In this post we will discuss important properties of multiplication.

These properties are very basic and simple in nature but are very helpful to solve complex problem faster.

I strongly urge all the kids to learn the properties and practice the relevant question for your Math Examination.

**What are Properties of Multiplication?**

**Commutative Property of Multiplication**

It says that in multiplication operation **even if you change the location of numbers, the end result will be the same**.

The commutative property is expressed as:

Where A & B can be any possible number

Let us understand the property with examples:

**Example 01**

Let A = 2 and B = 3

Calculate A x B

⟹ A x B

⟹ 2 x 3

⟹ 6

Calculate B x A

⟹ B x A

⟹ 3 x 2

⟹ 6

Note that on changing the location of number A & B, we get the same result 6

**Example 02**

Let A = -7 and B = 6

Calculate A x B

⟹ A x B

⟹ -7 x 6

⟹ – 42

Calculate B x A

⟹ B x A

⟹ 6 x -7

⟹ – 42

Hence we get same result -42 for both the calculation

**Conclusion**: In Multiplication, changing the order of digit will not effect the end result

**Associative Property of Multiplication**

It says that **forming different groups in multiplication operation will not change the end result**.

The associative property can be expressed as:

Where A, B & C can be any possible real number.

**Example 01**

Let A = 5, B = 2 & C = 4

Calculate (A x B) x C

⟹ (A x B) x C

⟹ (5 x 2) x 4

⟹ 10 x 4

⟹ 40

Now Calculate A x (B x C)

⟹ A x (B x C)

⟹ 5 x (2 x 4)

⟹ 5 x 8

⟹ 40

Note that both the calculation gives same result 40

**Example 02**

Let A = -11, B = 1 & C = 3

Calculate (A x B) x C

⟹ (A x B) x C

⟹ (-11 x 1) x 3

⟹ -11 x 3

⟹ -33

Now Calculate A x (B x C)

⟹ A x (B x C)

⟹ -11 x (1 x 3)

⟹ -11 x 3

⟹ -33

We get the same result -33 in both the calculation

**Conclusion**: In multiplication of numbers, forming different groups will not change the end result

**Distributive Property of Multiplication**

There are two different part for this property:

(i) Distributive Property of Multiplication over addition

(ii) Distributive Property of Multiplication over subtraction

Both are very similar properties but we will learn both.

**(i) Distributive property of multiplication over addition**

It says that **multiplication with sum of numbers is equal to sum of product of numbers**.

The property can be expressed as:

**(ii) Distributive Property of Multiplication over Subtraction**

It says that **multiplication with difference of numbers is equal to subtraction of product of numbers**.

The property can be expressed as:

Given below are some examples of distributive property.

**Example 01**

A = 6, B = 2 & C = 10

Verify distributive property of multiplication over addition

Calculate A x (B +C)

⟹ 6 x ( 2 + 10)

⟹ 6 x 12

⟹ 72

Calculate A.B + A.C

⟹6.2 + 6.10

⟹ 12 + 60

⟹ 72

hence, both the calculation give the same result 72

**Example 02**

If A = 9, B = 1 & C = 3

Verify distributive property of multiplication over subtraction

Calculate A x (B – C)

⟹ 9 x ( 1 – 3)

⟹ 9 x -2

⟹ -18

Calculate A.B – A.C

⟹ 9.1 – 9.3

⟹ 9 -27

⟹ -18

Both the calculation give -18 as a result

hence, distributive property of multiplication is verified

**Identity Property of Multiplication**

According to the property, **if you multiply any number with 1 you will get same number as a result**.

The property can be expressed as:

Where A can be any possible real number.

**Why the property is called “Identity” Property?**

The name “identity” is given as on multiplying number with 1, the identity of the number remain unchanged and we get the same number as a result

Given are some examples of identity property.**Example 01**

Let A = 4

Calculate A x 1

⟹ 4 x 1

⟹ 4

**Example 02**Let A = -17

Calculate A x 1

⟹ -17 x 1

⟹ -17

**Zero Property of Multiplication**

According to the property,** multiplication of any number with zero results in zero**.

Given below are some examples

(a) 12 x 0 = 0

(b) – 6 x 0 = 0

(c) 99 x 0 = 0

(d) 0 x 0 = 0

(e) 1000 x 0 = 0

**Inverse property of Multiplication**

According to the inverse property,** if we multiply any number with its reciprocal we get number 1**

The inverse property of multiplication is expressed as:

Where A can be any possible number and 1/A is its reciprocal

Here 1/A is also called Multiplicative Inverse

**How to find the reciprocal of any number?**

Just switch the numerator & denominator of any given number and you will get the reciprocal.

Given are some examples of reciprocal calculation

\mathtt{( i) \ 5\ =\ \frac{1}{5}}\\\ \\ \mathtt{( ii) \ \frac{2}{3} \ =\ \frac{3}{2}}\\\ \\ \mathtt{( iii) \ \frac{1}{7} \ =\ \frac{7}{1}}\\\ \\ \mathtt{( iv) \ \frac{1}{0} \ =\ not\ defined} \\\ \\

**Note**: keep two things in mind while finding reciprocal of a numbers:

(a) Reciprocal of number 0 is not defined

(b) Fraction with 0 in denominator has no reciprocal

**Multiplication Property of Equality**

The property states that** in a balanced equation if you multiply a number on both sides, the equation will remain balanced and valid**.

Suppose the given equation is 2y + 4 = 6

If we multiply number 5 on both sides, the equation will still be valid

5. (2y + 4) = 6. 5

10y + 20 = 30

Hence, the equation 10y + 20 = 30 is correct and valid.

In General Math terms, the multiplication property of equality can be expressed as:

Note: Remember to multiply on both sides, otherwise the equation will be invalid

**Question on Property of Multiplication**

(01) Find the property used in below expression

A + (B + C) = (A + B) + C

(a) Commutative Property

(b) Associative Property

(c) Distributive Property

(d) Inverse Property

Associative property is used in above expression**Option (b) is correct**

**(02) What is reciprocal of number 0?**

(a) 1/0

(b) 0

(c) Not defined

(d) 0/1

**Option (c) is correct**

**(03) According to commutative property, A x B = ?**

(a) – ( B x A )

(b) 1 / ( B x A )

(c) – 1 / ( B x A )

(d) B x A

**Option (d) is correct**

The general equation for commutative property is :

A x B = B x A

**(04) Which of the following is correct example of distribution property of multiplication over addition?**

(a) 2 ( 6 – 3 ) = 2 . 6 – 2 . 3

(b) 2 (6 + 3 ) = 2 . 6 + 2 . 3

(c) 2 (6 + 3 ) = 2 . 6 – 6 . 3

**Option (b) is correct**

The general expression for distribution property of multiplication over addition is :

A x ( B + C ) = A. B + A.C

**(05) Find multiplicative inverse of 1/6**

(a) 6

(b) -6

(c) -1/6

**Option (a) is correct**

The reciprocal of the number is also called multiplicative inverse.

The reciprocal of 1/6 is 6

According to inverse property, multiplication of number with its inverse results in number 1.

1/6 x 6 = 1

**(06) Find the name of property used by below expression**

A x 1 = A

(a) Identity Property

(b) Inverse property

(c) Commutative Property

**Option (a) is correct**