PROPERTIES OF WHOLE NUMBERS

In math, whole numbers are the numbers which are positive integers including zero.

They are, 

0, 1, 2, 3, 4, 5, 6, ........

There are some properties of whole 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) Closure Property : 

The sum of any two whole numbers is always a whole number. This is called ‘Closure property of addition’ of whole numbers. Thus, W is closed under addition

If a and b are any two whole numbers, then (a + b) is also a whole number. 

Example : 

2 + 4 = 6 is a whole number. 

(ii) Commutative Property : 

Addition of two whole numbers is commutative.

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

a + b  =  b + a

Example : 

2 + 4 = 6 

4 + 2 = 6 

Hence, 2 + 4 = 4 + 2.

(iii) Associative Property :

Addition of whole numbers is associative.

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

a + (b + c) = (a + b) + c

Example :

2 + (4 + 1) = 2 + (5) = 7 

(2 + 4) + 1 = (6) + 1 = 7

Hence, 2 + (4 + 1) = (2 + 4) + 1.

(iv) Additive Identity :

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

If a is any whole number, then

a + 0 = 0 + a = a

Zero is the additive identity for whole numbers.

Example : 

7 + 0 = 0 + 7 = 7

Subtraction

(i) Closure Property : 

The difference between any two whole numbers need not be a whole number.

Hence W is not closed under subtraction.

Example : 

2 - 5 = -3 is a not whole number. 

(ii) Commutative Property : 

Subtraction of two whole numbers is not commutative.

If a and b are any two whole numbers, then

(a - b)  (b - a)

Example : 

5 - 2 = 3

2 - 5 = -3

Hence, 5 - 2  2 - 5.

Therefore, Commutative property is not true for subtraction.

(iii) Associative Property :

Subtraction of whole numbers is not associative.

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

a - (b - d)  (a - b) - d

Example :

2 - (4 - 1) = 2 - 3 = -1 

(2 - 4) - 1 = -2 - 1 = -3 

Hence, 2 - (4 - 1)  (2 - 4) - 1.

Therefore, associative property is not true for subtraction.

Multiplication

(i) Closure Property :

The product of two whole numbers is always a whole number. Hence W is closed under multiplication.

If a and b are any two whole numbers, then

a x b = ab is also a whole number

Example : 

5 x 2 = 10 is a whole number

(ii) Commutative Property :

Multiplication of whole numbers is commutative.

If a and b are any two whole numbers, then

a x b = b x a

5 x 9 = 45

9 x 5 = 45

Hence, 5 x 9 = 9 x 5.

Therefore, commutative property is true for multiplication.

(iii) Associative Property :

Multiplication of whole numbers is associative.

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

a x (b x c) = (a x b) x c

Example :

2 x (4 x 5) = 2 x 20 = 40 

(2 x 4) x 5 = 8 x 5 = 40 

Hence, 2 x (4 x 5) = (2 x 4) x 5.

Therefore, Associative property is true for multiplication.

(iv) Multiplicative Identity :

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

If a is any whole number, then

a x 1 = 1 x a = a

Example : 

5 x 1 = 1 x 5 = 5

(v) Multiplication by 0 :

Every whole number multiplied with 0 gives 0.

If a is any whole number, then

a x 0 = 0 x a = 0

Example : 

5 x 0 = 0 x 5 = 0

Division

(i) Closure Property :

When we divide of a whole number by another whole number, the result does not need to be a whole number. 

Hence, W is not closed under multiplication. 

Example : 

When we divide the whole number 3 by another whole number 2, we get 1.5 which is not a whole number. 

(ii) Commutative Property :

Division of whole numbers is not commutative.

If a and b are two whole numbers, then

÷ b  ≠ b ÷ a

Example :

÷ 1 = 2

÷ 2 = 1.5

Hence, ÷ 1  1 ÷ 2.

Therefore, Commutative property is not true for division.

(iii) Associative Property :

Division of whole numbers is not associative.

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

÷ (b ÷ c)  (a ÷ b) ÷ c

Example :

÷ (4 ÷ 2) = 3 ÷ 2  =  1.5

(3 ÷ 4) ÷ 2 = 0.75 ÷ 2  =  0.375

Hence, ÷ (4 ÷ 2)  (÷ 4) ÷ 2

Therefore, Associative property is not true for division.

Distributive Property

(i) Distributive Property of Multiplication over Addition :

Multiplication of whole numbers is distributive over addition.

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

a x (b + c) = ab + ac

Example :

2 x (3 + 4) = (2 x 3) + (2 x 4) = 6 + 8 = 14

2 x (3 + 4) = 2 x (7) = 14

Hence, 2 x (3 + 4) = (2 x 3) + (2 x 4).

Therefore, Multiplication is distributive over addition.

(ii) Distributive Property of Multiplication over Subtraction :

Multiplication of whole numbers is distributive over subtraction.

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

a x (b - c) = ab - ac

Example :

2 x (4 - 1) = (2 x 4) - (2 x 1) = 8 - 2 = 6

2 x (4 - 1) = 2 x (3) = 6

Hence, 2 x (4 - 1) = (2 x 4) - (2 x 1).

Therefore, Multiplication is distributive over subtraction.

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