# PROPERTIES OF ADDITION AND MULTIPLICATION

In math, properties of addition and multiplication are nothing but certain principles are rules which will always be true for the two binary operations addition and multiplication. Knowing and following the properties of addition and multiplication will help us to solve math problems.

There are several properties of addition and multiplication.

 ADDITION PROPERTIES1. Commutative PropertyChanging the order of addends does not change the sum. a + b  =  b + aExample : 3 + 7  =  7 + 310  =  102. Associative propertyChanging the grouping of the addends does not change the sum.(a + b) + c  =  a + (b + c)Example :(5 + 6) + 4  =  5 + (6 + 4)11 + 4  =  5 + 1015  =  153.Identity PropertyThe sum of any number and zero is that number.a + 0  =  aExample :7 + 0  =  7 MULTIPLICATION PROPERTIES1. Commutative PropertyChanging the order of factors does not change the product.a x b  =  b x aExample : 3 x 7  =  7 x 321  =  212. Associative propertyChanging the grouping of the factors does not change the product.(a x b) x c  =  a x (b x c)Example :(5 x 6) x 4  =  5 x (6 x 4)30 x 4  =  5 x 24120  =  1203.Identity PropertyThe product of any number and one is that number.a x 1  =  aExample :7 x 1  =  7

From identity property of addition and multiplication, we can come to know the following important fact.

## Distributive property of multiplication over addition

The product of a factor and a sum is equal to the sum of the products.

a x (b + c)  =  (a x b) + (a x c)

Example :

3 x (5 + 8)  =  (3 x 5) + (3 x 8)

3 x 13  =  15 + 24

39  =  39

## Zero property

The product of zero and any number is zero.

a x 0  =  0

Example :

3 x 0  =  0

## Important hint on associative property

In associative property of addition and multiplication, always the order of numbers on the left side must be equal to the order of numbers on the right side.

Order of numbers on the left  =  Order of numbers on the right

(3 + 4) + 5  =  3 + (4 + 5)

Order -----------> 3, 4, 5  =  3, 4, 5

Example (Multiplication) :

(7 x 8) x 9  =  7 x (8 x 9)

Order -----------> 7, 8, 9  =  7, 8, 9

## Practice Problems

Problem 1 :

Identify the property

5 x 0  =  0

Solution :

The above expressions says that the product of 5 and 0 is 0.

This is "Zero property".

Because, in zero property, we have that the product of any number and zero is zero.

Let us look at the next problem on "Properties of addition and multiplication"

Problem 2 :

Identify the property

2(x + y)  =  2x + 2y

Solution :

The above expressions says that the product 2 and (x+y) is equal to the sum of the products 2x and 2y.

This is "Distributive property of multiplication over addition".

Because, in zero property, we have that the product of a factor and a sum is equal to the sum of the products.

Let us look at the next problem on "Properties of addition and multiplication"

Problem 3 :

Identify the property

101 + 99  =  99 + 101

Solution :

The above expressions says that the sum of 101 and 99 is equal to the sum of 99 and 101.

This is "Commutative property of addition".

Because, in commutative property of addition, we have that changing the order of addends does not change the sum.

Let us look at the next problem on "Properties of addition and multiplication"

Problem 4 :

Identify the property

3 + (4 + 5)  =  (3 + 4) + 5

Solution :

From the above expression, we have

Order of numbers on the left  =  Order of numbers on the right

That is,   3, 4, 5  =  3, 4, 5

Numbers on both sides are added with grouping different numbers on the left side and different numbers on the ride.

This is "Associative property of addition".

Because, in associative property of addition, we have that changing the grouping of the addends does not change the sum.

Let us look at the next problem on "Properties of addition and multiplication"

Problem 5 :

Identify the property

2 x (7 x 9)  =  (2 x 7) x 9

Solution :

From the above expression, we have

Order of numbers on the left  =  Order of numbers on the right

That is,   2, 7, 9  =  2, 7, 9

Numbers on both sides are multiplied with grouping different numbers on the left side and different numbers on the ride.

This is "Associative property of multiplication".

Because, in associative property of multiplication, we have that changing the grouping of the factors does not change the product.

Let us look at the next problem on "Properties of addition and multiplication"

Problem 6 :

Identify the property

2 + 0  =  0

Solution :

The above expression says that the sum of 2 and 0 is 2.

This is "Identity property of addition".

Because, in identity property of addition, we have that the sum of any number and zero is that number.

Let us look at the next problem on "Properties of addition and multiplication"

Problem 7 :

Identify the property

8 x 1  =  8

Solution :

The above expression says that the product of 8 and 1 is 8.

This is "Identity property of multiplication".

Because, in identity property of multiplication, we have that the product of any number and one is that number.

Let us look at the next problem on "Properties of addition and multiplication"

Problem 8 :

Identify the property

(2 + g) + 3   =  3 + (2 + g)

Solution :

As soon as students see this question, they will decide that associative property of addition is applied in the above expression.

Actually, it is not.

Because,

Order of numbers on the left  ≠  Order of numbers on the right

Order -----------> 2, g, 3  ≠  3, 2, g

And the above expressions says that the sum of (2+g) and 3 is equal to the sum of (2+g) and 3.

This is "Commutative property of addition".

Because, in commutative property of addition, we have that changing the order of addends does not change the sum.

Let us look at the next problem on "Properties of addition and multiplication"

Problem 9 :

Identify the property

(6x4) + (6x2)  =  (6x2) + (6x4)

Solution :

The above expressions says that the sum of  (6x4) and (6x2) is equal to the sum of (6x2) and (6x4).

This is "Commutative property of addition".

Because, in commutative property of addition, we have that changing the order of addends does not change the sum.

Let us look at the next problem on "Properties of addition and multiplication"

Problem 10 :

Identify the property

(1+2) x (3+4)  =  (3+4) x (1+2)

Solution :

The above expressions says that the product of (1+2) and (3+4) is   equal to  the product (3+4) and (1+2).

This is "Commutative property of multiplication".

Because, in commutative property of multiplication, we have that changing the order of factors does not change the product.

We hope that the stuff given above would be much helpful for the students who learn "Properties of addition and multiplication" in math.

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