Chapter 1 — Real Numbers

Chapter 1 of Class 10 Maths, "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.

Class 10 Maths Chapter 1, Real Numbers, builds on Class 9 knowledge to explore two key results. The Fundamental Theorem of Arithmetic states that every composite number can be expressed as a product of primes in a unique way (apart from order), and is used to find HCF and LCM by prime factorisation. The chapter then applies this theorem to prove that √2, √3, and √5 are irrational using proof by contradiction, and explains why the sum or product of a rational and an irrational number is irrational.

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.