NCERT Solutions for Class 10 Maths Chapter 1: Real Numbers

Chapter 1 Real Numbers is the opening chapter of CBSE Class 10 Mathematics and forms the foundation for understanding number theory. The chapter is part of the Number Systems unit, which carries 6 marks in the CBSE Class 10 board exam 2027. Real Numbers is considered one of the most scoring chapters because the question types are predictable and the proofs follow a fixed structure that can be memorised.

What Are Real Numbers?

Real numbers include all rational numbers (numbers that can be written as p/q where q is not zero) and all irrational numbers (numbers that cannot be written as p/q, such as the square root of 2, the square root of 3, and pi). Together, rational and irrational numbers make up the entire number line. Every point on the number line corresponds to a real number and every real number corresponds to a point on the number line.

In earlier classes, students learned about natural numbers, whole numbers, integers, and rational numbers. Class 10 Chapter 1 builds on this foundation by introducing two powerful tools: the Fundamental Theorem of Arithmetic and the concept of proof by contradiction for irrational numbers.

Fundamental Theorem of Arithmetic

The Fundamental Theorem of Arithmetic states that every composite number can be expressed as a product of prime numbers in a unique way, apart from the order of factors. For example, 140 can only be written as 2 squared times 5 times 7. No other combination of primes will give 140. This uniqueness is what makes the theorem so powerful.

This theorem is the backbone of Chapter 1. It is used in three major applications in this chapter. First, it is used to find the HCF (Highest Common Factor) and LCM (Lowest Common Multiple) of two or more numbers through prime factorisation. To find the HCF, we take the lowest power of each common prime factor. For LCM, we take the highest power of every prime factor that appears in any of the numbers.

A key property students must remember is that HCF multiplied by LCM equals the product of the two numbers. However, this identity holds only for exactly two numbers, not for three or more. This is one of the most common mistakes students make in board exams and is frequently tested in MCQs and assertion-reason questions.

Second, the Fundamental Theorem is used to determine whether the decimal expansion of a rational number is terminating or non-terminating repeating. A rational number p/q (in its lowest terms) has a terminating decimal expansion if and only if the prime factorisation of q is of the form 2 raised to n times 5 raised to m, where n and m are non-negative integers. If q has any prime factor other than 2 or 5, the decimal expansion will be non-terminating repeating.

Third, the theorem is used as a key step in proofs of irrationality. When we assume that the square root of a prime number is rational and arrive at a contradiction, the Fundamental Theorem guarantees that if a prime p divides a squared, then p must divide a. This logical step is essential for completing irrationality proofs.

Proofs of Irrationality

The second major topic in Chapter 1 is proving that certain numbers are irrational. Students learn to use the method of proof by contradiction, which is one of the most elegant techniques in mathematics.

The structure of an irrationality proof follows these steps. First, assume to the contrary that the number (say the square root of 5) is rational. Write it as p/q where p and q are co-prime integers and q is not zero. Second, square both sides to get p squared equals 5 times q squared. Third, conclude that 5 divides p squared, and by the Fundamental Theorem, 5 must divide p. Fourth, let p equal 5k for some integer k, substitute back, and show that 5 also divides q. Fifth, since both p and q are divisible by 5, this contradicts the assumption that they are co-prime. Sixth, conclude that the square root of 5 is irrational.

This proof structure appears in virtually every CBSE board exam as a 3-mark question. Students should memorise the six-step structure and practice writing it in under 5 minutes.

In recent years from 2023 to 2026, CBSE has evolved the irrationality question pattern. Instead of asking students to prove that the square root of 2 is irrational from scratch, they now provide that the square root of 5 or the square root of 2 or the square root of 3 is irrational and ask students to prove that a composite expression like 4 minus 2 times the square root of 5 or (3 plus the square root of 7) divided by 5 is irrational. This variation is actually simpler. It requires just 4 to 5 lines where you assume the composite expression is rational, rearrange to isolate the irrational term, show it equals a rational expression, and note the contradiction.

Euclid's Division Algorithm (For State Boards)

While Euclid's Division Algorithm has been removed from the CBSE 2026-27 syllabus, it remains part of the curriculum for state boards including UP Board, Bihar Board, MP Board, and Rajasthan Board. Euclid's Division Lemma states that for any two positive integers a and b, there exist unique integers q (quotient) and r (remainder) such that a equals b times q plus r, where r is greater than or equal to 0 and less than b.

The algorithm is a systematic way to find the HCF of two numbers by repeatedly applying the division lemma until the remainder becomes zero. The last non-zero remainder is the HCF. This method is particularly useful for large numbers where prime factorisation would be tedious.

Students studying under state boards should practice both Euclid's method and the prime factorisation method, as either can appear in their exams.

CBSE 2026-27 Syllabus Changes

For the CBSE 2026-27 academic session, Chapter 1 Real Numbers has been trimmed to contain only two exercises. Exercise 1.1 covers the Fundamental Theorem of Arithmetic with 7 questions on prime factorisation, HCF, LCM, composite numbers, and word problems. Exercise 1.2 covers proofs of irrationality with 3 questions.

The older exercises on Euclid's Division Algorithm (previously Exercise 1.1) and Decimal Expansions of Rational Numbers (previously Exercise 1.4) have been removed from the CBSE syllabus. These topics will not appear in the CBSE Class 10 board exam 2027. However, Padhaao provides solutions for these older exercises as well, since state board students in UP, Bihar, MP, Rajasthan, and Haryana may still follow the four-exercise format.

How This Chapter Is Tested in Board Exams

Real Numbers questions in the CBSE board exam typically follow this pattern. One MCQ worth 1 mark tests basic concepts like HCF, LCM, prime factorisation properties, or the nature of decimal expansions. One short answer question worth 2 marks asks students to find the HCF and LCM of two or three numbers using prime factorisation, or to verify that HCF times LCM equals the product of the numbers. One long answer question worth 3 marks requires proving the irrationality of a number using proof by contradiction.

Since 2023, CBSE has also introduced case study questions based on real-life scenarios involving prime factorisation. These carry 4 to 5 marks and have 3 to 4 sub-questions. Assertion-Reason questions, which test conceptual understanding without requiring calculations, also feature regularly. Students should practice both formats as they are relatively new additions to the exam pattern.

Tips for Scoring Full Marks

To score all 6 marks from this chapter, students should focus on three things. First, memorise the irrationality proof structure and practice writing proofs for the square root of 2, 3, 5, and composite forms. Second, practice finding HCF and LCM by prime factorisation quickly and accurately, always showing the complete factor tree. Third, solve at least 10 to 15 previous year board questions from this chapter to get familiar with how the examiner phrases questions and awards marks.

Common mistakes to avoid include using the HCF times LCM formula for three numbers (it only works for two), not mentioning the co-prime condition when starting an irrationality proof, confusing HCF (lowest powers of common primes) with LCM (highest powers of all primes), and not simplifying fractions to lowest terms before checking for terminating decimals.

Connection to Other Chapters

Real Numbers connects to several other chapters in Class 10 Maths. The concept of prime factorisation reappears in Chapter 2 (Polynomials) when discussing the relationship between zeros and coefficients. The proof by contradiction technique used for irrationality proofs is a general mathematical method that develops logical thinking required throughout higher mathematics. Understanding HCF and LCM is useful for solving word problems in Chapters 3 (Linear Equations) and 5 (Arithmetic Progressions).

For students preparing for competitive exams like NTSE, Olympiads, or JEE Foundation, the number theory concepts in this chapter form an essential base. Euclid's Algorithm, in particular, has applications in computer science (encryption algorithms) and is worth understanding even if it is not in the CBSE syllabus.