📘 Class 10 · Mathematics

Chapter 1: Real Numbers

6 lessons

Real Numbers opens Class 10 by proving results Class 9 only stated: Euclid's Division Lemma and its use in finding the HCF of two numbers, the Fundamental Theorem of Arithmetic (every number has one unique prime factorisation), why some decimal expansions terminate and others don't, and formal irrationality proofs for numbers like √2.

Introduction — Euclid's Division Lemma

Introduction of Euclid's Division Lemma and its applications.

Exercise 1.1 — Division Lemma Problems

Problems based on Euclid's Division Lemma.

Exercise 1.2 — Fundamental Theorem

Fundamental theorem of arithmetic, LCM and HCF.

Exercise 1.3 — Decimal Expansion

Rational numbers and their decimal expansion.

Exercise 1.4 — Irrationality Proofs

Proving irrationality of numbers using the contradictory method.

Exercise 1.5 — Logarithms

Logarithms, their properties and related problems.

Setting the tone for the year

This chapter is deliberately proof-heavy — confusing the Division Lemma (a=bq+r) with ordinary long division is a common early mix-up, since the lemma is a starting statement for a proof, not a calculation in itself. An irrationality proof also needs a clear "assume the opposite" contradiction structure to actually hold up: assume √2 is rational, show that assumption leads to a contradiction, and conclude it must be irrational after all. The prime-factorisation technique here is reused directly in this year's Polynomials and Quadratic Equations chapters.