Whole Numbers

Whole Numbers – CBSE Class 6 Maths Notes: Complete Guide with Examples & Practice Problems

Introduction to Whole Numbers

Whole numbers form the foundation of mathematics and are among the first number systems students encounter in their mathematical journey. Understanding whole numbers is essential for building strong arithmetic skills and progressing to more advanced mathematical concepts.

In CBSE Class 6 Mathematics, the chapter on Whole Numbers introduces students to fundamental concepts that are crucial for everyday calculations, problem-solving, and logical reasoning. This comprehensive guide covers everything you need to know about whole numbers, from basic definitions to advanced properties and real-world applications.

What Are Whole Numbers? 

Whole numbers are the set of numbers that include zero (0) and all positive counting numbers (1, 2, 3, 4, 5, ...). They can be represented mathematically as:

W = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...}

Whole numbers are also called non-negative integers because they include zero and all positive integers, but exclude negative numbers and fractions.

Core Points of Whole Numbers

• The smallest whole number is 0
• There is no largest whole number (whole numbers extend to infinity)
• Whole numbers do not include negative numbers
• Whole numbers do not include fractions or decimals
• Every natural number is a whole number, but not vice versa

Whole Numbers Definition and Examples

Formal Definition

A whole number is any number belonging to the set {0, 1, 2, 3, 4, 5, ...}, which represents zero along with all natural numbers. These numbers are used for counting discrete objects and representing quantities that cannot be divided into smaller parts.

Examples of Whole Numbers

Category	Examples
Single-digit whole numbers	0, 1, 2, 3, 4, 5, 6, 7, 8, 9
Two-digit whole numbers	10, 25, 47, 99
Three-digit whole numbers	100, 256, 999
Large whole numbers	1000, 50000, 999999

What Are NOT Whole Numbers?

Type	Examples	Reason
Negative numbers	-1, -5, -100	Whole numbers cannot be negative
Fractions	½, ¾, 2/3	Whole numbers must be complete
Decimals	0.5, 3.14, 2.7	Whole numbers have no decimal parts
Mixed numbers	1½, 2¾	Contains fractional component

Whole Numbers vs Natural Numbers

Understanding the difference between whole numbers and natural numbers is fundamental in mathematics.

Natural Numbers

Natural numbers are the counting numbers starting from 1. They are represented as:

N = {1, 2, 3, 4, 5, 6, ...}

Differences

Feature	Natural Numbers	Whole Numbers
Starting number	1	0
Includes zero	No	Yes
Representation	N = {1, 2, 3, ...}	W = {0, 1, 2, 3, ...}
Alternative name	Counting numbers	Non-negative integers
Smallest number	1	0

Important Relationship

• Every natural number is a whole number, but not every whole number is a natural number
• The only whole number that is NOT a natural number is 0
• Both sets extend infinitely in the positive direction

Whole Numbers vs Integers

While whole numbers include zero and positive numbers only, integers extend this concept to include negative numbers as well.

Integers Definition

Integers include all whole numbers AND their negative counterparts:

Z = {..., -3, -2, -1, 0, 1, 2, 3, ...}

Comparison Table

Feature	Whole Numbers	Integers
Includes zero	Yes	Yes
Includes positive numbers	Yes	Yes
Includes negative numbers	No	Yes
Set representation	W = {0, 1, 2, 3, ...}	Z = {..., -2, -1, 0, 1, 2, ...}
Extends in both directions	No (only positive)	Yes

Venn Diagram Relationship

Integers
├── Negative integers: ..., -3, -2, -1
└── Whole Numbers
 ├── Zero: 0
 └── Natural Numbers: 1, 2, 3, ...

Face Value and Place Value of Whole Numbers

Understanding face value and place value is essential for working with multi-digit whole numbers.

Face Value

The face value of a digit is the digit itself, regardless of its position in the number.

Place Value

The place value of a digit depends on its position in the number.

Example: Number 12345

Digit	Position	Face Value	Place Value Calculation	Place Value
5	Unit's place	5	5 × 1	5
4	Ten's place	4	4 × 10	40
3	Hundred's place	3	3 × 100	300
2	Thousand's place	2	2 × 1000	2000
1	Ten thousand's place	1	1 × 10000	10000

Properties of Whole Numbers

Whole numbers exhibit several important mathematical properties that make calculations easier and more predictable.

1. Closure Property

Addition: The sum of any two whole numbers is always a whole number.

• Example: 5 + 6 = 11 ✓ (whole number)

Multiplication: The product of any two whole numbers is always a whole number.

• Example: 5 × 6 = 30 ✓ (whole number)

Subtraction: NOT closed – the difference may not be a whole number.

• Example: 5 - 6 = -1 ✗ (not a whole number)

Division: NOT closed – the quotient may not be a whole number.

• Example: 5 ÷ 6 = 5/6 ✗ (not a whole number)

2. Commutative Property

The order of numbers does not affect the result in addition and multiplication.

Addition: a + b = b + a

• Example: 5 + 6 = 6 + 5 = 11

Multiplication: a × b = b × a

• Example: 5 × 6 = 6 × 5 = 30

Note: Subtraction and division are NOT commutative.

• 5 - 6 ≠ 6 - 5
• 5 ÷ 6 ≠ 6 ÷ 5

3. Associative Property

When adding or multiplying three or more numbers, the grouping does not affect the result.

Addition: (a + b) + c = a + (b + c)

• Example: (2 + 3) + 4 = 2 + (3 + 4) = 9

Multiplication: (a × b) × c = a × (b × c)

• Example: (2 × 3) × 4 = 2 × (3 × 4) = 24

4. Distributive Property

Multiplication distributes over addition and subtraction.

Over Addition: a × (b + c) = (a × b) + (a × c)

• Example: 2 × (3 + 4) = (2 × 3) + (2 × 4) = 6 + 8 = 14

Over Subtraction: a × (b - c) = (a × b) - (a × c)

• Example: 5 × (100 - 1) = (5 × 100) - (5 × 1) = 500 - 5 = 495

5. Identity Properties

Additive Identity: Zero (0) is the additive identity.

• a + 0 = a
• Example: 7 + 0 = 7

Multiplicative Identity: One (1) is the multiplicative identity.

• a × 1 = a
• Example: 7 × 1 = 7

Formulas for Whole Numbers

Formula Name	Mathematical Representation	Explanation
Closure (Addition)	a + b ∈ W, where a, b ∈ W	Sum of whole numbers is a whole number
Closure (Multiplication)	a × b ∈ W, where a, b ∈ W	Product of whole numbers is a whole number
Commutative (Addition)	a + b = b + a	Order doesn't matter in addition
Commutative (Multiplication)	a × b = b × a	Order doesn't matter in multiplication
Associative (Addition)	(a + b) + c = a + (b + c)	Grouping doesn't matter in addition
Associative (Multiplication)	(a × b) × c = a × (b × c)	Grouping doesn't matter in multiplication
Distributive Property	a × (b + c) = ab + ac	Multiplication distributes over addition
Additive Identity	a + 0 = a	Adding 0 gives the same number
Multiplicative Identity	a × 1 = a	Multiplying by 1 gives the same number
Successor	n + 1	Next whole number after n
Predecessor	n - 1	Previous whole number before n

How to Add, Subtract, Multiply, and Divide Whole Numbers

Addition of Whole Numbers

Addition combines two or more whole numbers to find their total.

Rules:

• Always results in a whole number (closure property)
• Order doesn't matter (commutative property)
• Can be grouped in any way (associative property)

Examples:

• 23 + 45 = 68
• 0 + 99 = 99 (adding zero)
• 156 + 234 + 110 = 500

Subtraction of Whole Numbers

Subtraction finds the difference between two whole numbers.

Rules:

• May NOT result in a whole number if the subtrahend is larger
• Order matters (not commutative)
• Use regrouping (borrowing) when needed

Examples:

• 45 - 23 = 22 ✓ (whole number)
• 23 - 45 = -22 ✗ (not a whole number)
• 100 - 0 = 100

Multiplication of Whole Numbers

Multiplication is repeated addition.

Rules:

• Always results in a whole number
• Order doesn't matter
• Multiplying by 0 gives 0
• Multiplying by 1 gives the same number

Examples:

• 12 × 8 = 96
• 25 × 0 = 0
• 99 × 1 = 99

Mental Math Trick using Distributive Property:

• 15 × 99 = 15 × (100 - 1) = 1500 - 15 = 1485
• 39 × 41 = (40 - 1)(40 + 1) = 1600 - 1 = 1599

Division of Whole Numbers

Division splits a number into equal parts.

Rules:

• May NOT result in a whole number
• Division by zero is undefined
• Order matters (not commutative)

Division Algorithm:Dividend = Divisor × Quotient + Remainder

Examples:

• 24 ÷ 6 = 4 ✓ (whole number)
• 25 ÷ 6 = 4 remainder 1 (or 4.166...)
• 0 ÷ 5 = 0
• 5 ÷ 0 = undefined

Successor and Predecessor

Successor

The successor of a whole number is the number that comes immediately after it.

Formula: Successor of n = n + 1

Examples:

• Successor of 99 = 100
• Successor of 999 = 1000
• Successor of 0 = 1

Predecessor

The predecessor of a whole number is the number that comes immediately before it.

Formula: Predecessor of n = n - 1

Examples:

• Predecessor of 100 = 99
• Predecessor of 1000 = 999
• Predecessor of 1 = 0

Important Note: The whole number 0 has no predecessor as a whole number because 0 - 1 = -1, which is not a whole number.

Difference Between Successor and Predecessor

For any natural number N:

• Successor = N + 1
• Predecessor = N - 1
• Difference = (N + 1) - (N - 1) = 2

Real-World Uses of Whole Numbers in Coding and Counting

Whole numbers are ubiquitous in daily life and technology. Here are some practical applications:

In Everyday Counting

• Counting students in a classroom: 25 students, 30 students
• Counting money: ₹100, ₹500, ₹2000 notes
• House numbers: 101, 102, 103
• Page numbers in books: 1, 2, 3, ... 500

In Computer Science and Coding

Array Indexing: Many programming languages use whole numbers (starting from 0 or 1) to index arrays.

Array[0], Array[1], Array[2], ...

Loop Counters: Whole numbers serve as counters in for loops and while loops.

for i = 0 to 10:
 print(i)

Memory Addresses: Computer memory locations are identified by whole number addresses.

Binary Numbers: Digital systems use whole numbers in binary (0s and 1s) for all computations.

In Science and Measurement

• Counting atoms in a molecule
• Population statistics
• Number of chromosomes (46 in humans)
• Discrete quantities (number of books, cars, people)

In Business and Finance

• Inventory counts
• Employee headcounts
• Transaction IDs
• Order numbers

In Sports and Games

• Scores and points
• Player jersey numbers
• Ranking positions
• Game levels

Practice Problems to Identify Whole Numbers

Problem Set 1: Identification

Question: Identify which of the following are whole numbers: -5, 0, 3.5, 7, -12, 100, ¾, 25, 0.001, 999

Answer: 0, 7, 100, 25, and 999 are whole numbers.

Problem Set 2: True or False

1. Every whole number has a successor. TRUE
2. Every whole number has a predecessor. FALSE (0 has no whole number predecessor)
3. 0 is the smallest whole number. TRUE
4. The largest whole number is 999999. FALSE (whole numbers are infinite)
5. All natural numbers are whole numbers. TRUE

Problem Set 3: Calculation Using Properties

Q1: Calculate 53 × 99 using the distributive property.

Solution: 53 × 99 = 53 × (100 - 1) = 53 × 100 - 53 × 1 = 5300 - 53 = 5247

Q2: Find the value of 25 × 37 × 4 using suitable rearrangement.

Solution: 25 × 37 × 4 = (25 × 4) × 37 = 100 × 37 = 3700

Q3: Verify: Is 2 + (3 + 4) = (2 + 3) + 4?

Solution: LHS = 2 + (3 + 4) = 2 + 7 = 9 RHS = (2 + 3) + 4 = 5 + 4 = 9 LHS = RHS ✓ (Associative property verified)

Problem Set 4: Word Problems

Q1: Sunil purchased a pen worth ₹59. If Anil purchased 99 such pens, find the amount paid by Anil.

Solution: Amount = 59 × 99 = 59 × (100 - 1) = 5900 - 59 = ₹5841

Q2: The largest 4-digit number is multiplied by 3. Find the product.

Solution: Largest 4-digit number = 9999 Product = 9999 × 3 = (10000 - 1) × 3 = 30000 - 3 = 29997

Solved Examples from CBSE Pattern

Example 1: Finding the Largest Two-Digit Number with a Condition

Problem: Find the largest two-digit number such that when the ten's digit is reduced by 1, it becomes twice the unit's digit.

Solution: Let the two-digit number be xy (where x is ten's digit and y is unit's digit)

Given: x - 1 = 2y Therefore: x = 2y + 1

Testing values:

• If y = 1 → x = 3 → Number = 31
• If y = 2 → x = 5 → Number = 52
• If y = 3 → x = 7 → Number = 73
• If y = 4 → x = 9 → Number = 94

The largest such number is 94.

Example 2: Division and Quotient Difference

Problem: Find the difference between the quotients when 41325 and 41333 are divided by 4.

Solution: Using dividend = divisor × quotient + remainder:

• 41325 = 4 × k₁ + 1 (since 41325 ÷ 4 leaves remainder 1)
• 41333 = 4 × k₂ + 1 (since 41333 ÷ 4 leaves remainder 1)

Subtracting: 41333 - 41325 = 4(k₂ - k₁) 8 = 4(k₂ - k₁) k₂ - k₁ = 2

Example 3: Non-Associativity of Division

Problem: Find P - Q given that P = 15 ÷ (4 ÷ 5) and Q = (15 ÷ 4) ÷ 5

Solution: P = 15 ÷ (4/5) = 15 × (5/4) = 75/4

Q = (15/4) ÷ 5 = 15/(4 × 5) = 15/20 = 3/4

P - Q = 75/4 - 3/4 = 72/4 = 18

This demonstrates that division is NOT associative.

Chapter Summary

Whole numbers are fundamental to mathematics and form the building blocks for all numerical operations. Starting from 0 and extending to infinity, they include all non-negative integers. Understanding their properties closure, commutativity, associativity, distributivity, and identity enables students to perform calculations efficiently and solve complex problems.

The distinction between whole numbers, natural numbers, and integers is crucial for mathematical precision. While natural numbers begin at 1, whole numbers include 0, and integers extend to include negative numbers. These concepts interconnect and build upon each other as students progress through mathematics.

Mastering whole numbers prepares students for more advanced topics including fractions, decimals, algebra, and beyond. The properties learned here especially the distributive property become essential tools for mental math and algebraic manipulation throughout one's mathematical journey.

Quick Revision Points

• Whole Numbers: W = {0, 1, 2, 3, 4, 5, ...}
• Smallest whole number: 0
• Largest whole number: None (infinite)
• 0 is NOT a natural number but IS a whole number
• Closed operations: Addition, Multiplication
• NOT closed operations: Subtraction, Division
• Additive identity: 0
• Multiplicative identity: 1
• Successor formula: n + 1
• Predecessor formula: n - 1
• 0 has no predecessor in whole numbers
• Distributive Property: Use for mental math shortcuts

"This comprehensive guide on Whole Numbers is designed for CBSE Class 6 students following the NCERT curriculum. For more CBSE Maths notes and practice questions, explore our complete collection of study materials."