NCERT maths || Chapter 1 || Concept overview ||

Chapter 1: Real Numbers — Concept Overview

1. Introduction

This chapter extends the study of real numbers by exploring foundational properties of positive integers, specifically focusing on prime factorisation, divisibility properties, and proofs of irrationality.

2. The Fundamental Theorem of Arithmetic

Core Theorem: Every composite number can be expressed (factorised) uniquely as a product of prime numbers, regardless of the order in which the prime factors occur.

Composite Number (x) = p₁^a × p₂^b × ... × p_n^k

Key Applications:

• Prime Factorisation: Expressing composite numbers as products of powers of primes.
• Calculating HCF and LCM:
  • HCF: Product of the smallest power of each common prime factor involved.
  • LCM: Product of the greatest power of each prime factor involved.
• Relationship for Two Numbers: For any two positive integers a and b:
  HCF(a, b) × LCM(a, b) = a × b
• Determining Ending Digits: Proving whether a number like 4ⁿ or 6ⁿ can end with the digit 0 (which requires prime factors 2 and 5).

3. Proofs of Irrationality

This section focuses on proving whether a given real number is irrational using the method of Proof by Contradiction.

Key Supporting Theorem:

Let p be a prime number. If p divides a² (where a is a positive integer), then p divides a.

Core Proof Types:

• Square Roots of Primes: Proving numbers like √2, √3, and √5 are irrational by assuming they can be written as coprime fractions a/b and deriving a contradiction regarding their common factors.
• Algebraic Combinations: Showing that expressions combining rational and irrational numbers (e.g., 5 - √3 or 3√2) are irrational.

4. Summary of Key Formulas & Identities

• Two-number HCF-LCM Relation: HCF(a, b) × LCM(a, b) = a × b
• Three-number Relation Note: HCF(p, q, r) × LCM(p, q, r) ≠ p × q × r