Class 6 Mathematics Notes Chapter 2 (Whole Numbers) – Mathematics Book
Detailed Notes with MCQs of Chapter 2: Whole Numbers from your NCERT Class 6 Mathematics book. This chapter builds upon your understanding of natural numbers and introduces a fundamental concept – the number zero – and explores the properties of this expanded set of numbers. These concepts are foundational for many topics in quantitative aptitude sections of government exams.
Chapter 2: Whole Numbers - Detailed Notes
1. Introduction: Natural Numbers and Whole Numbers
- Natural Numbers (N): These are the counting numbers: 1, 2, 3, 4, ... and so on, extending infinitely. They are used for counting objects. The smallest natural number is 1. There is no largest natural number.
- Need for Zero: Consider questions like "How many apples are left if you had 3 and ate 3?". The answer is 'none', which wasn't represented by natural numbers. This led to the introduction of Zero (0).
- Whole Numbers (W): When we include Zero (0) in the collection of Natural Numbers, we get the set of Whole Numbers.
- W = {0, 1, 2, 3, 4, ...}
- The smallest whole number is 0.
- Like natural numbers, there is no largest whole number.
- Key Point: All natural numbers are whole numbers, but not all whole numbers are natural numbers (because 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. Successor of 99 is 99 + 1 = 100.
- 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 100 is 100 - 1 = 99. Predecessor of 1 is 1 - 1 = 0.
- Important: The whole number 0 does not have a predecessor in the set of whole numbers. (Its predecessor, -1, is an integer, not a whole number). The smallest natural number, 1, has the predecessor 0.
3. The Number Line
- A visual representation of numbers. Draw a straight line, mark a point as 0. Mark points at equal distances to the right of 0 and label them 1, 2, 3, ...
- Properties on the Number Line:
- The distance between any two consecutive whole numbers is the same (unit distance).
- Numbers increase as we move to the right.
- The number on the right is always greater than the number on the left (e.g., 5 > 3 because 5 is to the right of 3).
- Operations on the Number Line:
- Addition: To add 'b' to 'a', start at 'a' and make 'b' jumps to the right. (e.g., 3 + 4: Start at 3, make 4 jumps right, reach 7).
- Subtraction: To subtract 'b' from 'a' (where a > b), start at 'a' and make 'b' jumps to the left. (e.g., 7 - 5: Start at 7, make 5 jumps left, reach 2).
- Multiplication: To multiply 'a' by 'b', start at 0 and make 'b' jumps, each of length 'a', to the right. (e.g., 3 x 4: Start at 0, make 4 jumps of length 3 each, reach 12).
4. Properties of Whole Numbers
These properties are extremely important for simplifying calculations and understanding number behaviour.
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(i) Closure Property:
- Addition: Whole numbers are closed under addition. If 'a' and 'b' are whole numbers, then a + b is also a whole number. (e.g., 5 + 8 = 13, which is a whole number).
- Multiplication: Whole numbers are closed under multiplication. If 'a' and 'b' are whole numbers, then a × b is also a whole number. (e.g., 5 × 8 = 40, which is a whole number).
- Subtraction: Whole numbers are NOT closed under subtraction. (e.g., 5 - 8 = -3, which is not a whole number).
- Division: Whole numbers are NOT closed under division. (e.g., 5 ÷ 8 = 5/8, which is not a whole number. Also, 8 ÷ 5 is not a whole number).
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(ii) Commutative Property: (Order doesn't matter)
- Addition: Addition is commutative for whole numbers. a + b = b + a. (e.g., 4 + 6 = 10 and 6 + 4 = 10).
- Multiplication: Multiplication is commutative for whole numbers. a × b = b × a. (e.g., 4 × 6 = 24 and 6 × 4 = 24).
- Subtraction: Subtraction is NOT commutative for whole numbers. a - b ≠ b - a (unless a=b). (e.g., 6 - 4 = 2, but 4 - 6 = -2).
- Division: Division is NOT commutative for whole numbers. a ÷ b ≠ b ÷ a (unless a=b and non-zero). (e.g., 6 ÷ 3 = 2, but 3 ÷ 6 = 1/2).
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(iii) Associative Property: (Grouping doesn't matter)
- Addition: Addition is associative for whole numbers. (a + b) + c = a + (b + c). (e.g., (2 + 3) + 4 = 5 + 4 = 9 and 2 + (3 + 4) = 2 + 7 = 9). This helps in adding numbers conveniently.
- Multiplication: Multiplication is associative for whole numbers. (a × b) × c = a × (b × c). (e.g., (2 × 3) × 4 = 6 × 4 = 24 and 2 × (3 × 4) = 2 × 12 = 24). This helps in multiplying numbers conveniently.
- Subtraction: Subtraction is NOT associative for whole numbers. (a - b) - c ≠ a - (b - c). (e.g., (8 - 4) - 2 = 4 - 2 = 2, but 8 - (4 - 2) = 8 - 2 = 6).
- Division: Division is NOT associative for whole numbers. (a ÷ b) ÷ c ≠ a ÷ (b ÷ c). (e.g., (16 ÷ 4) ÷ 2 = 4 ÷ 2 = 2, but 16 ÷ (4 ÷ 2) = 16 ÷ 2 = 8).
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(iv) Distributive Property of Multiplication over Addition:
- This property connects multiplication and addition: a × (b + c) = (a × b) + (a × c).
- Example: 6 × (5 + 3) = 6 × 8 = 48. Also, (6 × 5) + (6 × 3) = 30 + 18 = 48.
- This is very useful for mental calculations (e.g., 12 × 105 = 12 × (100 + 5) = (12 × 100) + (12 × 5) = 1200 + 60 = 1260).
- It also applies to subtraction: a × (b - c) = (a × b) - (a × c).
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(v) Identity Elements:
- Additive Identity: Zero (0) is the additive identity for whole numbers. Adding 0 to any whole number leaves it unchanged. a + 0 = 0 + a = a.
- Multiplicative Identity: One (1) is the multiplicative identity for whole numbers. Multiplying any whole number by 1 leaves it unchanged. a × 1 = 1 × a = a.
5. Special 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: Division of any number by 0 is undefined. a ÷ 0 is not defined.
- Division of Zero by a Non-Zero Number: Zero divided by any non-zero whole number is 0. 0 ÷ a = 0 (where a ≠ 0).
- Role of One (1): As mentioned, 1 is the multiplicative identity. Division by 1 leaves the number unchanged: a ÷ 1 = a.
6. Patterns in Whole Numbers (Brief Overview)
- Properties like associativity and distributivity help us find patterns and simplify calculations.
- Example: 96 × 25 = 96 × (100 ÷ 4) = (96 ÷ 4) × 100 = 24 × 100 = 2400.
- Example: 8 × 1769 × 125 = (8 × 125) × 1769 = 1000 × 1769 = 1769000 (using commutativity and associativity).
Key Takeaways for Exams:
- Know the difference between Natural and Whole numbers (0 is the key).
- Understand Predecessor and Successor (especially for 0 and 1).
- Master the properties: Closure, Commutativity, Associativity, Distributivity, Identity elements.
- Crucially, remember for which operations (Addition, Subtraction, Multiplication, Division) each property holds TRUE or FALSE for whole numbers. This is a common area for questions.
- Remember the special roles of 0 and 1, especially division by 0 (undefined).
- Practice using properties to simplify calculations quickly.
Multiple Choice Questions (MCQs)
Here are 10 MCQs based on the chapter 'Whole Numbers' for practice:
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Which of the following statements is TRUE?
A) All natural numbers are whole numbers.
B) All whole numbers are natural numbers.
C) The predecessor of 1 in natural numbers is 0.
D) The smallest natural number is 0. -
What is the additive identity for the set of whole numbers?
A) 1
B) 0
C) -1
D) Does not exist -
Which property is illustrated by the statement: 15 × (100 - 2) = (15 × 100) - (15 × 2)?
A) Associative Property of Multiplication
B) Commutative Property of Multiplication
C) Distributive Property of Multiplication over Subtraction
D) Closure Property of Multiplication -
The set of whole numbers is NOT closed under which operation?
A) Addition
B) Multiplication
C) Subtraction
D) Both Addition and Multiplication -
What is the result of 7 ÷ 0?
A) 7
B) 0
C) 1
D) Undefined -
Which of the following expressions is NOT equal to 0?
A) 0 × 15
B) 0 ÷ 15
C) (15 - 15) ÷ 5
D) 15 ÷ (3 - 3) -
What is the successor of the predecessor of 1000?
A) 999
B) 1000
C) 1001
D) 998 -
Which property allows us to compute 8 × 17 × 125 as (8 × 125) × 17?
A) Distributive Property
B) Closure Property
C) Associative Property of Multiplication
D) Additive Identity -
The smallest whole number is:
A) 1
B) 0
C) 2
D) Cannot be determined -
If 'a' and 'b' are two different whole numbers, then which statement is definitely TRUE?
A) a - b = b - a
B) a ÷ b = b ÷ a
C) a + b = b + a
D) a - b is always a whole number
Answer Key:
- A
- B
- C
- C (Also not closed under division, but Subtraction is listed as an option)
- D
- D (Because 15 ÷ (3-3) = 15 ÷ 0, which is undefined)
- B (Predecessor of 1000 is 999. Successor of 999 is 999 + 1 = 1000)
- C (It involves regrouping the multiplication)
- B
- C (Commutativity of Addition always holds for whole numbers)
Study these notes carefully, focusing on the definitions and properties. Practice applying them, and you'll build a strong foundation. Good luck!