Properties of Whole Numbers with FAQs

Properties of Whole Numbers

Whole numbers include the set of natural numbers along with the number 0, i.e., the set of whole numbers is 0, 1, 2, 3, 4, 5,6, 7,  … till infinity. The properties of whole numbers are described below with examples:

1. Closure Property of Whole Numbers

This property states that when mathematical operations like addition and multiplication are applied on any two whole numbers then the result is also a whole number.

For two Whole Numbers 4 and 7:

Closure under Addition
4 + 7 = 11, 11 is a whole number.

Hence, we say that whole numbers are closed under addition.

Therefore, if a and b are whole numbers \Rightarrow (a+b) is also a whole number.

Closure under Multiplication

4 \times 7=28, 28 is a whole number.

Hence, we say that whole numbers are closed under multiplication.

Therefore, if a and b are whole numbers \Rightarrow(\mathbf{a} \times \mathbf{b}) is a whole number.

Note:

Do you know why Subtraction and Division are not under Closure property?

Subtraction is not under closure property because for some numbers the subtraction result is not a whole number, for example, for two whole numbers 10 and 25:

10 – 25 = -15,  and ‘-15’ is not a whole number.

The division is not under closure property because division by zero is not defined and for some numbers division result is not a whole number, for example, for two whole numbers 2 and 6,

2 ÷ 6 = 0.333, which is not a whole number. 

2. Commutative Property 

This property states that when two whole numbers are added or multiplied in any order the outcome of the operation is equal.

For two whole numbers 11 and 17:

Commutative under Addition

11 + 17 = 28

17 + 11 = 28

Hence we say that whole numbers are commutative under addition.

Therefore, if a and b are whole numbers \Rightarrow(a+b)=(b+a). 

Commutative under Multiplication

11 \times 17=187
17 \times 11=187

Hence we say that whole numbers are commutative under multiplication.

Therefore, if a and b are whole numbers \Rightarrow(a \times b)=(b \times a).

3. Associative Property

This property states that when any three whole numbers are added or multiplied by grouping in any manner the outcome is equal.

For three whole numbers 6, 10 and 15:

Associative under Addition

(6 + 10) + 15 = 16 + 15 = 31

6 + (10 + 15) = 6 + 25 = 31

Hence we say that whole numbers are associative under addition.

Therefore, if a, b and c are whole numbers \Rightarrow(\mathbf{a}+\mathbf{b})+\mathbf{c}=\mathbf{a}+(\mathbf{b}+\mathbf{c}). 

Associative under Multiplication

(6 \times 10) \times 15=60 \times 15=900
6 \times(10 \times 15)=6 \times 150=900

Hence we say that whole numbers are associative under multiplication.

Therefore if a, b and c are whole numbers \Rightarrow(a \times b) \times c=a \times(b \times c).

4. Distributive Property

This property states that if a, b and c are whole numbers then:

Distributive Property of Multiplication under Addition

Distributive Property of Multiplication under Subtraction

Note: Distributive property of multiplication under subtraction holds true only if b \geq c.

For three whole numbers 3, 6 and 8:

3 \times(6+8)=3 \times 14=42
(3 \times 6)+(3 \times 8)=18+24=42
\therefore 3 \times(6+8)=(3 \times 6)+(3 \times 8)

Hence the distributive property of multiplication under addition holds true.

Similarly for three whole numbers 4, 11 and 5:

4 \times(11-5)=4 \times 6=24
(4 \times 11)-(4 \times 5)=44-20=24
\therefore 4 \times(11-5)=(4 \times 11)-(4 \times 5)

Hence the distributive property of multiplication under subtraction holds true.

5. Identity Property

For any whole number that exists the additive identity is 0 and the multiplicative identity is 1. When 0 is added to a whole number the result is the number itself and when 1 is multiplied to a whole number again the result is the number itself. 

For the whole number 11,

   11 + 0 = 11 

⇒ ‘0’ is the additive identity.

     11 × 1 = 11 

⇒ ‘1’ is the multiplicative identity.