# 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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