ADDITION AND MULTIPLICATION PROPERTIES

Properties of Addition

(i) Commutative Property : 

Changing the order of addends does not change the sum.

If 'a' and 'b' are any two numbers, then

a + b  =  b + a

Example : 

5 + 8  =  13  

8 + 5  =  13 

So,

5 + 8  =  8 + 5

(ii) Associative Property :

Changing the grouping of the addends does not change the sum.

If a, b and c are any three numbers, then

a + (b + c)  =  (a + b) + c

Example :

2 + (4 + 7)  =  2 + 11  =  13 

(2 + 4) + 7  =  6 + 7  =  13 

So,

2 + (4 + 7)  =  (2 + 4) + 7

(iii) Additive Identity :

The sum of any number and zero is the number itself.

If a is any number, then

a + 0  =  0 + a  =  a

So, zero is the additive identity.

Example : 

2 + 0  =  0 + 2  =  2

(iv) Additive Inverse :

- k is the negative or additive inverse of k.

If k is a number, then there exists a number -k such that

k + (-k)  =  (-k) + k  =  0

Examples : 

Additive inverse of 3 is -3.

Additive inverse of -5 is 5.

Additive inverse of 0 is 0 itself. 

Properties of Multiplication

(i) Commutative Property :

Changing the order of factors does not change the product.

If a and b are any two numbers, then

a x b  =  b x a

Example : 

5 x 2  =  10

2 x 5  =  10

So, 

5 x 2  =  2 x 5

Therefore, Commutative property is true for multiplication.

(ii) Associative Property :

Changing the grouping of the factors does not change the product.

If a, b and c are any three numbers, then

a x (b x c)  =  (a x b) x c

Example :

5 x (2 x 3)  =  5 x 6  =  30 

(5 x 2) x 3  =  10 x 3  =  30

So,

5 x (2 x 3)  =  (5 x 2) x 3

Therefore, Associative property is true for multiplication.

(iii) Multiplicative Identity :

The product of any number and 1 is the number itself. ‘One’ is the multiplicative identity for numbers.

If k is any number, then

k x 1 = 1 x k  =  k

Example : 

5 x 1  =  1 x 5  =  5

(iv) Multiplication by 0 :

Every number multiplied by 0 gives the result 0.

If k is any number, then

k x 0  =  0 x k  =  0

Example : 

7 x 0  =  0 x 7  =  0

(v) Multiplicative Inverse or Reciprocal :

For every number k, there exists a  number 1/k such that

k x 1/k  =  1

Then,

k and 1/k are multiplicative inverse of each other

That is,

k is the multiplicative inverse of 1/k

1/k is the multiplicative inverse of k

Examples : 

The multiplicative inverse of 2 is 1/2.

The multiplicative inverse of 1/3 is 3.

The multiplicative inverse 1 is 1.

The multiplicative inverse of 0 is undefined.

Distributive Property

(i) Distributive Property of Multiplication over Addition :

Multiplication of numbers is distributive over addition.

If a, b and c  are any three numbers, then

a x (b + c)  =  ab + ac

Example :

3 x (2 + 5)  =  3 x 7  =  21 -----(1)

3 x (2 + 5)  =  3x2 + 3x5  =  6 + 15  =  21 -----(2)

From (1) and (2), 

3 x (2 + 5)  =  3x2 + 3x5

Therefore, Multiplication of numbers is distributive over addition.

(ii) Distributive Property of Multiplication over Subtraction :

Multiplication of numbers is distributive over subtraction.

If a, b and c  are any three numbers, then

a x (b - c)  =  ab - ac

Example :

3 x (7 - 2)  =  3 x 5  =  15 -----(1)

3 x (7 - 2)  =  3x7 - 3x2  =  21 - 6  =  15 -----(2)

From (1) and (2), 

3 x (7 - 2)  =  3x7 - 3x2

Therefore, Multiplication of numbers is distributive over subtraction. 

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