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Mathematics: Class 10 — Chapter 1: Real Numbers

1.1 Introduction
1.2 The Fundamental Theorem of Arithmetic

Theorem 1.1 (Fundamental Theorem of Arithmetic)

  • 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.
  • Prime Factorisation: Writing a composite number as a product of powers of primes (e.g., 32760 = 2³ × 3² × 5 × 7 × 13).

HCF and LCM Relations

  • HCF (Highest Common Factor): Product of the smallest power of each common prime factor in the numbers.
  • LCM (Lowest Common Multiple): Product of the greatest power of each prime factor involved in the numbers.
  • Two-Number Formula: For any two positive integers a and b, HCF(a, b) × LCM(a, b) = a × b.
  • Three-Number Note: The product of three numbers is NOT equal to the product of their HCF and LCM (i.e., HCF(a,b,c) × LCM(a,b,c) ≠ a × b × c).
1.3 Revisiting Irrational Numbers

Theorems and Proofs

  • Irrational Number: A number that cannot be written in the form p/q, where p and q are integers and q ≠ 0.
  • Theorem 1.2: Let p be a prime number. If p divides a², then p divides a, where a is a positive integer.
  • Theorem 1.3: √2 is irrational (proven using proof by contradiction).

Properties of Irrational Numbers

  • The sum or difference of a rational and an irrational number is irrational (e.g., 5 - √3).
  • The product and quotient of a non-zero rational and irrational number is irrational (e.g., 3√2).
1.4 Summary & Advanced Formulas