Chapter 1: Real Numbers (वास्तविक संख्याएँ)

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Chapter 1: Real Numbers (वास्तविक संख्याएँ)

Class 10 – CBSE Basic Maths

Quick Summary for Exam: This chapter focuses on the Fundamental Theorem of Arithmetic, finding HCF/LCM using prime factorisation, and proving irrationality (specifically √2, √3).

1. Basic Concepts (मूल अवधारणाएँ)

Before we start the theorems, let’s revise two types of numbers essential for this chapter.

1. Prime Numbers (अभाज्य संख्याएँ)

  • English: Numbers that have only two factors: 1 and the number itself.
  • Hindi: वे संख्याएँ जिनके केवल दो गुणनखंड होते हैं: 1 और वह संख्या स्वयं।
  • Examples: 2, 3, 5, 7, 11, 13, 17…
    • Note: 1 is neither prime nor composite (1 न तो भाज्य है और न ही अभाज्य).

2. Composite Numbers (भाज्य संख्याएँ)

  • English: Numbers that have more than two factors.
  • Hindi: वे संख्याएँ जिनके दो से अधिक गुणनखंड होते हैं।
  • Examples: 4, 6, 8, 9, 10…

2. Fundamental Theorem of Arithmetic (अंकगणित की आधारभूत प्रमेय)

This is the main concept of Section 1.2.

Statement:

“Every composite number can be expressed (factorised) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur.”

Hindi Explanation: “हर भाज्य (Composite) संख्या को अभाज्य (Prime) संख्याओं के गुणनफल के रूप में लिखा जा सकता है। यह गुणनखंडन अद्वितीय (unique) होता है, बस हम संख्याओं का क्रम (order) बदल सकते हैं।”

Example: Factorise 140 (140 का अभाज्य गुणनखंडन करें):

140 = 2 × 70 = 2 × 2 × 35 = 2 × 2 × 5 × 7

Result: 140 = 2² × 5 × 7

3. HCF and LCM by Prime Factorisation

For Basic Maths students, this is the most important topic to practice.

A. How to find HCF (Highest Common Factor / म.स.प)

  • Definition: Product of the smallest power of each common prime factor in the numbers.
  • Hindi: संख्याओं में प्रत्येक उभयनिष्ठ (common) अभाज्य गुणनखंड की सबसे छोटी घात का गुणनफल।

B. How to find LCM (Lowest Common Multiple / ल.स.प)

  • Definition: Product of the greatest power of each prime factor involved in the numbers.
  • Hindi: संख्याओं में संबद्ध प्रत्येक अभाज्य गुणनखंड की सबसे बड़ी घात का गुणनफल।

Example:

Find HCF and LCM of 6 and 20.

  • 6 = 2¹ × 3¹
  • 20 = 2² × 5¹

HCF Calculation: Common factor is 2. The smallest power is 2¹. HCF = 2

LCM Calculation: All factors involved are 2, 3, 5. The greatest powers are 2², 3¹, 5¹. LCM = 2² × 3¹ × 5¹ = 4 × 3 × 5 = 60

4. Relationship between HCF and LCM (महत्वपूर्ण सूत्र)

Formula: For any two positive integers a and b:

HCF(a, b) × LCM(a, b) = a × b

Hindi: म.स.प × ल.स.प = पहली संख्या × दूसरी संख्या

Note: This formula is valid ONLY for two numbers. It does not work for three numbers.

Typical Question for Basic Maths: Given HCF(306, 657) = 9, find LCM(306, 657).

Solution: We know that LCM × HCF = Product of numbers.

LCM × 9 = 306 × 657 LCM = (306 × 657) / 9 LCM = 34 × 657 LCM = 22338

5. Irrational Numbers (अपरिमेय संख्याएँ)

Definition:

  • English: A number is called irrational if it cannot be written in the form p/q, where p and q are integers and q ≠ 0.
  • Hindi: वह संख्या जिसे p/q के रूप में नहीं लिखा जा सकता, अपरिमेय संख्या कहलाती है।
  • Examples: √2, √3, √5, π, 0.10110…

Important Theorems & Properties

  1. Theorem: Let p be a prime number. If p divides a², then p divides a. (अगर कोई अभाज्य संख्या p, a² को विभाजित करती है, तो वह a को भी विभाजित करेगी।)
  2. Operations:
    • Rational + Irrational = Irrational (e.g., 3 + √5)
    • Rational × Irrational = Irrational (e.g., 2√3)

Proof Strategy: √2 is Irrational

This is a guaranteed question type. Memorize these steps.

  1. Step 1: Assume √2 is rational (√2 = a/b). (माना कि √2 एक परिमेय संख्या है।)
  2. Step 2: Square both sides: 2 = a²/b² ⇒ 2b² = a².
  3. Step 3: This means 2 divides a², so 2 divides a. (इसका मतलब 2, a को विभाजित करता है।)
  4. Step 4: Let a = 2c. Substitute it back: 2b² = (2c)² 2b² = 4c² b² = 2c²
  5. Step 5: This means 2 divides b², so 2 divides b. (इसका मतलब 2, b को भी विभाजित करता है।)
  6. Conclusion: 2 is a common factor of both a and b. This contradicts our assumption that they are co-prime (have no common factor). Therefore, √2 is irrational. (अतः, हमारी मान्यता गलत थी। √2 एक अपरिमेय संख्या है।)

6. Summary for Exam (परीक्षा के लिए सारांश)

  1. Prime Factorisation: You must know how to make a factor tree.
  2. HCF vs LCM: Remember HCF is “Lowest powers” and LCM is “Highest powers”.
  3. Formula: Use HCF × LCM = Product to find the missing value.
  4. Proof: Learn the proof of √2 or √3 by heart.
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