India's #1 AI Tutortopic article · Mathematics

Real Numbers for Class 10: The Complete CBSE Guide (2026-27)

Name: CBSETUTOR.ai subscription
Brand: CBSETUTOR.ai
SKU: cbsetutor-ai-monthly
Price: 9999 INR
Availability: InStock
Rating: 4.8 (284 reviews)

Real Numbers Class 10 is the opening chapter of CBSE Class 10 Mathematics and arguably the most calculation-intensive topic you will encounter this year. Every year, the CBSE board exam sets exactly 6 marks from this chapter—one short answer (2 marks), one long answer (3 marks), and one MCQ or assertion-reason (1 mark). The chapter revolves around four core topics from NCERT: the Fundamental Theorem of Arithmetic, properties and proofs of irrational numbers, decimal expansion patterns of rational numbers, and finding HCF and LCM through prime factorisation. Mastering Real Numbers Class 10 is critical not just for these 6 marks, but because prime factorisation, LCM, and HCF reappear in polynomials, algebraic identities, and even coordinate geometry later in the syllabus.

Your child's private AI tutor — trained on NCERT.

3-day free trial · ₹1 to start · Cancel anytime.

Start 3-day free trial →

Key takeaways

✓Real Numbers Class 10 carries 6 marks in the CBSE board exam with predictable question patterns on HCF-LCM, decimal expansions, and irrationality proofs.
✓The Fundamental Theorem of Arithmetic states every composite number has exactly one prime factorisation (order doesn't matter), which is the basis for all HCF-LCM problems.
✓Euclid's division lemma (a = bq + r, 0 ≤ r < b) is the fastest method to find HCF of large numbers and appears in 80% of Real Numbers Class 10 board questions.
✓A rational number p/q has a terminating decimal expansion only if the prime factorisation of q (in lowest terms) contains only 2s and 5s.
✓To prove a number is irrational, assume it is rational (p/q in lowest terms), derive a contradiction showing p and q share a common factor, thus violating the assumption.
✓HCF × LCM = Product of two numbers is the most tested formula in Real Numbers Class 10, appearing in 3-mark application problems.
✓CBSETUTOR.ai's 24×7 AI tutor has solved every NCERT Real Numbers exercise with step-by-step working at ₹999/month for all classes 6-12, helping students upload and solve any worksheet instantly.

What are Real Numbers? The NCERT Class 10 Definition

Real Numbers Class 10 builds on what you learned in Class 9. Real numbers include all rational numbers (numbers expressible as p/q where p and q are integers and q ≠ 0) and all irrational numbers (numbers that cannot be expressed as p/q, like √2, √3, π). The union of these two sets forms the complete real number system. NCERT Class 10 Maths Chapter 1 does not dwell on the definition itself—it assumes you know rational and irrational classifications—and instead focuses on properties and structure. The key insight is that between any two real numbers, there exist infinitely many rational and infinitely many irrational numbers. This density property is what makes the real number line continuous. In CBSE exams, you will never be asked to define real numbers; instead, you will be asked to prove whether a given number is rational or irrational, find its decimal expansion type, or compute HCF and LCM.

Rational numbers: integers, fractions, terminating decimals (0.75), and non-terminating repeating decimals (0.333...)
Irrational numbers: non-terminating non-repeating decimals such as √2 = 1.41421356... or π = 3.14159265...
Real numbers = Rational ∪ Irrational (every point on the number line)
CBSE board exams test classification, decimal expansion patterns, and proof of irrationality—not the definition itself

Fundamental Theorem of Arithmetic: The Backbone of Real Numbers Class 10

The Fundamental Theorem of Arithmetic is the single most important concept in Real Numbers Class 10. It states: 'Every composite number can be expressed (factorised) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur.' For example, 60 = 2 × 2 × 3 × 5 or 60 = 2² × 3 × 5. Whether you write 2 × 3 × 2 × 5 or 3 × 5 × 2 × 2, the prime factors remain the same. This uniqueness is what allows us to define HCF and LCM unambiguously. CBSE examiners love to test this theorem indirectly: you will be given two or three numbers and asked to find their HCF or LCM using prime factorisation, or prove that a number like √2 or √3 is irrational by leveraging the uniqueness of prime factorisation. In the 2024 CBSE Class 10 Maths paper, a 3-mark question asked students to use the Fundamental Theorem to show that 7√5 is irrational—precisely the type of application you must practice.

Prime Factorisation Method: How to Break Down Any Number

Prime factorisation is the process of expressing a composite number as a product of prime numbers. For Real Numbers Class 10, you must master the factor tree or division method. Start by dividing the number by the smallest prime (2), continue dividing by 2 until the quotient is odd, then move to the next prime (3, 5, 7,...). Stop when you reach 1. Write the number as the product of all divisors. This method is the foundation for finding HCF and LCM. In CBSE exams, you will often see questions like 'Find the HCF and LCM of 72 and 120 using prime factorisation'—the first step is always to write each number in its prime factorised form. NCERT Real Numbers Class 10 Exercise 1.2 has six such problems, and this pattern accounts for roughly 40% of the marks from this chapter.

Step 1: Divide the number by 2 repeatedly until you get an odd quotient.
Step 2: Divide by 3 repeatedly, then by 5, 7, 11,... (only primes).
Step 3: Stop when the quotient is 1. The product of all divisors is the prime factorisation.
Write in exponential form: e.g. 360 = 2³ × 3² × 5.

HCF and LCM by Prime Factorisation: The CBSE Formula-Based Approach

Once you have the prime factorisations of two or more numbers, finding HCF (Highest Common Factor) and LCM (Lowest Common Multiple) becomes a matter of applying two simple rules. HCF is the product of the smallest power of each common prime factor. LCM is the product of the highest power of each prime factor present in any of the numbers. For example, if you have 72 = 2³ × 3² and 120 = 2³ × 3 × 5, the HCF is 2³ × 3¹ = 24 (take the lower exponent for each common prime), and the LCM is 2³ × 3² × 5 = 360 (take the higher exponent for each prime). The relationship HCF × LCM = Product of the two numbers is tested in almost every CBSE Real Numbers Class 10 board paper. In 2023, a 3-mark question asked students to verify this relation for 56 and 72—expect similar application problems.

Euclid's Division Algorithm: The Fastest Way to Find HCF

Euclid's division lemma states that for any two positive integers a and b (a > b), there exist unique integers q and r such that a = bq + r, where 0 ≤ r < b. If r = 0, then b is the HCF. Otherwise, apply the lemma again with b and r. Repeat until the remainder is zero. The divisor at that step is the HCF. This algorithm is faster than prime factorisation for large numbers and is the method CBSE examiners prefer for 2-mark and 3-mark HCF questions in Real Numbers Class 10. NCERT presents this in Exercise 1.1 with detailed examples. For instance, to find HCF of 867 and 255: 867 = 255 × 3 + 102, then 255 = 102 × 2 + 51, then 102 = 51 × 2 + 0. Hence HCF = 51. Practice this method until you can execute it without error under exam pressure.

Proving Irrationality: The Contradiction Method for Real Numbers Class 10

A number is irrational if it cannot be expressed as p/q (where p and q are integers with no common factors and q ≠ 0). The standard CBSE proof technique is proof by contradiction. Assume the number is rational, express it as p/q in lowest terms, manipulate the equation to show that both p and q must share a common factor (violating the assumption that they are coprime), hence the assumption is false and the number is irrational. NCERT Real Numbers Class 10 proves √2, √3, √5 are irrational using this method. In board exams, you may be asked to prove numbers like 3 + 2√5, 7√5, or 1/√2 are irrational. The key is to assume rationality, square or manipulate to isolate the surd, then show a rational equals an irrational (contradiction). These proofs typically carry 3 marks and require every logical step to be written clearly.

Decimal Expansions of Rational Numbers: Terminating vs Non-Terminating Repeating

Every rational number p/q (in lowest terms) has either a terminating decimal expansion or a non-terminating repeating (recurring) decimal expansion. The NCERT theorem states: p/q has a terminating decimal if and only if the prime factorisation of q is of the form 2ⁿ5ᵐ (where n, m ≥ 0). If q has any prime factor other than 2 or 5, the decimal expansion is non-terminating repeating. For example, 7/8 = 7/(2³) = 0.875 (terminating), but 7/12 = 7/(2² × 3) has a factor of 3, so it is non-terminating repeating: 0.58333... This concept accounts for 1–2 marks in every Real Numbers Class 10 board exam, typically as an MCQ or a 2-mark justify-your-answer question. You must reduce p/q to lowest terms before checking the prime factors of q.

Real Numbers Class 10 Formulas: The Must-Know List

Although Real Numbers Class 10 is more proof and logic-heavy than formula-heavy, there are a few relationships you must memorise. First, HCF(a, b) × LCM(a, b) = a × b for any two positive integers a and b—this is tested in application problems where you are given HCF and one number, and asked to find the other number or LCM. Second, for Euclid's division lemma: a = bq + r (0 ≤ r < b). Third, the condition for a terminating decimal: q (in lowest terms) must be 2ⁿ5ᵐ. Fourth, √p is irrational if p is not a perfect square and is prime. These are not formulas in the algebraic sense, but they are the rules you will apply mechanically in CBSE exams. Write them on the first page of your answer sheet for quick reference during the exam.

HCF × LCM = Product of two numbers (most tested relation)
Euclid's lemma: a = bq + r, 0 ≤ r < b
Terminating decimal iff q = 2ⁿ5ᵐ in lowest terms
√p is irrational if p is prime or not a perfect square
For HCF by prime factorisation: take minimum exponent of common primes
For LCM by prime factorisation: take maximum exponent of all primes

Real Numbers Class 10 Notes: Chapter Summary and Revision Points

Your Real Numbers Class 10 notes should be a one-page reference sheet covering definitions, theorems, methods, and sample questions. Write the Fundamental Theorem statement, the Euclid division algorithm steps, the terminating decimal test, and a template for proving irrationality. Include one worked example of HCF-LCM by prime factorisation, one Euclid's algorithm example, and one irrationality proof. Annotate common errors: forgetting to reduce p/q to lowest terms before checking decimal type, or omitting the contradiction step in irrationality proofs. Many students lose 1–2 marks because they write 'assume √2 is rational' but never explicitly state 'this contradicts our assumption'. Your notes should highlight these pitfalls. CBSETUTOR.ai users can upload their handwritten notes or any worksheet, and the AI tutor will instantly verify working, point out gaps, and provide alternate methods—all at ₹999/month for Classes 6–12 with a 3-day free trial (no card needed).

Section 1: Fundamental Theorem of Arithmetic—statement and two examples
Section 2: HCF and LCM by prime factorisation—method and verification formula
Section 3: Euclid's algorithm—step-by-step template with one large-number example
Section 4: Decimal expansions—2ⁿ5ᵐ rule and three test fractions
Section 5: Proving irrationality—contradiction structure and √2, √3, √5 proofs
Section 6: Common mistakes—forgetting coprime assumption, skipping contradiction line

NCERT Real Numbers Class 10 Exercise-Wise Breakdown

NCERT Real Numbers Class 10 contains four exercises: Exercise 1.1 (Euclid's division algorithm, 5 questions), Exercise 1.2 (Prime factorisation, HCF, LCM, 7 questions), Exercise 1.3 (Decimal expansions, 3 questions), and Exercise 1.4 (Proving irrationality and miscellaneous, 3 questions). Exercise 1.1 builds fluency with Euclid's method—every student should solve all five problems without a calculator to develop speed. Exercise 1.2 is the most important for board exams; it includes multi-step HCF-LCM word problems (e.g. two numbers have HCF 17 and product 714, find the numbers). Exercise 1.3 tests your understanding of the 2ⁿ5ᵐ rule—expect one direct question here in the board paper. Exercise 1.4 is where you practise irrationality proofs and apply the Fundamental Theorem creatively. Solve every exercise twice: once open-book to learn the method, once closed-book under 40-minute time pressure.

Real Numbers Class 10 Important Questions for Board Exams

CBSE Real Numbers Class 10 board papers follow a predictable pattern. One 1-mark MCQ on decimal expansion (e.g. 'Which of the following has a terminating decimal?'). One 2-mark question on Euclid's algorithm or HCF-LCM verification. One 3-mark question on proving irrationality or a two-step HCF-LCM application problem (e.g. two numbers are in the ratio 3:4, their LCM is 180, find the numbers). Practice these question types from the last five years' CBSE papers and NCERT exemplar. The 2023 board paper asked: 'Prove that 5 – √3 is irrational.' The 2022 paper had: 'Find HCF of 96 and 404 using Euclid's algorithm, then express HCF as 96x + 404y.' These are the templates. Drill them until you can write a full solution in under 4 minutes for a 2-mark question and under 7 minutes for a 3-mark question.

1-mark MCQ: Decimal expansion type or HCF-LCM property (annual fixture)
2-mark: Euclid's algorithm or verify HCF × LCM = product (80% probability)
3-mark: Prove irrationality using contradiction OR two-part HCF-LCM word problem (alternates yearly)
Common traps: Not simplifying fraction to lowest terms, arithmetic errors in Euclid steps, skipping 'contradiction' line in proofs

Common Mistakes in Real Numbers Class 10 and How to Avoid Them

Students lose easy marks in Real Numbers Class 10 because of careless errors, not conceptual gaps. Mistake 1: Forgetting to reduce p/q to lowest terms before applying the decimal test—this causes wrong answers in Exercise 1.3. Mistake 2: Writing 'assume √2 is rational' but never writing 'this contradicts our assumption that p and q are coprime'—examiners deduct 1 mark for incomplete logic. Mistake 3: In Euclid's algorithm, writing the wrong quotient q (rushing the division)—leads to wrong HCF. Mistake 4: Confusing HCF and LCM rules: taking maximum for HCF instead of minimum. Mistake 5: In HCF × LCM = a × b problems, forgetting that a and b must be the two numbers (not HCF and LCM themselves). Create a checklist and tick each step as you solve practice problems. CBSETUTOR.ai's AI tutor flags these exact errors in real-time when students upload solution photos, explaining precisely where the logic breaks—available for ₹999/month across Classes 6–12.

Always reduce p/q to simplest form before checking prime factors of q
Write the contradiction statement explicitly in every irrationality proof
Double-check division quotients in Euclid's algorithm (use a scratch column)
HCF = minimum exponent, LCM = maximum exponent (never reverse)
In word problems, identify what is given and what is asked before writing equations

How CBSETUTOR.ai Helps You Master Real Numbers Class 10 in 2026-27

CBSETUTOR.ai is a 24×7 AI tutor designed exclusively for CBSE students in Classes 6–12. It has ingested every NCERT textbook—including every solved example and exercise in Real Numbers Class 10—and can explain any step, any method, any proof in natural language. Upload a photo of your worksheet or a tricky HCF-LCM problem from your school notes, and the AI tutor will solve it step-by-step, showing Euclid's algorithm working or prime factorisation trees. Ask 'Why is √7 irrational?' and get a full proof with every logical step annotated. The tutor also generates unlimited practice questions at your level: if you are weak on decimal expansions, it creates 20 fractions for you to classify. All of this runs at ₹999 per month flat for every class from 6 to 12—one subscription, one price. There is a 3-day free trial with no credit card required, so parents can test whether their child actually uses it before committing. Thousands of families across India rely on CBSETUTOR.ai to fill gaps left by crowded classrooms and expensive coaching centres.

Instant step-by-step solutions to any Real Numbers Class 10 NCERT or worksheet problem via photo upload
On-demand explanations of the Fundamental Theorem, Euclid's algorithm, irrationality proofs—just ask in plain English
Unlimited practice question generation tailored to your weak areas (HCF-LCM, decimal expansions, proofs)
₹999/month flat for Classes 6–12, 3-day free trial, no card required—one subscription for your entire school journey

Frequently asked questions

How many marks does Real Numbers Class 10 carry in the CBSE board exam?+

Real Numbers Class 10 carries exactly 6 marks in the CBSE Class 10 Maths board exam: typically one 1-mark MCQ, one 2-mark short answer (often Euclid's algorithm or HCF-LCM verification), and one 3-mark long answer (proving irrationality or a two-step application problem). This weightage has been consistent since 2020.

What is the Fundamental Theorem of Arithmetic and why is it important?+

The Fundamental Theorem of Arithmetic states that every composite number can be uniquely expressed as a product of prime numbers (ignoring the order of factors). This theorem underpins all HCF-LCM calculations and irrationality proofs in Real Numbers Class 10. It guarantees that when you write 60 = 2² × 3 × 5, that is the only prime factorisation possible. Board exams test this indirectly through HCF-LCM questions and proofs of irrational numbers.

How do I know if a fraction will have a terminating or non-terminating decimal?+

First, reduce the fraction p/q to its lowest terms (cancel all common factors). Then look at the prime factorisation of the denominator q. If q = 2ⁿ × 5ᵐ (only 2s and 5s), the decimal terminates. If q has any other prime factor (like 3, 7, 11), the decimal is non-terminating repeating. For example, 7/20 = 7/(2² × 5) terminates, but 7/12 = 7/(2² × 3) does not because of the 3.

What is Euclid's division algorithm and when should I use it instead of prime factorisation?+

Euclid's division algorithm finds HCF by repeatedly applying the division lemma a = bq + r until the remainder is zero. Use it for large numbers (like 867 and 255) where prime factorisation is tedious. It is faster and less error-prone. Use prime factorisation when you need both HCF and LCM, or when the numbers are small and easily factorisable. CBSE examiners often specify which method to use in the question.

How do I prove a number like √5 or 3 + 2√5 is irrational?+

Use proof by contradiction. Step 1: Assume the number is rational, so it can be written as p/q in lowest terms (p and q coprime). Step 2: Manipulate the equation (square both sides if needed) to isolate the irrational part. Step 3: Show that this forces both p and q to share a common factor, contradicting the assumption they are coprime. Step 4: Conclude the number is irrational. Write every step clearly—examiners deduct marks for skipped logic.

Is the formula HCF × LCM = Product of two numbers always true?+

Yes, for any two positive integers a and b, HCF(a,b) × LCM(a,b) = a × b. This is one of the most tested relations in Real Numbers Class 10. Typical question: Given HCF = 12 and one number = 84, find the other number and the LCM. Use 12 × LCM = 84 × (other number). Be careful—this formula applies only to two numbers, not three or more.

My child's school uses RS Aggarwal instead of NCERT for Real Numbers Class 10. Will they fall behind?+

No. RS Aggarwal, RD Sharma, and other reference books cover the same NCERT syllabus with more practice problems. However, CBSE board exams are set strictly from NCERT content, examples, and theorem statements. Your child should solve NCERT Exercises 1.1–1.4 thoroughly even if the school uses a different book. CBSETUTOR.ai supports NCERT and all major reference books, so students get explanations for any problem they encounter—RS Aggarwal, NCERT Exemplar, or school worksheets.

Which exercise in NCERT Real Numbers Class 10 is most important for the board exam?+

Exercise 1.2 (HCF and LCM by prime factorisation) is most important—it accounts for roughly 3 marks every year. Exercise 1.4 (proving irrationality) is second, contributing another 3 marks. Exercise 1.1 (Euclid's algorithm) appears as a 2-mark question about 70% of the time. Exercise 1.3 (decimal expansions) is usually tested as a 1-mark MCQ. Prioritise 1.2 and 1.4 if you are short on time.

How can I improve my speed in Euclid's division algorithm questions?+

Practice with a timer. Set a target of 3 minutes for a 2-mark Euclid's algorithm question. Write the steps in a fixed format: 'a = bq + r' on one line, then immediately write the next step with the new a and b. Use mental math for simple divisions (e.g. 255 ÷ 102 = 2 remainder 51) instead of long division. After 20 practice problems, you will recognize common quotients and remainders, drastically cutting time.

What are the most common errors students make in Real Numbers Class 10 proofs?+

Three errors cost the most marks: (1) Not stating 'Let p and q be coprime integers' at the start of an irrationality proof—examiners expect this. (2) Forgetting to write the contradiction statement: 'This contradicts our assumption that p and q have no common factor.' (3) Arithmetic mistakes when squaring or simplifying (e.g. writing 9m² = 3q² ⇒ q² = 9m² instead of q² = 3m²). Write slowly and double-check algebra.

Can CBSETUTOR.ai help if my child is stuck on a specific Real Numbers Class 10 question during homework?+

Yes. Your child can photograph the question from the textbook or worksheet, upload it to CBSETUTOR.ai, and get a full step-by-step solution within seconds. The AI tutor explains which theorem or method to apply (Euclid's algorithm vs prime factorisation, for example), shows all working, and highlights common mistakes. If your child uploads incorrect working, the tutor identifies exactly where the error occurred. This works 24×7, even at 11 pm before an exam. ₹999/month for all classes 6–12, 3-day free trial available.

Will understanding Real Numbers Class 10 help in other chapters later in the year?+

Absolutely. HCF and LCM reappear when simplifying algebraic fractions in Polynomials (Chapter 2). The Fundamental Theorem is used in factorising quadratic expressions. Decimal expansions and rational vs irrational classification return in Real Numbers applications in coordinate geometry (rational coordinates vs irrational distances). Mastering Real Numbers Class 10 now saves you re-learning time in March and April when the syllabus pressure peaks.

Related resources

Pair of Linear Equations in Two Variables for Class 10: The Complete CBSE Guide (2026-27)Quadratic Equations for Class 10: The Complete CBSE Guide (2026-27)Triangles for Class 10: The Complete CBSE Guide (2026-27)Polynomials for Class 10: The Complete CBSE Guide (2026-27)AI Tutor for Class 10: The Smart Alternative to Tuition AI Tutor for Class 10 Maths: Learn Faster with Instant Help Class 9 Coordinate Geometry Chapter 3: Cartesian System & Plotting Points Number Systems for Class 9: The Complete CBSE Guide (2026-27)

Keep learning — related guides

Class 10Mathematics

Important Questions: CBSE Class 10 Mathematics Chapter 14 Probability

Class 10Mathematics

CBSE Class 10 Mathematics Chapter 14 Probability Worksheet with Answers

Class 10Mathematics

Class 10 Mathematics Chapter 14 Probability — Formulas & Key Points

Class 10Mathematics

CBSE Class 10 Mathematics Chapter 13 Statistics Worksheet with Answers

Class 10Mathematics

Important Questions: CBSE Class 10 Mathematics Chapter 13 Statistics

Class 10Mathematics

Class 10 Mathematics Chapter 13 Statistics — Formulas & Key Points

Ready to give your Class 10 child the tutor that never sleeps?

CBSETUTOR.ai covers every chapter in the Class 10 NCERT syllabus — Maths, Science, Social Science, English, Hindi and more. 24×7. Patient. Unlimited. 3-day free trial.

Start your child's 3-day free trial →