Real Numbers (Class 10) — SOF IMO Study Notes

Overview

Real numbers form the foundation of Class 10 mathematics and are crucial for SOF IMO success. This topic introduces rigorous number theory concepts that underpin algebra, geometry and higher mathematics. You must master three core pillars: Euclid's division lemma (a tool for finding HCF and proving divisibility), the fundamental theorem of arithmetic (unique prime factorization), and properties of irrational numbers (non-terminating, non-repeating decimals).

IMO questions test these concepts through proof-based problems, HCF-LCM calculations, rationality tests and multi-step reasoning puzzles. Unlike school exams that focus on direct formula application, IMO emphasizes understanding why algorithms work and applying them to unfamiliar scenarios. Strong command of divisibility arguments and prime factorization logic will help you tackle both Mathematical Reasoning and Achievers Section problems.

Expect 2–4 questions directly from this topic in the IMO paper, plus indirect applications in number puzzles and logical reasoning sections. Master the theory first, then practice variations to build speed and accuracy.

Key Concepts

• **Euclid's Division Lemma**: For any two positive integers a and b, there exist unique integers q (quotient) and r (remainder) such that a = bq + r, where 0 ≤ r < b. This is the foundation of the division algorithm and HCF computation.

• **Euclid's Division Algorithm**: To find HCF of two numbers, repeatedly apply the lemma: divide the larger by the smaller, replace the larger with the smaller and the smaller with the remainder, until remainder becomes zero. The last non-zero remainder is the HCF.

• **Fundamental Theorem of Arithmetic**: Every composite number can be expressed as a product of primes in exactly one way (ignoring order). This uniqueness property is central to solving HCF, LCM and divisibility problems.

• **Prime Factorization Method**: Express numbers as products of prime powers (e.g. 72 = 2³ × 3²). HCF is the product of lowest powers of common primes; LCM is the product of highest powers of all primes present.

• **Rational Numbers**: Numbers expressible as p/q where p, q are integers and q ≠ 0. Their decimal expansions either terminate or repeat periodically.

• **Irrational Numbers**: Numbers that cannot be expressed as p/q. Their decimal expansions are non-terminating and non-repeating. Examples: √2, √3, π, e.

• **Proving Irrationality**: Use contradiction — assume the number is rational (p/q in lowest terms), derive that both p and q share a common factor, contradicting the assumption that p/q is in lowest terms.