Properties of Whole Numbers

The set of whole numbers consists of all the natural numbers, including zero. Whole numbers are a subset of real numbers, which includes only positive integers and zero. Whole numbers do not include negative numbers, fractions, or decimals. Whole numbers are represented by W. Whole numbers begin with zero and continue with 1, 2, 3, 4, and so on. Since whole numbers start from 0, it is the smallest whole number. Arithmetic operations like addition, subtraction, multiplication, and division can be performed on whole numbers.

The basic arithmetic operations are applicable on whole numbers, resulting in five main properties: closure property, commutative property, associative property, and distributive property.

Closure Property

The Closure property of numbers states that when you add or multiply any two whole numbers, the result will always be a whole number.

For example, 4 × 6 = 24

When we multiply 4 and 6, the product is 24, which is also a whole number. 

The closure property is not applicable to the subtraction and division of whole numbers.

Commutative Property of Addition and Multiplication

The commutative property says that the sum and product of whole numbers will not change even if you change the order of the numbers. 

For example, consider that ‘a’  and ‘b’ are two whole numbers. According to this property 
 

a + b = b + a 
a × b = b × a

Example: Consider a = 13 and b = 2

13 + 2 = 2 + 13
13 × 2 = 2 × 13

 
Associative Property of Addition and Multiplication

The associative property refers to the grouping of three or more whole numbers in addition or multiplication without changing the result.
 

For example, consider a, b, c are three whole numbers. According to the associative property:

a + (b + c) = (a + b) + c 
a × (b × c) = (a × b) × c 
 
Example: For Addition

4 + (3 + 5) = (4 + 3) + 5
4 + 8 = 7 + 5
12 = 12
 

For Multiplication

2 × (3 × 4) = (2 × 3) × 4
2 × 12 = 6 × 4 
24 = 24 

Distributive Property of Multiplication Over Addition

It states that when you multiply a number by a sum, it is the same as multiplying that same number by each part of the sum separately. It can be written as :

a × (b + c) = (a × b)  + (a × c)
 

For example, a = 3, b = 4, c = 5

3 × (4 + 5) = (3 × 4)  + (3 × 5)
3 × 9 = 12 + 15
27 = 27

Identity Property
 

• Additive Identity: 
  
  The property states that when you add 0 to any whole number, the answer will be the whole number itself. 
  For example, consider ‘a’ as a whole number 
  a + 0 = a
   
• Multiplicative Identity:
  
  This property states that when we multiply a whole number by 1 then it results in the whole number itself. 
  For example, consider ‘b’ as a whole number
  1 × b = b

Here are some real-life examples where we use properties of whole numbers 
 

• Closure property is used in Banking Transactions 
  
  This states that when we add or multiply two whole numbers, the result is always a whole number. This is used in the banking sector. When depositing and withdrawing money from any account, these transactions involve whole numbers. So, banks use this property to keep track of deposits and withdrawals.
   
• Shopping and Budgeting
  
  During shopping, we use the commutative property of addition and multiplication to simplify the calculations. For example, a person purchasing two items costs ₹10 and ₹20 each. To find the total bill, the person uses the commutative property. The total bill will be ₹30.
   
• Shopping Discounts 
  
  In this particular application, we use the distributive property, which states that multiplication distributes over addition. This is commonly used for calculating discounts that a particular store is providing. For example, if a store offers a 20% discount on ₹50 and ₹30 items, you can first add the prices:  (₹50 +  ₹30) and find 20% of ₹80. This makes it easier to find the total discount.