Chapter 1 — Real Numbers

Chapter 1 of the Class 10 Maths NCERT textbook, "Real Numbers", covers the Fundamental Theorem of Arithmetic — which states every composite number has a unique prime factorisation — and uses it to prove that numbers such as √2 and √3 are irrational.

• Unique prime factorisation — The Fundamental Theorem of Arithmetic guarantees that every composite number breaks down into primes in exactly one way, apart from order. This uniqueness is what makes prime factorisation a reliable tool for finding HCF and LCM.
• HCF and LCM through primes — Once numbers are written as products of primes, the HCF collects the smallest shared prime powers and the LCM the largest powers. For two numbers this ties together neatly as HCF × LCM = the product of the numbers.
• Proving irrationality — By assuming a number like √2 is rational and reaching a contradiction with unique factorisation, the chapter shows such roots cannot be expressed as fractions. The same reasoning extends to combinations like 5 − √3.

1. 01The Fundamental Theorem of Arithmetic: every composite number has a unique prime factorisation (apart from the order of factors).
2. 02HCF of two numbers equals the product of the smallest powers of all common prime factors; LCM equals the product of the greatest powers of all prime factors involved.
3. 03For any two positive integers a and b, HCF(a, b) × LCM(a, b) = a × b (this does NOT extend to three numbers).
4. 04If a prime p divides a², then p divides a (Theorem 1.2) — the key lemma used to prove irrationality.
5. 05√2 and √3 are proved irrational using proof by contradiction together with the Fundamental Theorem of Arithmetic.
6. 06Expressions such as 5 − √3 and 3√2 are irrational because assuming otherwise leads to a contradiction with the irrationality of √3 or √2.