CBSE Class 10 Mathematics – Chapter 1: Real
Numbers
Complete Study Notes
Introduction
Real numbers include rational and irrational numbers. They form the foundation of mathematics.
Euclid's Division Lemma
For any two positive integers a and b, there exist unique whole numbers q and r such that a = bq +
r, where 0 ≤ r < b.
Euclid's Division Algorithm
Used to find the HCF of two numbers by repeated division.
Fundamental Theorem of Arithmetic
Every composite number can be expressed as a product of prime numbers, and this factorization is
unique.
HCF and LCM
HCF × LCM = Product of the two numbers.
Irrational Numbers
Numbers such as √2, √3 and π cannot be expressed as p/q.
Decimal Expansion of Rational Numbers
A rational number has a terminating decimal expansion only if its denominator is of the form 2■ ×
5■.
Important Questions
1. Find HCF using Euclid's Algorithm. 2. Prove √5 is irrational. 3. Find HCF and LCM using prime
factorization.
Numbers
Complete Study Notes
Introduction
Real numbers include rational and irrational numbers. They form the foundation of mathematics.
Euclid's Division Lemma
For any two positive integers a and b, there exist unique whole numbers q and r such that a = bq +
r, where 0 ≤ r < b.
Euclid's Division Algorithm
Used to find the HCF of two numbers by repeated division.
Fundamental Theorem of Arithmetic
Every composite number can be expressed as a product of prime numbers, and this factorization is
unique.
HCF and LCM
HCF × LCM = Product of the two numbers.
Irrational Numbers
Numbers such as √2, √3 and π cannot be expressed as p/q.
Decimal Expansion of Rational Numbers
A rational number has a terminating decimal expansion only if its denominator is of the form 2■ ×
5■.
Important Questions
1. Find HCF using Euclid's Algorithm. 2. Prove √5 is irrational. 3. Find HCF and LCM using prime
factorization.
