Class 8 Notes

Class 8 Mathematics Notes Chapter 1 (Rational Numbers) – Mathematics Book

Philoid

Philoid

— 7 min read

Mathematics
Alright class, let's dive deep into Chapter 1: Rational Numbers. This is a foundational chapter, and understanding it well is crucial not just for your Class 8 exams, but also as a building block for many concepts you'll encounter in government exam preparations.

Chapter 1: Rational Numbers - Detailed Notes

1. What are Rational Numbers?

  • A number that can be expressed in the form p/q, where 'p' and 'q' are integers and q ≠ 0 (q is not equal to zero), is called a rational number.
  • Examples: 1/2, -3/4, 5 (since 5 can be written as 5/1), 0 (since 0 can be written as 0/1, 0/5 etc.), -2 (since -2 can be written as -2/1).
  • Why q ≠ 0? Division by zero is undefined in mathematics.
  • The numerator 'p' can be any integer (positive, negative, or zero).
  • The denominator 'q' can be any integer except zero (positive or negative).

2. Relationship with Other Number Systems:

  • Natural Numbers (N): {1, 2, 3, ...}. All natural numbers are rational numbers (e.g., 3 = 3/1).
  • Whole Numbers (W): {0, 1, 2, 3, ...}. All whole numbers are rational numbers (e.g., 0 = 0/1).
  • Integers (Z): {..., -3, -2, -1, 0, 1, 2, 3, ...}. All integers are rational numbers (e.g., -4 = -4/1).
  • So, we have the relationship: N ⊂ W ⊂ Z ⊂ Q (where Q represents the set of Rational Numbers).

3. Properties of Rational Numbers:

Understanding these properties is key to solving problems quickly and accurately. Let 'a', 'b', and 'c' be any rational numbers.

  • (i) Closure Property:

    • Addition: Rational numbers are closed under addition (a + b is always a rational number).
      • Example: 1/2 + 1/3 = (3+2)/6 = 5/6 (which is rational).
    • Subtraction: Rational numbers are closed under subtraction (a - b is always a rational number).
      • Example: 1/2 - 1/3 = (3-2)/6 = 1/6 (which is rational).
    • Multiplication: Rational numbers are closed under multiplication (a × b is always a rational number).
      • Example: (1/2) × (-3/4) = -3/8 (which is rational).
    • Division: Rational numbers are closed under division, except when dividing by zero (a ÷ b is rational, provided b ≠ 0).
      • Example: (1/2) ÷ (3/4) = (1/2) × (4/3) = 4/6 = 2/3 (which is rational).
      • Counter-example: 5 ÷ 0 is undefined.

  • (ii) Commutativity: Does the order matter?

    • Addition: Addition is commutative for rational numbers (a + b = b + a).
      • Example: 1/2 + 1/3 = 5/6 and 1/3 + 1/2 = 5/6.
    • Subtraction: Subtraction is NOT commutative for rational numbers (a - b ≠ b - a, unless a=b).
      • Example: 1/2 - 1/3 = 1/6, but 1/3 - 1/2 = -1/6.
    • Multiplication: Multiplication is commutative for rational numbers (a × b = b × a).
      • Example: (1/2) × (-3/4) = -3/8 and (-3/4) × (1/2) = -3/8.
    • Division: Division is NOT commutative for rational numbers (a ÷ b ≠ b ÷ a, unless a=b and a≠0).
      • Example: (1/2) ÷ (1/4) = 2, but (1/4) ÷ (1/2) = 1/2.

  • (iii) Associativity: Does the grouping matter?

    • Addition: Addition is associative for rational numbers (a + (b + c) = (a + b) + c).
      • Example: 1/2 + (1/3 + 1/4) = 1/2 + 7/12 = 13/12. And (1/2 + 1/3) + 1/4 = 5/6 + 1/4 = 13/12.
    • Subtraction: Subtraction is NOT associative for rational numbers (a - (b - c) ≠ (a - b) - c).
    • Multiplication: Multiplication is associative for rational numbers (a × (b × c) = (a × b) × c).
      • Example: 1/2 × (1/3 × 1/4) = 1/2 × 1/12 = 1/24. And (1/2 × 1/3) × 1/4 = 1/6 × 1/4 = 1/24.
    • Division: Division is NOT associative for rational numbers (a ÷ (b ÷ c) ≠ (a ÷ b) ÷ c).

  • (iv) The Role of Zero (0):

    • Zero is the Additive Identity for rational numbers. Adding zero to any rational number leaves it unchanged (a + 0 = 0 + a = a).
    • Example: -3/7 + 0 = -3/7.

  • (v) The Role of One (1):

    • One is the Multiplicative Identity for rational numbers. Multiplying any rational number by one leaves it unchanged (a × 1 = 1 × a = a).
    • Example: 5/8 × 1 = 5/8.

  • (vi) Negative or Additive Inverse:

    • For any rational number a (or p/q), there exists a rational number -a (or -p/q) such that their sum is the additive identity (0).
    • a + (-a) = 0.
    • -a is called the additive inverse (or negative) of a.
    • Example: The additive inverse of 2/3 is -2/3, because 2/3 + (-2/3) = 0.
    • Example: The additive inverse of -5/7 is -(-5/7) = 5/7, because -5/7 + 5/7 = 0.

  • (vii) Reciprocal or Multiplicative Inverse:

    • For any non-zero rational number a = p/q, there exists a rational number 1/a = q/p such that their product is the multiplicative identity (1).
    • a × (1/a) = 1.
    • 1/a (or q/p) is called the multiplicative inverse (or reciprocal) of a (or p/q).
    • Important: Zero (0) does not have a reciprocal, because 1/0 is undefined.
    • Example: The reciprocal of 4/9 is 9/4, because (4/9) × (9/4) = 1.
    • Example: The reciprocal of -5/8 is -8/5, because (-5/8) × (-8/5) = 1.
    • The reciprocal of 1 is 1. The reciprocal of -1 is -1.

  • (viii) Distributivity of Multiplication over Addition and Subtraction:

    • Multiplication distributes over addition: a × (b + c) = (a × b) + (a × c).
    • Multiplication distributes over subtraction: a × (b - c) = (a × b) - (a × c).
    • This property is very useful for simplifying calculations.
    • Example: 1/2 × (1/3 + 1/4) = 1/2 × (7/12) = 7/24.
    • Using distributivity: (1/2 × 1/3) + (1/2 × 1/4) = 1/6 + 1/8 = (4+3)/24 = 7/24.

4. Representation of Rational Numbers on the Number Line:

  • Rational numbers can be precisely located on the number line.
  • Steps:
    1. Draw a line and mark integers (0, 1, -1, 2, -2, etc.) at equal intervals.
    2. To represent p/q (where q > 0):
      • Divide the segment between two consecutive integers into 'q' equal parts.
      • If p is positive, count 'p' parts to the right of 0.
      • If p is negative, count 'p' parts to the left of 0.
  • Example: To represent 3/4: Divide the segment between 0 and 1 into 4 equal parts. The 3rd mark to the right of 0 is 3/4.
  • Example: To represent -5/3: -5/3 = -1 ⅔. Divide the segment between -1 and -2 into 3 equal parts. The 2nd mark to the left of -1 represents -5/3.

5. Rational Numbers Between Two Rational Numbers:

  • There are infinitely many rational numbers between any two distinct rational numbers.
  • Method 1: The Mean Method
    • If 'a' and 'b' are two rational numbers, then (a + b) / 2 is a rational number lying between 'a' and 'b'.
    • You can repeat this process with the new number and one of the original numbers to find more rational numbers.
    • Example: Find a rational number between 1/4 and 1/2.
      • (1/4 + 1/2) / 2 = ((1+2)/4) / 2 = (3/4) / 2 = 3/8. So, 1/4 < 3/8 < 1/2.
  • Method 2: Equivalent Fractions Method
    • Find equivalent fractions for the given rational numbers with a common denominator.
    • Choose integers between the numerators of these equivalent fractions.
    • Example: Find 5 rational numbers between 1/3 and 1/2.
      1. Find a common denominator (LCM of 3 and 2 is 6): 1/3 = 2/6, 1/2 = 3/6. (Only one integer, none between 2 and 3).
      2. Choose a larger common denominator. Let's multiply the denominator by 6 (to get enough space):
        • 1/3 = 2/6 = (2 × 6) / (6 × 6) = 12/36
        • 1/2 = 3/6 = (3 × 6) / (6 × 6) = 18/36
      3. Now, the numbers between 12/36 and 18/36 are: 13/36, 14/36, 15/36, 16/36, 17/36. These are 5 rational numbers between 1/3 and 1/2.

Multiple Choice Questions (MCQs)

Here are 10 MCQs based on the concepts discussed, suitable for government exam preparation:

  1. Which of the following is NOT a rational number?
    (a) 0
    (b) -5/7
    (c) √2
    (d) 22/7

  2. The additive inverse of -6/(-5) is:
    (a) 6/5
    (b) -6/5
    (c) 5/6
    (d) -5/6

  3. The multiplicative inverse of -3/8 × (-4/15) is:
    (a) -10
    (b) 10
    (c) 12/120
    (d) -12/120

  4. Which property is illustrated by the statement: (2/3 + 4/5) + 1/6 = 2/3 + (4/5 + 1/6)?
    (a) Commutative property of addition
    (b) Associative property of addition
    (c) Distributive property
    (d) Additive Identity

  5. The rational number that does not have a reciprocal is:
    (a) 1
    (b) -1
    (c) 0
    (d) 1/2

  6. A rational number between 1/5 and 1/4 is:
    (a) 9/40
    (b) 1/9
    (c) 1/3
    (d) 4/15

  7. What is the result of (-7/8) × 1?
    (a) 1
    (b) 0
    (c) -7/8
    (d) 8/7

  8. Which operation is NOT commutative for rational numbers?
    (a) Addition
    (b) Multiplication
    (c) Subtraction
    (d) Both (a) and (b)

  9. The expression (-5/4) × (8/5 + 16/15) simplifies using the distributive property as:
    (a) (-5/4 × 8/5) + 16/15
    (b) (-5/4 × 8/5) × (-5/4 × 16/15)
    (c) (-5/4 × 8/5) + (-5/4 × 16/15)
    (d) -5/4 + (8/5 + 16/15)

  10. If 'a' is the additive inverse of 'b', then:
    (a) a × b = 1
    (b) a + b = 0
    (c) a - b = 0
    (d) a / b = -1

Answer Key for MCQs:

  1. (c) √2 (Square root of 2 is an irrational number as it cannot be expressed as p/q)
  2. (b) -6/5 (First, -6/(-5) = 6/5. The additive inverse of 6/5 is -6/5)
  3. (b) 10 (First calculate the product: (-3/8) × (-4/15) = 12/120 = 1/10. The multiplicative inverse (reciprocal) of 1/10 is 10/1 = 10)
  4. (b) Associative property of addition (The grouping of numbers being added is changed)
  5. (c) 0 (Division by zero is undefined, so 0 has no reciprocal)
  6. (a) 9/40 (Using mean method: (1/5 + 1/4)/2 = ((4+5)/20)/2 = (9/20)/2 = 9/40. Check: 1/5 = 8/40, 1/4 = 10/40. 9/40 lies between them.)
  7. (c) -7/8 (1 is the multiplicative identity)
  8. (c) Subtraction (and Division) are not commutative for rational numbers.
  9. (c) (-5/4 × 8/5) + (-5/4 × 16/15) (This directly applies a × (b + c) = ab + ac)
  10. (b) a + b = 0 (Definition of additive inverse)

Study these notes thoroughly. Focus on understanding the why behind the properties and definitions. Practice applying them, and you'll build a strong foundation. Good luck!

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