PROPERTIES OF ADDITION

Following are the properties of addition :

(i)  Commutative property

(ii)  Associative property

(iii)  Identity property

(iv)  Property of additive inverse

(v)  Distributive property

(vi)  Property of equality  

Commutative Property of Addition

Two real numbers can be added in any order.

In other words, addition is commutative for real numbers.

We have 8 + (- 3)  =  5 and (- 3) + 8  =  5

So,

8 + (- 3)  =  (- 3) + 8

In general, for any two real numbers a and b we can say,

a + b  =  b + a

Therefore addition of real numbers is commutative.

Associative Property of Addition

Observe the following example :

Consider the real numbers 5, – 4 and 7.

Look at

5 + [(– 4) + 7]  =  5 + 3  =  8

and 

[5 + (– 4)] + 7  =  1 + 7  =  8

Therefore,

5 + [(– 4) + 7]  =  [5 + (– 4)] + 7

In general, for any real numbers a, b and c, we can say,

a + (b + c)  =  (a + b) + c.

Therefore addition of real numbers is associative.

Identity Property of Addition

When we add zero to any real number, we get the same real number.

Observe the examples :

5 + 0  =  5

0 + 7  =  7

-2 + 0  =  -2

0 + (-3)  =  -3

In general, for any real number a,

a + 0  =  a

0 + a  =  a

Therefore, zero is the additive identity for real numbers.

Property of Additive Inverse

The sum of a real number and its additive inverse is zero. 

3 + (-3)  =  (-3) + 3  =  0

Note : 

(i) If a number is positive, then its additive inverse is the same number with negative sign.

(ii) If a number is negative, then its additive inverse is the same number with positive sign.  

Examples : 

-5 is the additive inverse of 5

7 is the additive inverse of -7

Distributive Property of Addition

Consider the real numbers 12, 9, 7.

Look at

12 x (9 + 7)  =  12 x 16  =  192

12 x (9 + 7)  =  12 x 9  +  12 x 7  =  108 + 84  =  192

Thus

12 x (9 + 7)  =  (12 x 9) + (12 x 7)

In general, for any real numbers a, b, c.

a x (b + c)  =  (a x b) + (a x c)

Therefore, multiplication is distributive over addition.

Addition Property of Equality

Two sides of an equation remain equal, if the same number is added to each side.

For example, if x = y, then

x + z  =  y + z

Example : 

3  =  3

Add 5 to each side. 

3 + 5  =  3 + 5

8  =  8

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