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Select Medium / माध्यम चुनें:
Eng (English) Hindi (हिन्दी)
JAC • Class 6 • Mathematics • Ch 2
Estimated Time: 45 Mins
Study Progress: In Progress

Whole Numbers

In JAC Board Class 6 Mathematics, "Whole Numbers" introduces the collection $W = \{0, 1, 2, 3, \dots\}$, natural numbers, predecessor and successor, representation and arithmetic on the number line, fundamental properties of operations (Closure, Commutativity, Associativity, Distributivity over addition, and Additive/Multiplicative Identity), patterns in numbers, and the mathematical undefined nature of division by zero.

View All Chapters JCERT Hindi Textbook
Overview Core Concepts Formulas & Identities Solved Examples Common Traps Summary Self Check Take MCQ Test

⭕ Have You Ever Wondered?

Why does dividing 12 by 3 equal 4, but dividing 12 by 0 breaks all math calculators and is called "undefined"?

Zero is one of humanity's greatest intellectual inventions, pioneered in ancient India by Brahmagupta! Explore the complete realm of Whole Numbers and discover the arithmetic laws that keep mathematics consistent.

Why This Chapter Matters

In JAC Board Class 6 Mathematics, "Whole Numbers" introduces the collection $W = \{0, 1, 2, 3, \dots\}$, natural numbers, predecessor and successor, representation and arithmetic on the number line, fundamental properties of operations (Closure, Commutativity, Associativity, Distributivity over addition, and Additive/Multiplicative Identity), patterns in numbers, and the mathematical undefined nature of division by zero.

Before You Begin (Prerequisites)

  • Natural numbers 1, 2, 3...
  • Counting objects
  • Basic addition and multiplication

What You Will Learn (Core Objectives)

  • Differentiate Natural Numbers $\mathbb{N}$ from Whole Numbers $\mathbb{W}$ by the inclusion of $0$.
  • Find predecessor ($n - 1$) and successor ($n + 1$) for any whole number.
  • Perform addition, subtraction, and multiplication visually on the number line.
  • Verify Closure, Commutative, Associative, and Distributive properties for whole numbers.
  • Identify Additive Identity ($0$) and Multiplicative Identity ($1$).
  • Explain why division by zero is mathematically undefined.

Chapter Roadmap & Progression

1 Syllabus and book alignment
2 1. Natural Numbers, Whole Numbers &...
3 2. Properties of Operations on Whol...
4 3. Division by Zero: Why is it Unde...

Complete Concept Guide (100% Curriculum Coverage)

Syllabus and book alignment

JCERT 2026–27: Regular schools (Non-SOE) — Chapter 2 · Whole Numbers. Syllabus pages 17–41. These notes follow this topic. Official syllabus. Official JCERT Hindi textbook. Some source PDFs are damaged. The supplementary reference uses different chapter numbering.

1. Natural Numbers, Whole Numbers & The Number Line

Natural Numbers (Counting Numbers): $\mathbb{N} = \{1, 2, 3, 4, 5, \dots\}$. There is no greatest natural number.

Whole Numbers: The collection of natural numbers along with zero: $\mathbb{W} = \{0, 1, 2, 3, 4, \dots\}$. Every natural number is a whole number, but $0$ is a whole number that is not a natural number.

  • Predecessor: The number just before a given number: $\text{Predecessor of } n = n - 1$. (Note: In whole numbers, $0$ has no predecessor!).
  • Successor: The number just after: $\text{Successor of } n = n + 1$.

Arithmetic on the Number Line

  • Addition: Moving rightwards on the number line. (e.g. $3 + 4$: start at 3, take 4 steps to the right $\to 7$).
  • Subtraction: Moving leftwards. (e.g. $7 - 4$: start at 7, take 4 steps to the left $\to 3$).
  • Multiplication: Repeated steps of equal length to the right. (e.g. $3 \times 2$: start at 0, take 3 steps of 2 units $\to 6$).

2. Properties of Operations on Whole Numbers

PropertyAdditionMultiplicationSubtraction & Division
Closure$a + b \in \mathbb{W}$ (Closed)$a \times b \in \mathbb{W}$ (Closed)NOT closed ($3 - 5 \notin \mathbb{W}$, $5 \div 2 \notin \mathbb{W}$)
Commutative$a + b = b + a$$a \times b = b \times a$NOT commutative ($a - b \ne b - a$)
Associative$(a + b) + c = a + (b + c)$$(a \times b) \times c = a \times (b \times c)$NOT associative
DistributiveDistributivity of multiplication over addition:
$a \times (b + c) = (a \times b) + (a \times c)$		Simplifies mental arithmetic significantly!
Identity Element$a + 0 = a$ ($0$ is Additive Identity)$a \times 1 = a$ ($1$ is Multiplicative Identity)$a \times 0 = 0$

3. Division by Zero: Why is it Undefined?

Division is mathematically defined as repeated subtraction:

  • $12 \div 3 = 4$ because $12 - 3 - 3 - 3 - 3 = 0$ (takes 4 subtractions).
  • Now consider $12 \div 0$: $12 - 0 = 12, \; 12 - 0 = 12, \; 12 - 0 = 12 \dots$ We can subtract zero infinitely many times and the remainder never reaches zero!
  • Alternatively, if $a \div 0 = x$, then by definition $x \times 0 = a$. But any number multiplied by 0 is 0! No number $x$ can satisfy $x \times 0 = 12$.
  • Therefore, division by zero is undefined in mathematics.

Key Formulas, Identities & Theorems

Distributive Law
$$a \times (b + c) = (a \times b) + (a \times c)$$
Distributivity of multiplication over addition.
Division by Zero
$$a \div 0 = \text{Undefined} \quad (\forall a \in \mathbb{W})$$
Division by 0 is invalid in mathematics.

Conceptual Solved Examples & Case Studies

Example 1
Find the value of 297 × 17 + 297 × 3 using suitable properties.
Step-by-Step Solution:
Using the Distributive Property of multiplication over addition: $a \times b + a \times c = a \times (b + c)$.
Here $a = 297, b = 17, c = 3$.
$$297 \times (17 + 3) = 297 \times 20 = \mathbf{5,940}$$.
Example 2
Find the product by suitable rearrangement: 4 × 166 × 25.
Step-by-Step Solution:
Using Commutative and Associative properties, group numbers that produce powers of 10:
$(4 \times 25) \times 166 = 100 \times 166 = \mathbf{16,600}$.

Common Misconceptions & Examiner Traps

Common Misconception

Assuming subtraction is commutative (e.g. 5 - 3 = 3 - 5).

Scientific Reality & Correction

$5 - 3 = 2$, but $3 - 5 = -2$, which is not even a whole number! Subtraction is NOT commutative.

Common Misconception

Claiming that 0 has a predecessor in whole numbers.

Scientific Reality & Correction

In the set of Whole Numbers $\mathbb{W}$, $0$ is the smallest number and has NO predecessor.

Visual Learning & Conceptual Map

Properties of Whole Number Operations

Algebraic Invariants Governing Arithmetic

Commutative

a + b = b + a
a × b = b × a

Associative

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

Distributive

a × (b + c) =
(a × b) + (a × c)

Identities

a + 0 = a (Additive)
a × 1 = a (Multiplicative)

Chapter Summary & 10 Key Takeaways

Takeaway 1
Whole numbers = Natural numbers + 0: $W = \{0, 1, 2, 3, \dots\}$.
Takeaway 2
Predecessor is $n - 1$; Successor is $n + 1$. 0 has no predecessor in whole numbers.
Takeaway 3
Whole numbers are closed under addition and multiplication, but not subtraction or division.
Takeaway 4
Addition and multiplication are commutative and associative.
Takeaway 5
Distributive property: $a(b + c) = ab + ac$.
Takeaway 6
Division by zero is undefined.

Check Your Understanding (Diagnostic Practice Questions)

Diagnostic questions testing core conceptual clarity. Answers are hidden initially — solve each problem first, then click to reveal the step-by-step verified solution.

1
Find using distributive property: 728 × 101.
Reveal Answer & Explanation
Answer: $728 \times (100 + 1) = 728 \times 100 + 728 \times 1 = 72,800 + 728 = \mathbf{73,528}$.
Break 101 into 100 + 1.

Chapter practice

CBT tests for this chapter are not published yet. Solve the self-check questions above and compare your answers.

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