PROPERTIES OF NUMBERS

There are some properties of numbers like closure property, commutative property and associative property. 

Let us explore these properties on the four binary operations (Addition, subtraction, multiplication and division) in mathematics.

Addition

(i) Commutative Property : 

Addition of two numbers is commutative.

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 :

Addition of numbers is associative.

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/b is any rational number, then

a + 0  =  0 + a  =  a

Zero is the additive identity for rational numbers.

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. 

Subtraction

(i) Commutative Property : 

Subtraction of two numbers is not commutative.

If a and b are any two numbers, then

a - b    b - c 

Example : 

9 - 2  =  7

2 - 9  =  -7 

And,

9 - 2  ≠  2 - 9

Therefore, Commutative property is not true for subtraction of numbers.

(ii) Associative Property :

Subtraction of numbers is not associative.

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

a - (b - c)    (a - b) - c

Example :

9 - (5 - 2)  =  9 - 3  =  6 

(9 - 5) - 2  =  4 - 2  =  2 

And,

9 - (5 - 2)  ≠  (9 - 5) - 2

Therefore, Associative property is not true for subtraction of numbers.

Multiplication

(i) Commutative Property :

Multiplication of numbers is commutative.

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 :

Multiplication of numbers is associative.

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.

Division

(i) Commutative Property :

Division of numbers is not commutative.

If a and b are two numbers, then

÷ b  ≠  b ÷ a

Example :

÷ 3  =  2

÷ 6  =  1/2

So, 

÷ 3    3 ÷ 6

Therefore, Commutative property is not true for division of numbers.

(ii) Associative Property :

Division of numbers is not associative.

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

÷ (c/÷ e/f)    (a/b ÷ c/d) ÷ e/f

Example :

÷ (4 ÷ 2)  =  8 ÷ 2  =  4

(8 ÷ 4) ÷ 2  =  4 ÷ 2  =  2

So,

÷ (4 ÷ 2)  ≠  (8 ÷ 4) ÷ 2

Therefore, Associative property is not true for division of numbers.

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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