Class 9 Mathematics Notes Chapter 1 (Number Systems) – Mathematics Book
Alright class, let's begin our preparation for Chapter 1: Number Systems. This is a foundational chapter, and understanding it well is crucial not just for your exams but for all future mathematics. We will cover the key concepts precisely as needed for competitive government exams.
Chapter 1: Number Systems - Detailed Notes
1. Introduction
- Number systems deal with different types of numbers, their properties, and how they relate to each other. We use numbers constantly, so classifying them helps in understanding mathematical operations and concepts.
2. Types of Numbers
- Natural Numbers (N):
- These are the counting numbers.
- N = {1, 2, 3, 4, ...}
- The smallest natural number is 1. There is no largest natural number.
- Whole Numbers (W):
- Natural numbers along with zero.
- W = {0, 1, 2, 3, ...}
- The smallest whole number is 0.
- Note: All natural numbers are whole numbers, but 0 is a whole number that is not natural. (N ⊂ W)
- Integers (Z):
- Whole numbers and their negative counterparts.
- Z = {..., -3, -2, -1, 0, 1, 2, 3, ...}
- Includes positive integers (1, 2, 3,...), negative integers (-1, -2, -3,...), and zero (which is neither positive nor negative).
- There is no smallest or largest integer.
- Note: All whole numbers are integers. (W ⊂ Z)
- Rational Numbers (Q):
- Definition: Any number that can be expressed in the form p/q, where 'p' and 'q' are integers, and q ≠ 0.
- Examples: 1/2, -3/4, 5 (since 5 = 5/1), 0 (since 0 = 0/1), -2 (since -2 = -2/1), 2/3, 4/5.
- Key Properties:
- Between any two given rational numbers, there exist infinitely many rational numbers.
- Finding rationals between 'a' and 'b': A simple method is to calculate (a+b)/2. Another method is to make the denominators equal and find intermediate numerators.
- Decimal Expansion: The decimal expansion of a rational number is either:
- Terminating: The division process ends after a finite number of steps. Example: 1/4 = 0.25, 5/8 = 0.625. This happens when the prime factors of the denominator 'q' (in the simplest p/q form) are only 2s or 5s or both.
- Non-terminating Recurring (Repeating): The division process does not end, but a sequence of digits repeats periodically. Example: 1/3 = 0.333..., 2/7 = 0.285714285714... (the block 285714 repeats). This happens when the prime factors of the denominator 'q' (in the simplest p/q form) include prime numbers other than 2 or 5.
- Note: All integers are rational numbers. (Z ⊂ Q)
- Irrational Numbers:
- Definition: Any number that cannot be expressed in the form p/q, where 'p' and 'q' are integers, and q ≠ 0.
- Examples: √2, √3, √5, √7, π (pi), 0.10110111011110... (non-repeating, non-terminating pattern).
- Key Properties:
- Their decimal expansion is non-terminating and non-recurring (non-repeating).
- The sum, difference, product, or quotient of a non-zero rational number and an irrational number is always irrational. (e.g., 2 + √3, 5√2, √7 / 3).
- The sum, difference, product, or quotient of two irrational numbers may be rational or irrational. (e.g., √2 + (-√2) = 0 (rational); √2 * √2 = 2 (rational); √2 + √3 (irrational); √2 * √3 = √6 (irrational)).
- Real Numbers (R):
- Definition: The collection of all rational and irrational numbers together.
- Every real number can be represented by a unique point on the number line.
- Conversely, every point on the number line represents a unique real number.
- R = Q ∪ (Irrational Numbers)
3. Representing Numbers on the Number Line
- Integers, rational numbers (like 1/2, 3/4) can be precisely marked.
- Irrational numbers like √2, √3 can be represented using Pythagoras' theorem. (e.g., For √2, construct a right triangle with base 1 and height 1 on the number line starting from 0. The hypotenuse length is √2. Use a compass to mark this length on the number line).
- Real numbers with non-terminating decimals can be visualized using the process of successive magnification.
4. Operations on Real Numbers
- Real numbers obey standard arithmetic laws (commutative, associative, distributive).
- Rationalizing the Denominator: The process of converting an expression with an irrational number in the denominator to an equivalent expression whose denominator is a rational number.
- If the denominator is of the form √a, multiply numerator and denominator by √a.
Example: 1/√7 = (1 * √7) / (√7 * √7) = √7 / 7. - If the denominator is of the form a + √b or √a + √b, multiply numerator and denominator by its conjugate. The conjugate of a + √b is a - √b, and vice-versa. The conjugate of √a + √b is √a - √b, and vice-versa. 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: 1 / (√5 - √2) = [1 * (√5 + √2)] / [(√5 - √2) * (√5 + √2)] = (√5 + √2) / ((√5)² - (√2)²) = (√5 + √2) / (5 - 2) = (√5 + √2) / 3.
- If the denominator is of the form √a, multiply numerator and denominator by √a.
5. Laws of Exponents for Real Numbers
Let 'a' > 0 be a real number and 'p', 'q' be rational numbers. Then:
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i) aᵖ ⋅ a<0xE1><0xB5><0xA1> = aᵖ⁺<0xE1><0xB5><0xA1> (Product Law)
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ii) (aᵖ)<0xE1><0xB5><0xA1> = aᵖ<0xE1><0xB5><0xA1> (Power of a Power Law)
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iii) aᵖ / a<0xE1><0xB5><0xA1> = aᵖ⁻<0xE1><0xB5><0xA1> (Quotient Law)
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iv) aᵖ ⋅ bᵖ = (ab)ᵖ (Same Exponent Law)
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v) a⁰ = 1 (Zero Exponent)
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vi) a⁻ᵖ = 1 / aᵖ (Negative Exponent)
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vii) nth root: √[n]{a} = a¹ᐟⁿ. More generally, √[n]{aᵐ} = aᵐᐟⁿ.
- Example: (27)²ᐟ³ = (3³) ²ᐟ³ = 3³ˣ²ᐟ³ = 3² = 9.
- Example: (16)⁻¹ᐟ⁴ = 1 / (16)¹ᐟ⁴ = 1 / (2⁴)¹ᐟ⁴ = 1 / 2¹ = 1/2.
Key Takeaways for Exams:
- Know the definitions and differences between N, W, Z, Q, and Irrational numbers.
- Understand the decimal representation of rational (terminating/recurring) and irrational (non-terminating, non-recurring) numbers.
- Be proficient in finding rational numbers between two given numbers.
- Master the technique of rationalizing the denominator.
- Memorize and apply the laws of exponents for real numbers with rational powers.
- Remember that π is irrational, while 22/7 is a rational approximation.
Multiple Choice Questions (MCQs)
-
Which of the following is an irrational number?
a) √16
b) √(12/3)
c) √12
d) 1.5 -
The decimal expansion of 1/7 is:
a) Terminating
b) Non-terminating recurring
c) Non-terminating non-recurring
d) None of the above -
Which of the following is a rational number between 1/4 and 1/2?
a) 1/8
b) 3/8
c) 5/8
d) 1/5 -
The value of (√5 + √2)(√5 - √2) is:
a) 7
b) 3
c) √3
d) 25 -
The simplest form of rationalizing the denominator of 1/√8 is:
a) √8 / 8
b) √2 / 4
c) √2 / 2
d) 4√2 -
The value of (64)¹ᐟ⁶ is:
a) 1
b) 2
c) 4
d) 8 -
Which statement 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. -
The product of any two irrational numbers is:
a) Always irrational
b) Always rational
c) Always an integer
d) Sometimes rational, sometimes irrational -
The decimal representation 0.323232... can be expressed in p/q form as:
a) 32/100
b) 32/99
c) 32/90
d) 29/90 -
The value of (3 + √3)(2 + √2) is:
a) 6 + 3√2 + 2√3 + √6
b) 6 + 5√5
c) 5 + √5 + √6
d) 11 + 5√6
Answers to MCQs:
- c) √12 (since √16=4, √(12/3)=√4=2, 1.5=3/2)
- b) Non-terminating recurring (denominator 7 has prime factors other than 2 or 5)
- b) 3/8 (1/4 = 2/8, 1/2 = 4/8; 3/8 lies between them)
- b) 3 (Using (a+b)(a-b) = a² - b² => (√5)² - (√2)² = 5 - 2 = 3)
- b) √2 / 4 (1/√8 = 1/(2√2) = (1*√2)/(2√2*√2) = √2/(2*2) = √2/4)
- b) 2 ((64)¹ᐟ⁶ = (2⁶)¹ᐟ⁶ = 2⁶ˣ¹ᐟ⁶ = 2¹)
- c) Every irrational number is a real number.
- d) Sometimes rational, sometimes irrational (Examples: √2 * √2 = 2 (rational); √2 * √3 = √6 (irrational))
- b) 32/99 (Let x = 0.3232...; 100x = 32.3232...; 100x - x = 32; 99x = 32; x = 32/99)
- a) 6 + 3√2 + 2√3 + √6 (Using distributive multiplication: 32 + 3√2 + √32 + √3√2)
Study these notes thoroughly. Practice applying the concepts, especially rationalization and laws of exponents. Good luck with your preparation!