Class 6 Mathematics Notes Chapter 2 (Chapter 2) – Exemplar Problem (English) Book

Detailed Notes with MCQs of Chapter 2: Whole Numbers from your NCERT Class 6 Mathematics Exemplar. This chapter lays the foundation for many arithmetic concepts crucial for competitive exams. Pay close attention to the definitions and properties.

Chapter 2: Whole Numbers - Detailed Notes for Exam Preparation

1. Introduction: Natural Numbers and Whole Numbers

• Natural Numbers (N): These are the counting numbers: 1, 2, 3, 4, ... and so on, extending infinitely.
  • The smallest natural number is 1.
  • There is no largest natural number.
• Whole Numbers (W): These include all natural numbers and the number zero (0): 0, 1, 2, 3, 4, ... and so on.
  • The smallest whole number is 0.
  • There is no largest whole number.
• Key Relationship: All natural numbers are whole numbers. However, 0 is a whole number but not a natural number.

2. Predecessor and Successor

• Successor: The successor of a whole number is the number obtained by adding 1 to it.
  • Example: Successor of 15 is 15 + 1 = 16. Successor of 0 is 0 + 1 = 1.
  • Every whole number has a successor.
• Predecessor: The predecessor of a whole number (except 0) is the number obtained by subtracting 1 from it.
  • Example: Predecessor of 15 is 15 - 1 = 14. Predecessor of 1 is 1 - 1 = 0.
  • The whole number 0 does not have a predecessor within the set of whole numbers. (0 - 1 = -1, which is not a whole number).

3. Number Line

• A visual representation of numbers marked at equal intervals.
• Starts from 0 for whole numbers and extends indefinitely to the right.
• Operations on Number Line:
  • Addition: Move to the right. (e.g., 3 + 4: Start at 3, move 4 steps right to reach 7).
  • Subtraction: Move to the left. (e.g., 7 - 5: Start at 7, move 5 steps left to reach 2).
  • Multiplication: Represents repeated addition (jumps of equal size). (e.g., 3 x 4: Start at 0, take 3 jumps of 4 units each to the right, reaching 12).

4. Properties of Whole Numbers (Very Important for Exams)

These properties help simplify calculations and understand number behaviour. They apply to specific operations (Addition '+', Multiplication '×', Subtraction '-', Division '÷').

• (a) Closure Property:

  • Addition: If 'a' and 'b' are whole numbers, then a + b is always a whole number. (Whole numbers are closed under addition).
    • Example: 5 + 8 = 13 (Whole number)
  • Multiplication: If 'a' and 'b' are whole numbers, then a × b is always a whole number. (Whole numbers are closed under multiplication).
    • Example: 6 × 7 = 42 (Whole number)
  • Subtraction: If 'a' and 'b' are whole numbers, a - b is not always a whole number. (Whole numbers are not closed under subtraction).
    • Example: 4 - 9 = -5 (Not a whole number)
  • Division: If 'a' and 'b' (b≠0) are whole numbers, a ÷ b is not always a whole number. (Whole numbers are not closed under division).
    • Example: 5 ÷ 2 = 2.5 (Not a whole number)

• (b) Commutative Property: (Order of numbers doesn't matter)

  • Addition: a + b = b + a for any whole numbers a, b. (Addition is commutative).
    • Example: 7 + 9 = 16; 9 + 7 = 16
  • Multiplication: a × b = b × a for any whole numbers a, b. (Multiplication is commutative).
    • Example: 4 × 6 = 24; 6 × 4 = 24
  • Subtraction: a - b ≠ b - a (unless a=b). (Subtraction is not commutative).
    • Example: 10 - 3 = 7, but 3 - 10 = -7
  • Division: a ÷ b ≠ b ÷ a (unless a=b and not zero). (Division is not commutative).
    • Example: 8 ÷ 2 = 4, but 2 ÷ 8 = 0.25

• (c) Associative Property: (Grouping of numbers doesn't matter)

  • Addition: (a + b) + c = a + (b + c) for any whole numbers a, b, c. (Addition is associative).
    • Example: (2 + 3) + 5 = 5 + 5 = 10; 2 + (3 + 5) = 2 + 8 = 10
  • Multiplication: (a × b) × c = a × (b × c) for any whole numbers a, b, c. (Multiplication is associative).
    • Example: (4 × 2) × 3 = 8 × 3 = 24; 4 × (2 × 3) = 4 × 6 = 24
  • Subtraction: (a - b) - c ≠ a - (b - c). (Subtraction is not associative).
    • Example: (10 - 5) - 2 = 5 - 2 = 3; 10 - (5 - 2) = 10 - 3 = 7
  • Division: (a ÷ b) ÷ c ≠ a ÷ (b ÷ c). (Division is not associative).
    • Example: (16 ÷ 4) ÷ 2 = 4 ÷ 2 = 2; 16 ÷ (4 ÷ 2) = 16 ÷ 2 = 8

• (d) Distributive Property of Multiplication over Addition: (Crucial for simplifying calculations)

  • a × (b + c) = (a × b) + (a × c) for any whole numbers a, b, c.
  • Example: 5 × (3 + 4) = 5 × 7 = 35
    Also, (5 × 3) + (5 × 4) = 15 + 20 = 35
  • This property also applies to multiplication over subtraction: a × (b - c) = (a × b) - (a × c).
  • Example: 6 × (10 - 2) = 6 × 8 = 48
    Also, (6 × 10) - (6 × 2) = 60 - 12 = 48
  • Useful for calculations like: 12 × 105 = 12 × (100 + 5) = (12 × 100) + (12 × 5) = 1200 + 60 = 1260.

• (e) Identity Elements:

  • Additive Identity (0): Adding 0 to any whole number leaves it unchanged.
    • a + 0 = 0 + a = a.
    • Zero (0) is the additive identity for whole numbers.
  • Multiplicative Identity (1): Multiplying any whole number by 1 leaves it unchanged.
    • a × 1 = 1 × a = a.
    • One (1) is the multiplicative identity for whole numbers.

5. Properties of Zero (0) and One (1)

• Multiplication by Zero: Any whole number multiplied by 0 results in 0. (a × 0 = 0 × a = 0).
• Division by Zero: Dividing any whole number by 0 is undefined. (a ÷ 0 is not defined). Remember this!
• Division of Zero: Dividing 0 by any non-zero whole number results in 0. (0 ÷ a = 0, for a ≠ 0).
• Division by One: Dividing any whole number by 1 results in the number itself. (a ÷ 1 = a).

6. Patterns in Whole Numbers (Less common in exams but good to know)

• Numbers can be arranged in elementary shapes using dots:
  • Line: Every number can be arranged as a line (except 1, trivially).
  • Rectangle: Some numbers like 6 (2x3), 8 (2x4), etc.
  • Square: Numbers like 1 (1x1), 4 (2x2), 9 (3x3), 16 (4x4)... These are square numbers.
  • Triangle: Numbers like 1, 3, 6, 10, 15... These are triangular numbers (formed by adding consecutive natural numbers: 1, 1+2, 1+2+3, 1+2+3+4...).
• Observing patterns can help in quick calculations and understanding number sequences. Example: 99 × 5 = (100 - 1) × 5 = 500 - 5 = 495.

Key Takeaways for Exams:

• Know the difference between Natural and Whole numbers.
• Understand Predecessor and Successor, especially the case for 0.
• Master the properties (Closure, Commutative, Associative, Distributive, Identity) and know which operations they apply to.
• The Distributive property is frequently used for simplification.
• Remember the special rules for operations involving 0 and 1, especially that division by 0 is UNDEFINED.

Multiple Choice Questions (MCQs)

Here are 10 MCQs based on Chapter 2 - Whole Numbers:

1. What is the smallest whole number?
   (A) 1
   (B) 0
   (C) -1
   (D) Not defined

2. Which of the following statements is FALSE?
   (A) 0 is the additive identity for whole numbers.
   (B) Whole numbers are closed under subtraction.
   (C) Multiplication is commutative for whole numbers.
   (D) The predecessor of 1 in whole numbers is 0.

3. The property used in 8 × (5 + 9) = (8 × 5) + (8 × 9) is:
   (A) Associative property
   (B) Commutative property
   (C) Distributive property of multiplication over addition
   (D) Closure property

4. What is the value of 173 × 15 + 173 × 5?
   (A) 1730
   (B) 3460
   (C) 17300
   (D) 2595

5. Which of the following expressions is undefined?
   (A) 0 ÷ 5
   (B) 5 ÷ 1
   (C) 5 - 0
   (D) 5 ÷ 0

6. The successor of the largest 4-digit number is:
   (A) 9998
   (B) 1000
   (C) 10000
   (D) 9999

7. Which operation is NOT associative for whole numbers?
   (A) Addition
   (B) Multiplication
   (C) Subtraction
   (D) Both (A) and (B)

8. The multiplicative identity for whole numbers is:
   (A) 0
   (B) 1
   (C) -1
   (D) Any number

9. If 'a' and 'b' are two whole numbers, then which of the following may NOT be a whole number?
   (A) a + b
   (B) a × b
   (C) a - b
   (D) Additive identity (0)

10. The expression (7 + 5) + 3 = 7 + (5 + 3) demonstrates:
    (A) Commutativity of addition
    (B) Associativity of addition
    (C) Distributive property
    (D) Closure property of addition

Answer Key for MCQs:

1. (B)
2. (B)
3. (C)
4. (B) [Hint: Use distributive property: 173 × (15 + 5) = 173 × 20 = 3460]
5. (D)
6. (C) [Largest 4-digit number is 9999. Successor is 9999 + 1 = 10000]
7. (C) [Subtraction and Division are not associative]
8. (B)
9. (C) [Example: 3 - 5 = -2, which is not a whole number]
10. (B)

Study these notes and properties thoroughly. Understanding them well will help you solve problems quickly and accurately in your exams. Good luck!