Real Numbers – Class 10 Maths Chapter 1 Notes & NCERT PDF

Class 10MathematicsChapter 1NCERT book: Mathematics - Class 10

Notes and a simple summary of Chapter 1 of the NCERT Class 10 Mathematics book, with the official chapter PDF, flashcards, an MCQ quiz and an AI tutor for your doubts.

🎧 Listen to the content

Chapter 1: Real Numbers

Introduction

This chapter continues the Class IX study of real numbers — the rationals and irrationals together. It focuses on one powerful idea, the Fundamental Theorem of Arithmetic, and uses it for two things: finding HCF and LCM by prime factorisation, and proving that numbers such as √2, √3 and √5 are irrational. The chapter also recalls, from Class IX, that the sum or difference of a rational and an irrational number is irrational, and that the product or quotient of a non-zero rational and an irrational number is irrational.

Key Concepts & Theorems

The Fundamental Theorem of Arithmetic

Theorem 1.1: 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.

Example from the book (factor tree): 32760 = 2 × 2 × 2 × 3 × 3 × 5 × 7 × 13 = 2³ × 3² × 5 × 7 × 13.

Another: 123456789 = 3² × 3803 × 3607.

If we agree to write the primes in ascending order, the factorisation is exactly one.

Historical note from the book: an equivalent version appears as Proposition 14 of Book IX of Euclid's Elements; the first correct proof was given by Carl Friedrich Gauss (1777 – 1855), the "Prince of Mathematicians", in Disquisitiones Arithmeticae.

HCF and LCM by prime factorisation

HCF = product of the smallest power of each common prime factor.

LCM = product of the greatest power of each prime factor involved.

For two positive integers a and b: HCF(a, b) × LCM(a, b) = a × b.

This does not hold for three numbers: 6 × 72 × 120 ≠ HCF × LCM (= 6 × 360).

Irrationality

Theorem 1.2: Let p be a prime number. If p divides a², then p divides a, where a is a positive integer.

Theorem 1.3: √2 is irrational.

A number s is irrational if it cannot be written as p/q with p, q integers and q ≠ 0. The proofs use proof by contradiction.

Methods Step by Step

Finding HCF and LCM

Write each number as a product of powers of primes.

HCF: take each prime common to all numbers with its lowest power.

LCM: take every prime that appears, with its highest power.

For two numbers you may find LCM = (a × b) ÷ HCF.

Proving √p is irrational (p prime)

Assume, to the contrary, √p = a/b with a, b coprime integers, b ≠ 0.

Square: p·b² = a², so p divides a², hence p divides a (Theorem 1.2).

Write a = p·c; substitute: p·b² = p²c², so b² = p·c², hence p divides b.

Now p divides both a and b — contradicting coprimality. So √p is irrational.

Proving (rational ± irrational) or (rational × irrational) is irrational

Assume the expression equals a/b (coprime, b ≠ 0).

Rearrange so that the known irrational (e.g. √3) equals an expression in a, b and integers.

That expression is rational — contradiction.

Worked Examples

1. Can 4ⁿ end with 0? For 4ⁿ to end in 0 its prime factorisation must contain 5. But 4ⁿ = 2²ⁿ, and by uniqueness the only prime is 2. So no natural n works. (By the same reasoning 6ⁿ = 2ⁿ × 3ⁿ never ends in 0.)

2. HCF and LCM of 96 and 404. 96 = 2⁵ × 3, 404 = 2² × 101. HCF = 2² = 4. LCM = (96 × 404) ÷ 4 = 9696.

3. HCF and LCM of 6, 72 and 120. 6 = 2 × 3, 72 = 2³ × 3², 120 = 2³ × 3 × 5. HCF = 2 × 3 = 6; LCM = 2³ × 3² × 5 = 360.

4. Given HCF(306, 657) = 9, find LCM. LCM = 306 × 657 ÷ 9 = 22338.

5. Why is 7 × 11 × 13 + 13 composite? It equals 13 × (7 × 11 + 1) = 13 × 78 = 1014 — it has a factor other than 1 and itself. Similarly 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 = 5 × (7 × 6 × 4 × 3 × 2 + 1) = 5 × 1009.

6. Show 5 − √3 is irrational. Suppose 5 − √3 = a/b. Then √3 = 5 − a/b = (5b − a)/b, which is rational — contradicting the irrationality of √3. So 5 − √3 is irrational. Likewise 3√2 = a/b gives √2 = a/(3b), a contradiction.

Word problem (Exercise 1.1): Sonia takes 18 minutes and Ravi 12 minutes per round. They meet again at the start after LCM(18, 12) = 36 minutes.

Board Exam Focus

1–2 mark: express a number as a product of primes; find HCF/LCM; use HCF × LCM = product.

2–3 mark: prove √5, 3 + 2√5, 1/√2, 7√5 or 6 + √2 irrational — examiners award marks for (i) the "assume to the contrary" line, (ii) the coprime assumption, (iii) using Theorem 1.2 correctly, (iv) stating the contradiction and conclusion.

Word problems on "meeting again", "bells ringing together", "largest tile/container" → LCM or HCF.

Formula Sheet

a = p₁ × p₂ × … × pₙ (unique prime factorisation)

HCF(a, b) × LCM(a, b) = a × b (two numbers only)

HCF = smallest powers of common primes; LCM = greatest powers of all primes

If p is prime and p | a², then p | a

Common Mistakes

Using HCF × LCM = product for three numbers — it fails in general.

Forgetting to assume a and b are coprime in irrationality proofs, which is what the contradiction relies on.

Writing 1 as a prime, or missing a prime factor (7429 = 17 × 19 × 23 — all three are prime).

In "meet again" problems, using HCF instead of LCM.

💡 Key Learning Points

✓State and apply the Fundamental Theorem of Arithmetic: every composite number has a unique prime factorisation (apart from order).
✓Find HCF (smallest powers of common primes) and LCM (greatest powers of all primes) by prime factorisation.
✓Use HCF(a, b) × LCM(a, b) = a × b for two positive integers, and know it fails for three numbers.
✓Prove √2, √3, √5 (and √p for prime p) irrational by contradiction using Theorem 1.2.
✓Prove that expressions like 5 − √3, 3√2 or 3 + 2√5 are irrational.

👨‍🏫 Teaching Tips

→Build the factor tree of 32760 on the board and let students check that every route gives 2³ × 3² × 5 × 7 × 13 — this makes 'uniqueness' concrete.
→Use a two-column layout (common primes / all primes) so students see why HCF uses the smallest power and LCM the greatest.
→Give the counter-example 6, 72, 120 so students discover that HCF × LCM = product works only for two numbers.
→Model the √2 proof line by line and have students copy the 4-step skeleton, then adapt it to √3 and √5 themselves.
→Pose the 4ⁿ and 6ⁿ 'ending in zero' questions as puzzles before revealing the prime-factorisation argument.
→Link LCM to real life with the Sonia–Ravi circular track problem and bells ringing together.
→Mark mock proofs using a board-style rubric so students learn which lines carry marks.

📋 Assessment Questions

Can the student express 3825 and 5005 as products of primes correctly?
Can the student find HCF and LCM of three numbers (e.g. 12, 15, 21) by prime factorisation?
Can the student write a complete proof that √5 is irrational, including the coprime assumption and contradiction?
Can the student explain why 7 × 11 × 13 + 13 is composite?
Can the student decide whether 6ⁿ can end with the digit 0, giving a reason?

Ask the AI tutor about Class 10 Mathematics Chapter 1

Get question answers, important questions, a quick MCQ quiz or a simple explanation of “Real Numbers”. Free — sign in with Google.

How the AI tutor works →

👩‍🏫 Class 10 Mathematics classes near you — online or at home

Prefer a real teacher? TalentJR teachers take live online classes anywhere in India and visit homes in selected cities. Tell us what you need and our team will call you with timings and fees.

Live online classesSmall groups or 1:1 with a teacher, from home, any city.
Teacher at homeA TalentJR teacher visits your home (selected cities).
At a centreClasses at a TalentJR partner centre near you, where available.