = 0

(ii) 9 × 0 = 0 + 0 + 0 = 0


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

(vi) 78160 × 0 = 0 × 78160 = 0

(vii) 51999 × 0 = 0 × 51999 = 0


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



IV. Multiplicative Identity of  Whole Numbers / Identity Property of  Whole Numbers:

When a number is multiplied by 1, the product is the number itself.

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.

When 1 is multiplied by any number the product is always the number itself.

For example:

(i) 1 × 2 = 1 + 1 = 2

(ii) 1 × 6 = 1 + 1 + 1 + 1 + 1 + 1 = 6


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

(v) 4295620 × 1 = 4295620

(vi) 108519 × 1 = 108519


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.


V. Associativity Property of Multiplication of Whole Numbers:

We can multiply three or more numbers in any order. The product remains the same.

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.

When three or more numbers are multiplied, the product remains the same regardless of their group or place. We can multiply three or more numbers in any order, the product remains the same.

For example:

(i) (6 × 5) × 3 = 90

(ii) 6 × (5 × 3) = 90

(iii) (6 × 3) × 5 = 90



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).

(v) (1462 × 1250) × 421 = 1462 × (1250 × 421) = (1462 × 421) × 1250

(vi) (7902 × 810) × 1725 = 7902 × (810 × 1725) = (7902 × 1725) × 810


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

Thus, the multiplication of whole numbers is associative.


VI. Distributive Property of Multiplication of Whole Numbers / Distributivity of Multiplication over Addition of Whole Numbers:

When multiplier is the sum of two or more numbers the product is equal to the sum of products.

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.


(vi) 50 × (325 + 175) = 50 × 3250 + 50 × 175

(v) 1007 × (310 + 798) = 1007 × 310 + 1007 × 798

Properties of Multiplication of Whole Numbers


These are the important properties of multiplication of whole numbers.

Let us first review the properties of multiplication learnt earlier:

1. The product of two numbers does not change when we change the order of the numbers.

Example: 25 × 13 = 13 × 25


2. The product of three numbers does not change when we change the groupings of the numbers.

Example: 20 × (8 × 4) = (20 × 8) × 4

                                   = (20 × 4) × 8


3. The product of a number by 1 is the number itself.

Example: 143 × 1 = 143 and 2535 × 1 = 2535


4. The product of a number by zero is always zero

Example: 563 × 0 = 0 and 6984 × 0 = 0


5. The product of a number by the sum of two numbers is the same as the sum of the products of that number by the two numbers separately

Example: 15 × (45 + 38) = 15 × 45 + 5 × 38


REMEMBER: In a multiplication question, the number to be multiplied is called multiplicand, the number try which we multiply is called multiplier and the result of multiplication is called product.

For example,

                  15          ×          10          =          150

           Multiplicand            Multiplier               Product



Worksheet on Properties of Multiplication:

1. Fill in the Blanks.

(i) Number × 0 = __________

(ii) 54 × __________ = 54000

(iii) Number × __________ = Number itself

(iv) 8 × (5 × 7) = (8 × 5) × __________

(v) 7 × _________ = 9 × 7

(vi) 5 × 6 × 12 = 12 × __________

(vii) 62 × 10 = __________

(viii) 6 × 32 × 100 = 6 × 100 × __________


Answers:

(i) 0

(ii) 1000

(iii) 1

(iv) 7

(v) 79

(vi) 5 × 6

(vii) 620

(viii) 32


2. Fill in the blanks using Properties of Multiplication:

(i) 62 × ………… = 5 × 62

(ii) 31 × ………… = 0

(iii) ………… × 9 = 332 × 9

(iv) 134 × 1 = …………

(v) 26 × 16 × 78 = 26 × ………… × 16

(vi) 43 × 34 = 34 × …………

(vii) 540 × 0 = …………

(viii) 29 × 4 × ………… = 4 × 15 × 29

(ix) 47 × ………… = 47


Answer:

2. (i) 5

(ii) 0

(iii) 332

(iv) 134

(v) 78

(vi) 43

(vii) 0

(viii) 15

(ix) 



You might like these

● Whole Numbers






Numbers Page 

6th Grade Page 

From Properties of Multiplication to HOME PAGE


New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.



Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



Share this page: What’s this?

Recent Articles

  1. Multiply a Number by a 2-Digit Number | Multiplying 2-Digit by 2-Digit

    Mar 27, 24 05:21 PM

    Multiply 2-Digit Numbers by a 2-Digit Numbers
    How to multiply a number by a 2-digit number? We shall revise here to multiply 2-digit and 3-digit numbers by a 2-digit number (multiplier) as well as learn another procedure for the multiplication of…

    Read More

  2. Multiplication by 1-digit Number | Multiplying 1-Digit by 4-Digit

    Mar 26, 24 04:14 PM

    Multiplication by 1-digit Number
    How to Multiply by a 1-Digit Number We will learn how to multiply any number by a one-digit number. Multiply 2154 and 4. Solution: Step I: Arrange the numbers vertically. Step II: First multiply the d…

    Read More

  3. Multiplying 3-Digit Number by 1-Digit Number | Three-Digit Multiplicat

    Mar 25, 24 05:36 PM

    Multiplying 3-Digit Number by 1-Digit Number
    Here we will learn multiplying 3-digit number by 1-digit number. In two different ways we will learn to multiply a two-digit number by a one-digit number. 1. Multiply 201 by 3 Step I: Arrange the numb…

    Read More

  4. Multiplying 2-Digit Number by 1-Digit Number | Multiply Two-Digit Numb

    Mar 25, 24 04:18 PM

    Multiplying 2-Digit Number by 1-Digit Number
    Here we will learn multiplying 2-digit number by 1-digit number. In two different ways we will learn to multiply a two-digit number by a one-digit number. Examples of multiplying 2-digit number by

    Read More

  5. Worksheet on Multiplying 1-Digit Numbers |Multiplying One Digit Number

    Mar 25, 24 03:39 PM

    Multiplication tables will help us to solve the worksheet on multiplying 1-digit numbers. The questions are based on multiplying one digit number and word problems on multiplying one digit number.

    Read More