Real Numbers

Board: GSEB Class: 10

Chapter Overview

Real numbers include rational and irrational numbers. Euclid's division lemma states: for positive integers a and b, there exist unique integers q and r such that a = bq + r, where 0 <= r < b. This lemma is used to compute HCF of two numbers. The Fundamental Theorem of Arithmetic states that every composite number can be expressed uniquely as a product of primes, up to order. This theorem is used to find LCM and HCF of positive integers. The chapter also covers the irrationality of numbers like √(2), √(3), √(5) using proof by contradiction, and the decimal expansions of rational numbers (terminating and non-terminating repeating). GSEB board exams frequently ask proof of irrationality and HCF/LCM word problems.

Topics Covered

Euclid Division Lemma
Euclid Algorithm for HCF
Fundamental Theorem of Arithmetic
Prime Factorization Method
LCM and HCF
Irrational Numbers Proof
Decimal Expansions
Rational vs Irrational

Key Formulas

a = bq + r, 0 <= r < b

HCF x LCM = a x b

LCM = product of highest powers of primes

HCF = product of lowest powers of common primes

Real-World Applications

Applications: Computer algorithms (GCD), cryptography (prime factorization), number theory foundations.

Study Tips

Tip: Practice Euclid division algorithm step by step

Tip: Master prime factorization method for HCF/LCM

Tip: Solve GSEB previous year irrationality questions