Properties of Addition

The properties of addition are useful in adding group of numbers. So, if we want to add two numbers or more than two , then we can use these properties. Simplification will become easy.

The Properties of Addition are,

1) Closure Property.

2) Commutative Property

3) Associative Property

4) Distributive Property

5) Identity Property

6) Inverse Property

Closure Property:

If we add two real numbers , then we get always a real number. This Property is called Closure Property.

Examples:

1) 2 + 3 = 5 .

Here 2 and 3 are real numbers. By adding 2 and 3 , the result is 5 which is also a real number.

2) 7 + 4 = 11.

We are adding 7 and 4 which are real numbers , then we have 11 which is a real number.

Commutative Property:

If A and B are two real numbers , then according to Commutative Property,

A + B = B + A.

That is result of adding A and B is same as adding B and A . When you are

adding two numbers , you can change the way of adding the numbers , here way

of adding numbers means order of adding numbers.

Examples:

1) 2 + 3 = 5

3 + 2 = 5

Here you get 5 when you add 2 and 3 . The same you get when you add 3 and

2. So, you can add 2 and 3 in any order , the result is 5 only.

2) 7 + 4 = 11

4 + 7 = 11.

11 is the result when you add 7 and 4 .The same result you get when you add 4 and 7. Here order is not considered while adding two numbers.

Associative Property:

If A , B , C are three real numbers , then according to Associative Property

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

Grouping B and C and adding the result to A is same as grouping A and B and

adding the result to C . When you need to add three numbers, then you can add any two

numbers and then add the result to the third number.

Example:

1) 1 + (3 + 4) = 1 + 7 = 8

(1 + 3 ) + 4 = 4 + 4 = 8

(1 + 4 ) + 3 = 5 + 3 = 8

So , We can group any two numbers and add the result to third number , the result is same.

Grouping is indicated by the parentheses .

Distributive Property :

If A , B , C are three real numbers , then according to Distributive Property

A(B + C) = AB + AC

The name itself indicates that A is distributed between B and C .

Example:

6(9+5) = 6(14)

= 6 x 14

= 84.

6 x 9 + 6 x 5 = 54 + 30

= 84.

Therefore 6(9+5) = 6 x 9 + 6 x 5.

Here 6 is distributed between 9 and 5.

Identity Property:

If A is a real number , then

0 + A = A + 0 = A . This is called Identity Property of Addition. 0 is called Additive identity . Here we have same number A . When we add zero to it . There is no change in the number .

Examples :

1) 0 + 5 = 5 + 0 = 5 .

There is no change in the number 5 when we add zero to 5 . We get the same number 5.

2) 0 + 11 = 11 + 0 = 11

The number is same when you add 0 to 11 .There is no change in the number 11 .

Inverse Property:

If A is a real number , then -A is called Additive Inverse of A .

So , A + (-A) = (-A) + A = 0. This Property is called Inverse Property.

Examples:

1) Additive inverse of 13 is -13

2) Additive inverse of 29 is -29.

So , 29 + (-29 ) = (-29) + 29 = 0.

Solved Problems:

1) ( 25 + 6 ) + 5 = ?

Solution:

By Associative property ,

(25 + 6 ) + 5 = ( 25 + 5 ) + 6

= 30 + 6

= 36.

2) 34 + 27 + 16 = ?

Solution :

By Commutative Property,

34 + 27 + 16 = 34 + 16 + 27

= 50 + 27

= 77.

3) 46 + 60 + 54 + 30 = ?

Solution :

By Commutative Property,

46 + 60 + 54 + 30 = 46 + 54 + 60 + 30

By Associative Property,

46 + 54 + 60 + 30 = (46 + 54 ) + (60 + 30)

= 100 + 90

= 190.