Class 9 Notes

Class 9 Mathematics Notes Chapter 1 (Chapter 1) – Examplar Problem (Englisha) Book

Philoid

Philoid

— 5 min read

Examplar Problem (Englisha)
Alright class, let's focus on Chapter 1: Number Systems from your NCERT Exemplar. This chapter forms the bedrock for many mathematical concepts, and mastering it is crucial, especially for competitive government exams where foundational clarity is key. We'll go through the essential points systematically.

Chapter 1: Number Systems - Detailed Notes for Exam Preparation

1. Introduction to Number Systems:

  • Natural Numbers (N): Counting numbers: 1, 2, 3, 4, ...
  • Whole Numbers (W): Natural numbers including zero: 0, 1, 2, 3, ... (Note: All natural numbers are whole numbers).
  • Integers (Z): Whole numbers and their negatives: ..., -3, -2, -1, 0, 1, 2, 3, ... (Note: All whole numbers are integers).
  • Rational Numbers (Q):
    • Numbers that can be expressed in the form p/q, where 'p' and 'q' are integers, and crucially, q ≠ 0.
    • Examples: 1/2, -3/4, 5 (since 5 = 5/1), 0 (since 0 = 0/1), 2.5 (since 2.5 = 25/10 = 5/2).
    • Includes all integers, whole numbers, and natural numbers.
    • Important Property: Between any two given rational numbers, there exist infinitely many rational numbers. (Methods to find them: Mean method, equivalent fractions method).
  • Irrational Numbers:
    • Numbers that cannot be expressed in the form p/q, where p and q are integers and q ≠ 0.
    • Examples: √2, √3, √5, π (pi), 0.101101110... (non-repeating, non-terminating decimals).
    • Key Distinction: √m is irrational if 'm' is a positive integer that is not a perfect square. √4 = 2 is rational, but √3 is irrational.
  • Real Numbers (R):
    • The collection of all rational and irrational numbers together.
    • Every real number can be represented by a unique point on the number line, and conversely, every point on the number line represents a unique real number.
    • Relationship: N ⊂ W ⊂ Z ⊂ Q ⊂ R.

2. Decimal Expansions of Real Numbers:

  • Rational Numbers: Have decimal expansions that are either:

    • Terminating: The division ends after a finite number of steps. Example: 1/4 = 0.25, 5/8 = 0.625.
      • Exam Tip: A rational number p/q (in simplest form) has a terminating decimal expansion if and only if the prime factorization of the denominator 'q' consists only of powers of 2 and/or 5 (i.e., q is of the form 2ⁿ5ᵐ, where n, m are non-negative integers).
    • Non-terminating Repeating (or Recurring): The division does not end, but a sequence of digits repeats indefinitely. Example: 1/3 = 0.333... (written as 0.3̅), 1/7 = 0.142857142857... (written as 0.1̅4̅2̅8̅5̅7̅).

  • Irrational Numbers: Have decimal expansions that are:

    • Non-terminating Non-repeating: The division never ends, and there is no repeating block of digits. Example: π ≈ 3.14159265..., √2 ≈ 1.41421356...

  • Converting Recurring Decimals to p/q Form: (Essential Skill)

    • Example: Convert 0.7̅ to p/q.
      • Let x = 0.777... (1)
      • Multiply by 10 (since one digit repeats): 10x = 7.777... (2)
      • Subtract (1) from (2): 10x - x = 7.777... - 0.777... => 9x = 7 => x = 7/9.
    • Example: Convert 0.23̅5̅ to p/q.
      • Let x = 0.23535... (1)
      • Multiply by 10 (to get non-repeating part before decimal): 10x = 2.3535... (2)
      • Multiply by 1000 (to get one full repeating block before decimal): 1000x = 235.3535... (3)
      • Subtract (2) from (3): 1000x - 10x = 235.3535... - 2.3535... => 990x = 233 => x = 233/990.

3. Operations on Real Numbers:

  • Rational + Rational = Rational
  • Rational × Rational = Rational
  • Rational ± Irrational = Irrational
  • Rational (≠ 0) × Irrational = Irrational
  • Irrational ± Irrational = May be Rational or Irrational (e.g., √2 + (-√2) = 0 [Rational]; √2 + √3 [Irrational])
  • Irrational × Irrational = May be Rational or Irrational (e.g., √2 × √2 = 2 [Rational]; √2 × √3 = √6 [Irrational])
  • Irrational / Irrational = May be Rational or Irrational (e.g., √8 / √2 = √4 = 2 [Rational]; √6 / √2 = √3 [Irrational])

4. Rationalizing the Denominator:

  • Process of converting an expression with an irrational denominator into an equivalent expression with a rational denominator.
  • Type 1: Denominator is √a. Multiply numerator and denominator by √a.
    • Example: 1/√7 = (1 × √7) / (√7 × √7) = √7 / 7.
  • Type 2: Denominator is a ± √b or √a ± √b. Multiply numerator and denominator by the conjugate.
    • Conjugate of (a + √b) is (a - √b).
    • Conjugate of (√a + √b) is (√a - √b).
    • Use the identity: (x + y)(x - y) = x² - y².
    • Example: 1 / (2 + √3) = [1 × (2 - √3)] / [(2 + √3) × (2 - √3)] = (2 - √3) / (2² - (√3)²) = (2 - √3) / (4 - 3) = (2 - √3) / 1 = 2 - √3.
    • Example: 5 / (√5 - √2) = [5 × (√5 + √2)] / [(√5 - √2) × (√5 + √2)] = 5(√5 + √2) / ((√5)² - (√2)²) = 5(√5 + √2) / (5 - 2) = 5(√5 + √2) / 3.

5. Laws of Exponents for Real Numbers:

Let a > 0 be a real number and p, q be rational numbers.

  • aᵖ ⋅ a<0xE1><0xB5><0xA1> = aᵖ⁺<0xE1><0xB5><0xA1> (Product Rule)

  • (aᵖ)<0xE1><0xB5><0xA1> = aᵖ<0xE1><0xB5><0xA1> (Power of a Power Rule)

  • aᵖ / a<0xE1><0xB5><0xA1> = aᵖ⁻<0xE1><0xB5><0xA1> (Quotient Rule)

  • aᵖ bᵖ = (ab)ᵖ (Same Exponent Rule)

  • a⁰ = 1 (Zero Exponent)

  • a⁻ᵖ = 1/aᵖ (Negative Exponent)

  • ⁿ√a = a¹/ⁿ (Radical as Fractional Exponent)

  • ⁿ√aᵐ = aᵐ/ⁿ = (a¹/ⁿ)ᵐ = (aᵐ)¹/ⁿ

  • Exam Focus: Be comfortable simplifying expressions involving fractional and negative exponents.

    • Example: Simplify (64)^(2/3) = (4³)^(2/3) = 4^(3 × 2/3) = 4² = 16.
    • Example: Simplify (81)^(-1/4) = 1 / (81)^(1/4) = 1 / (3⁴)^(1/4) = 1 / 3¹ = 1/3.
    • Example: Simplify 7^(1/2) ⋅ 8^(1/2) = (7 × 8)^(1/2) = 56^(1/2) = √56.

Multiple Choice Questions (MCQs)

Here are 10 MCQs to test your understanding. Remember to apply the concepts we just discussed.

  1. Which of the following is an irrational number?
    (a) √16
    (b) √(12/3)
    (c) √12
    (d) √100

  2. The decimal expansion of an irrational number is always:
    (a) Terminating
    (b) Non-terminating repeating
    (c) Terminating repeating
    (d) Non-terminating non-repeating

  3. Which of the following is a rational number between 1/4 and 1/3?
    (a) 7/24
    (b) 5/12
    (c) 1/5
    (d) 8/24

  4. The value of (√5 + √2)² is:
    (a) 7
    (b) 10
    (c) 7 + 2√10
    (d) 5 + 2√10

  5. After rationalizing the denominator of 7 / (3√3 - 2√2), we get the denominator as:
    (a) 19
    (b) 5
    (c) 35
    (d) 1

  6. The number 1.272727... (or 1.2̅7̅) can be expressed in the form p/q as:
    (a) 127/100
    (b) 127/99
    (c) 14/11
    (d) 11/14

  7. The value of (256)^0.16 × (256)^0.09 is:
    (a) 4
    (b) 16
    (c) 64
    (d) 256.25

  8. Which of the following statements is TRUE?
    (a) Every integer is a whole number.
    (b) Every rational number is an integer.
    (c) Every irrational number is a real number.
    (d) Every real number is an irrational number.

  9. If x = 2 + √3, then the value of (x - 1/x) is:
    (a) 2√3
    (b) 2
    (c) 4
    (d) √3

  10. The product √6 × √8 is equal to:
    (a) 2√3
    (b) 4√3
    (c) √14
    (d) 3√4

Answers to MCQs:

  1. (c) √12 (since 12 is not a perfect square)
  2. (d) Non-terminating non-repeating
  3. (a) 7/24 (1/4 = 6/24, 1/3 = 8/24. 7/24 lies between them)
  4. (c) 7 + 2√10 ((√5)² + (√2)² + 2(√5)(√2) = 5 + 2 + 2√10)
  5. (a) 19 (Denominator = (3√3)² - (2√2)² = (9×3) - (4×2) = 27 - 8 = 19)
  6. (c) 14/11 (Let x = 1.2727... 100x = 127.2727... 99x = 126, x = 126/99 = 14/11)
  7. (a) 4 ((256)^(0.16+0.09) = (256)^0.25 = (256)^(1/4) = (4⁴)^(1/4) = 4)
  8. (c) Every irrational number is a real number.
  9. (a) 2√3 (1/x = 1/(2+√3) = (2-√3)/((2+√3)(2-√3)) = (2-√3)/(4-3) = 2-√3. So, x - 1/x = (2+√3) - (2-√3) = 2√3)
  10. (b) 4√3 (√6 × √8 = √(6×8) = √48 = √(16×3) = 4√3)

Keep practicing these concepts, especially rationalization and laws of exponents, as they are frequently tested. Good luck with your preparation!

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