The chapter Real Numbers extends the concept of numbers studied in earlier classes. It discusses the Euclidean Division Lemma, the Fundamental Theorem of Arithmetic, and their applications to problems involving HCF and LCM, as well as the representation of rational numbers as terminating/non-terminating decimals. It also includes the proof of the irrationality of numbers like $sqrt{2}, quad sqrt{3}$, etc. This chapter sets the base for number theory and algebraic thinking.
II. Key Concepts Covered
Concept
Description
Euclid’s Division Lemma
For any two positive integers a and b, there exist unique integers q and r such that:
$a = bq + r, quad text{where } 0 leq r < b$
Euclidean Algorithm
Used to find the HCF of two numbers using Euclid’s Division Lemma.
Fundamental Theorem of Arithmetic
Every composite number can be expressed uniquely (apart from the order) as a product of prime numbers.
LCM and HCF
Product of two numbers = LCM × HCF.
Irrational Numbers
Numbers that cannot be expressed as $frac{p}{q}$ form. E.g., $sqrt{2}, sqrt{3}, pi$.
Proof of Irrationality
Proofs for the irrationality of numbers like $sqrt{2}, sqrt{3}$, and $sqrt{2}$ using contradiction method.
Decimal Expansions
– Terminating decimals → when denominator is in the form $2^m times 5^n$
III. Important Questions
(A). Multiple Choice Questions (1 Mark)
Q1. Which of the following is an irrational number? a) $frac{22}{7}$ b) $sqrt{2}$✅ (PYQ 2020) c) 0.25 d) 5
Q2. If the decimal expansion of a rational number terminates, then the prime factorization of the denominator has: a) Only 2 b) Only 5 c) Only 2 and/or 5 ✅ (PYQ 2021) d) Neither 2 nor 5
Q3. What is the HCF of 16 and 24 using Euclid’s division lemma? a) 4 ✅ b) 6 c) 8 d) 2
Q4. Which of the following numbers has a non-terminating, repeating decimal expansion? a) $frac{1}{2}$ b) $frac{3}{5}$ c) $frac{1}{7}$ ✅ (PYQ 2019) d) $frac{5}{10}$
(B). Short Answer Questions (2/3 Marks)
Use Euclid’s division lemma to find the HCF of 4052 and 12576.
Show that $sqrt{3}$is an irrational number.(PYQ 2021)
Find the LCM and HCF of 6 and 20 using the prime factorization method. Also verify that $text{LCM} times text{HCF} =$ Product of the two numbers.
Check whether $frac{7}{80}$ has a terminating or non-terminating decimal expansion.
(C). Long Answer Questions (5 Marks)
Prove that $sqrt{5}$ is irrational. Use a method of contradiction.
Using Euclid’s division algorithm, find the HCF of 960 and 4320.
Express 180 as a product of its prime factors. Using the result, find whether the decimal expansion of $frac{7}{180}$ is terminating or non-terminating.
Find the HCF and LCM of 60 and 72 using prime factorization and verify the identity: LCM × HCF = Product of two numbers
(D). HOTS (Higher Order Thinking Skills)
Prove that the square of any positive integer is either divisible by 3 or leaves a remainder 1 when divided by 3.
Can two irrational numbers be added to give a rational number? Justify your answer with an example.
IV. Key Formulas/Concepts
Concept/Formula
Application
Euclid’s Division Lemma:
$a = bq + r, quad text{where } 0 leq r < b$
Used to find HCF of two numbers
Fundamental Theorem of Arithmetic
Unique prime factorization of composite numbers
LCM × HCF = Product of Numbers
To cross-check correctness of HCF and LCM
Terminating Decimal
If denominator of rational number (in simplest form) is of the form $2^m times 5^n$
Irrational Numbers
Cannot be expressed as a fraction of two integers and have non-repeating, non-terminating decimal expansion
V. Deleted Portions (CBSE 2025–2026)
No portions have been deleted from this chapter as per the rationalized NCERT textbooks.
VI. Chapter-Wise Marks Bifurcation (Estimated – CBSE 2025–2026)
Unit/Chapter
Estimated Marks
Types of Questions Typically Asked
Chapter 1: Real Numbers
6–8 Marks
1 Long Answer, 1 Short Answer, 1–2 MCQs
VII. Previous Year Questions (PYQs)
Year
Marks
Question
2021
2
Show that $sqrt{3}$ is irrational.
2020
1
Which of the following is irrational? $(sqrt{2})$
2019
1
Which of the following is non-terminating repeating? $frac{1}{7}$
2018
5
Using Euclid’s algorithm, find the HCF of 4052 and 12576.
VIII. Real-World Application Examples to Connect with Topics
Concept
Real-Life Application
Prime Factorization
Cryptography, data compression
HCF & LCM
Synchronization of repeating events like traffic lights, calendar problems
Irrational Numbers
Geometry (e.g., diagonal of square = √2), engineering design
