Real Numbers — Class 10 Mathematics (CBSE) explained

What you’ll learn

1. 1

   Lesson 1: Prime Factors and the Fundamental Theorem of Arithmetic

   5 min
   • From Positive Integers to Irrational Numbers
   • Read Numbers as Products of Prime Factors
   • Construct Larger Numbers from Primes
   • Follow the Complete Factor Tree of 32760
   • State the Fundamental Theorem of Arithmetic
   • The Textbook's Historical NoteNot assessed
   • Changing Order Does Not Change Prime Factorisation
2. 2

   Lesson 2: Applications: HCF, LCM and Prime Factorisation

   5 min
   • Example 1: Can a Power of Four End in Zero?
   • Example 2: Find the HCF and LCM of 6 and 20
   • The Product Relationship for Two Positive Integers
   • Example 3: HCF First, Then LCM
   • Example 4: HCF and LCM of Three Integers
   • The Two-Number Identity Does Not Extend Unchanged
   • Exercise 1.1: Choose the Appropriate Method
3. 3

   Lesson 3: Why Root Two and Root Three Are Irrational

   5 min
   • Recall What an Irrational Number Is
   • Theorem 1.2: From a Square to Its Base
   • Optional Enrichment: Proof of Theorem 1.2Not assessed
   • Theorem 1.3: Assume Root Two Is Rational
   • Root Two: Reach the Contradiction
   • Example 5: Begin the Proof for Root Three
   • Example 5: Complete the Proof for Root Three
4. 4

   Lesson 4: Irrational Expressions, Three-Number Relationships and Practice

   6 min
   • Combine Rational and Irrational Numbers Carefully
   • Example 6: Five Minus Root Three
   • Example 7: Three Times Root Two
   • Exercise 1.2: Plan Your Irrationality Proofs
   • Section 1.4: Retrieve the Chapter's Three Summary Points
   • A Note to the Reader: Relationships for Three Integers
   • Chapter Recap: Factorise, Justify and Practise

Key concepts

Prime products and factorisation

Instead of starting with a number and finding its factors, start with prime numbers and multiply them.

Constructing integers from prime products

A prime may also be repeated, with a power recording those repeated copies.

A complete prime factor tree

The highlighted leaves now give the complete prime factorisation.

The meaning of uniqueness

Once this order is fixed, the written factorisation is unique.

Revisiting Irrational Numbers

Rational and irrational numbers together make up the real numbers.

Properties of rational–irrational combinations

The sum or difference of a rational number and an irrational number is irrational.

Exam focus

Board exam

CBSE assesses this chapter in the 2026-27 year-end board examination.

Syllabus

REAL NUMBERS: 1. Fundamental Theorem of Arithmetic - statements after reviewing work done earlier and after illustrating and motivating through examples. 2. Proofs of irrationality of root 2, root 3, root 5. Learning outcomes: describes Fundamental Theorem of Arithmetic with examples; proves algebraically the irrationality of numbers like root 2, root 3, root 5, 3 + 2 root 5 etc.; applies the Fundamental Theorem of Arithmetic to solve problems related to real-life contexts.

Not assessed

These parts go beyond the 2026-27 syllabus and are taught for understanding only: The Textbook's Historical Note and Optional Enrichment: Proof of Theorem 1.2.

Exam note

The NCERT proof of Theorem 1.2 is marked 'not from the examination point of view' in the textbook.

Paper pattern

Mathematics Standard: 80 marks, 3 hours, 38 questions. Section A: 18 MCQs + 2 assertion-reason (1 mark each); B: 5 VSA (2 marks); C: 6 SA (3 marks); D: 4 LA (5 marks); E: 3 case-study questions (4 marks). No calculators.

Question style

About 50% of board questions are competency-focused.

Practice in the app

84 board-pattern questions with marking schemes: 29 multiple choice (1 mark), 13 assertion–reason (1 mark), 16 very short answer (2 marks), 14 short answer (3 marks), 3 long answer (5 marks) and 9 case-based (4 marks).

Sample questions with marking points

Board-pattern questions from the AI Lessons practice set for this chapter. The marking points show what an examiner looks for; full model answers are in the app.

1. Very short answer2 marks

   A student accepts 253 = 11 × 23 but rejects 4 = 2 × 2, saying that a prime may occur only once. Correct the student's reasoning.

   Marking points

   • Identifies both representations as valid prime factorisations.1
   • Explains that prime factors may repeat, as 2 does in the factorisation of 4.1

2. Short answer3 marks

   A calculation record correctly lists HCF(96, 404) = 4 but gives LCM(96, 404) = 96 × 404. Identify the missing operation and correct the LCM, showing the identity used.

   Marking points

   • States the two-integer HCF–LCM product identity.1
   • Identifies division by the HCF and substitutes (96 × 404)/4.1
   • Obtains the correct LCM, 9696.1

3. Long answer5 marks

   Prove that √5 is irrational. Hence explain why 7 √5 is irrational.

   Marking points

   • Assumes √5 = a/b with coprime positive integers a and b.0.5
   • Obtains b √5 = a and 5b² = a².0.5
   • Uses the prime-divisibility theorem to infer that 5 divides a.0.5
   • Writes a = 5c and substitutes to obtain 5b² = 25c².0.5
   • Simplifies to b² = 5c².0.5
   • Uses the theorem again to infer that 5 divides b.0.5
   • Identifies the common factor 5 as contradicting coprimality.0.5
   • Rejects the assumption and concludes that √5 is irrational.0.5
   • Uses the non-zero rational multiplier 7 to justify that 7 √5 is irrational.1

Frequently asked questions

More Class 10 Mathematics chapters

• Chapter 2Polynomials4 lessons · 21 min→
• Chapter 3Pair of Linear Equations in Two Variables4 lessons · 24 min→

All CBSE Class 10 Mathematics guides →