Properties of Whole Numbers

The set of whole numbers includes all the natural numbers as well as zero. Whole numbers are a subset of the real numbers, consisting of only positive integers and zero. They do not include negative numbers, fractions, or decimals. W. Whole numbers represent the set of whole numbers start at zero and continue as 1, 2, 3, 4, and so on, with 0 being the smallest whole number. Arithmetic operations such as addition, subtraction, multiplication, and division can all be performed with whole numbers.

Examples of Whole Numbers:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, and so on...
 

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 tips and tricks for remembering the properties of whole numbers that can help make math easier and more intuitive:
 

• Commutative property of addition: The order in which you add whole numbers doesn't affect the result, e.g.,\( 3+5=5+3\).
   
• Commutative property of multiplication: The order of multiplication doesn't change the product, e.g.,\( 4×6=6×4.\)
   
• Associative property of addition: You can group numbers in any way when adding, and the sum stays the same, e.g., \((2+3)+4=2+(3+4).\)
   
• Associative property of multiplication: Grouping numbers differently when multiplying doesn't affect the product, e.g.,\((2×3)×4=2×(3×4).\)
   
• Identity property of addition: Adding zero to any number leaves the number unchanged; e.g., \(5 + 0 = 5.\)

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. 

• Organizing or Counting Items: When grouping items say, pencils in boxes it doesn’t matter how you group them: (2 + 3) + 4 = 2 + (3 + 4). The associative property makes counting and inventory management faster and more reliable.

• Attendance and Records: When two whole numbers (like number of students in two classes) are added or multiplied, the result is always a whole number. This closure property ensures consistency in data like attendance, production counts, or survey results.