Here, we are going to see the following four properties of addition.

(i)  Commutative property

(ii)  Associative property

(iv)  Distributive property

Let us look at these properties of addition in detail.

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.

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.

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

Observe the example: 5 + 0 = 5.

In general, for any real number a, a + 0 = a.

Therefore, zero is the additive identity for real numbers.

## Distributive property

Distributive property of multiplication over 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.

After having gone through the stuff given above, we hope that the students would have understood "Properties of adding real numbers".

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