Real Numbers Class 10 Notes PDF 2026–27

Real Numbers Class 10 notes cover prime factorisation, the Fundamental Theorem of Arithmetic, HCF and LCM applications, and proofs that √2, √3 and √5 are irrational. These CBSE and NCERT-aligned revision notes also include solved examples, important questions, MCQs, competency-based questions and older syllabus topics clearly marked for reference.

These notes are based on CBSE Class 10 Maths Syllabus, classification of real numbers, prime factorisation, HCF and LCM applications, irrationality proofs, solved examples, competency-based questions, revision plans and frequently asked questions.

Real Numbers Class 10 Notes: What Is in the 2026–27 Syllabus?

The CBSE 2026–27 Real Numbers syllabus includes the Fundamental Theorem of Arithmetic and algebraic proofs of the irrationality of √2, √3 and √5.

Students should be able to:

1. State and understand the Fundamental Theorem of Arithmetic.
2. Express composite numbers as products of prime factors.
3. Apply prime factorisation to mathematical and real-life problems.
4. Prove algebraically that √2, √3 and √5 are irrational.
5. Extend the proof method to expressions such as 3 + 2√5.

The current NCERT chapter contains four principal sections:

1. Introduction
2. The Fundamental Theorem of Arithmetic
3. Revisiting Irrational Numbers
4. Summary

Current and Older Syllabus Topics

A school may assign additional material for an internal examination. For CBSE board preparation, prioritise the concepts explicitly listed in the current official curriculum.

Is the Chapter Different for Basic and Standard Maths?

The central Real Numbers concepts are shared by Mathematics Basic and Standard, but CBSE publishes separate question-paper designs for the two subjects.

Mathematics Basic generally emphasises direct applications. Mathematics Standard can require more multi-step reasoning and proof analysis. Students should follow the practice set matching their registered subject code.

What Are Real Numbers?

Real numbers are all rational and irrational numbers together, and every real number corresponds to a point on the number line.

Examples of real numbers include:

• −8
• 0
• 3/5
• 1.75
• √2
• π

An imaginary number such as √−1 is not a real number.

Classification of Real Numbers

Natural numbers (N): The counting numbers 1, 2, 3, …

Whole numbers (W): Natural numbers together with zero: 0, 1, 2, 3, …

Integers (Z): Negative integers, zero and positive integers:

…, −3, −2, −1, 0, 1, 2, 3, …

Rational numbers (Q): Numbers that can be written as p/q, where p and q are integers and q ≠ 0.

Irrational numbers: Real numbers that cannot be written as p/q.

Real numbers (R): Rational and irrational numbers considered together.

The main relationship is:

N ⊂ W ⊂ Z ⊂ Q ⊂ R

Every natural number is a whole number, every whole number is an integer, every integer is rational, and every rational number is real. The reverse of each statement is not always true.

Number-System Concept Map

Real numbers

• Rational numbers
  • Non-integer rational numbers
  • Integers
    • Negative integers
    • Whole numbers
      • Zero
      • Natural numbers
• Irrational numbers

Rational Numbers vs Irrational Numbers

Common Misconceptions

Every square root is irrational: False. √2 is irrational, but √4 = 2 is rational.

Every non-terminating decimal is irrational: False. A repeating decimal such as 0.333… = 1/3 is rational.

π is equal to 22/7: False. The fraction 22/7 is a rational approximation to π.

Two irrational numbers always produce an irrational result: False. For example:

√2 + (−√2) = 0

and:

√2 × √2 = 2

What Is the Fundamental Theorem of Arithmetic?

The Fundamental Theorem of Arithmetic states that every composite number has a unique prime factorisation, apart from the order of its prime factors.

For example:

60 = 2 × 2 × 3 × 5

Using powers:

60 = 2² × 3 × 5

Writing the factors in another order does not create a different prime factorisation. The prime factors and their powers remain unchanged.

Important Terms

Prime number: A natural number greater than 1 with exactly two positive factors—1 and itself.

Examples: 2, 3, 5, 7 and 11.

Composite number: A natural number greater than 1 with more than two positive factors.

Examples: 4, 6, 8, 9 and 10.

Coprime numbers: Two positive integers whose HCF is 1.

For example, 8 and 15 are coprime even though neither number is prime.

Prime factorisation: Expressing a composite number as a product of prime numbers.

The number 1 is neither prime nor composite because it has only one positive factor.

How to Find Prime Factorisation

Use repeated division or a factor tree until every remaining factor is prime.

Example: Find the prime factorisation of 420.

1. 420 = 2 × 210
2. 210 = 2 × 105
3. 105 = 3 × 35
4. 35 = 5 × 7
5. Combine the prime factors:

420 = 2 × 2 × 3 × 5 × 7

1. Write repeated factors using powers:

420 = 2² × 3 × 5 × 7

Check:

4 × 3 × 5 × 7 = 420

Common Prime-Factorisation Errors

Quick Practice

Find the prime factorisation of 84.

Answer: 84 = 2² × 3 × 7

Find the prime factorisation of 540.

Answer: 540 = 2² × 3³ × 5

A student writes 72 = 2² × 3². Identify the missing factor.

Answer: The correct factorisation is 72 = 2³ × 3², so one factor of 2 is missing.

Explain why 5 × 14 is not the prime factorisation of 70.

Answer: Fourteen is composite. Since 14 = 2 × 7, the prime factorisation is:

70 = 2 × 5 × 7

How Are HCF and LCM Found Using Prime Factorisation?

HCF is found using the lowest powers of common prime factors, while LCM is found using the highest powers of all prime factors present.

How to Find HCF

Find the HCF of 72 and 120.

Prime-factorise each number:

72 = 2³ × 3²

120 = 2³ × 3 × 5

The common prime factors are 2 and 3.

Select their lowest powers:

HCF = 2³ × 3

HCF = 8 × 3

HCF = 24

Therefore:

HCF(72, 120) = 24

How to Find LCM

Using the same numbers:

72 = 2³ × 3²

120 = 2³ × 3 × 5

Select the highest power of every prime present:

LCM = 2³ × 3² × 5

LCM = 8 × 9 × 5

LCM = 360

Therefore:

LCM(72, 120) = 360

HCF vs LCM

These clues are helpful, but the mathematical relationship in the problem should determine the method.

Relationship Between HCF and LCM

For two positive integers a and b:

HCF(a, b) × LCM(a, b) = a × b

Check the relationship for 72 and 120:

24 × 360 = 8,640

72 × 120 = 8,640

The two products are equal.

Do not apply this simple formula directly to three or more numbers.

Solved Example: Find an Unknown Number

The HCF of two positive integers is 6, their LCM is 180, and one number is 30. Find the other number.

Let the unknown number be x.

HCF × LCM = Product of the two numbers

6 × 180 = 30x

1,080 = 30x

x = 36

Therefore, the other number is 36.

Check:

HCF(30, 36) = 6

LCM(30, 36) = 180

Solved Example: Equal Groups

A teacher has 48 red counters and 72 blue counters. She wants to form the greatest possible number of identical groups without leaving any counter unused.

Prime-factorise the quantities:

48 = 2⁴ × 3

72 = 2³ × 3²

HCF = 2³ × 3

HCF = 24

The teacher can make 24 identical groups.

Each group contains:

48 ÷ 24 = 2 red counters

72 ÷ 24 = 3 blue counters

What Is an Irrational Number?

An irrational number is a real number that cannot be written as p/q, where p and q are integers and q ≠ 0.

The decimal expansion of an irrational number neither terminates nor repeats. Examples include:

• √2
• √3
• √5
• π

A decimal approximation does not prove irrationality. The Class 10 proofs use algebra and contradiction.

What Is Proof by Contradiction?

Proof by contradiction assumes that the statement to be proved is false and then shows that the assumption creates an impossibility.

To prove that a number is irrational:

1. Assume that the number is rational.
2. Write it as p/q in lowest terms.
3. Use algebra to derive divisibility facts about p and q.
4. Show that p and q have a common factor.
5. Contradict the assumption that p/q was in lowest terms.
6. Reject the assumption and conclude that the number is irrational.

Why Must p and q Be Coprime?

Every rational number can be written as a fraction in lowest terms. Therefore, an irrationality proof may assume that p and q have no common factor other than 1.

The proof then shows that the same prime divides both p and q. This contradicts the lowest-term assumption.

Prime-Divisibility Fact

If a prime number divides p², then it divides p.

The condition that the divisor is prime matters. The same inference cannot be applied carelessly to a composite number.

How Do You Prove That √2 Is Irrational?

To prove that √2 is irrational, assume it is a rational fraction in lowest terms and show that its numerator and denominator must both be even.

Assume, for contradiction, that:

√2 = p/q

Here, p and q are coprime integers and q ≠ 0.

Square both sides:

2 = p²/q²

Therefore:

p² = 2q² … (1)

Equation (1) shows that p² is divisible by 2. Since 2 is prime, p is also divisible by 2.

Let:

p = 2k

Substitute p = 2k into Equation (1):

(2k)² = 2q²

4k² = 2q²

q² = 2k²

Therefore, q is also divisible by 2.

Both p and q are divisible by 2. This contradicts the assumption that p and q are coprime.

Therefore:

√2 is irrational.

Reasoning Checklist

Common Errors

• Omitting the condition q ≠ 0
• Failing to say that p/q is in lowest terms
• Showing that p is even but not proving that q is even
• Failing to identify the contradiction
• Ending without rejecting the original assumption

How Do You Prove That √3 and √5 Are Irrational?

The proofs for √3 and √5 use the same contradiction structure as the proof for √2.

Proof That √3 Is Irrational

Assume:

√3 = p/q

Here, p and q are coprime integers and q ≠ 0.

Squaring gives:

p² = 3q²

Therefore, 3 divides p². Since 3 is prime, 3 divides p.

Let:

p = 3k

Substitute:

9k² = 3q²

q² = 3k²

Therefore, 3 also divides q.

Both p and q are divisible by 3. This contradicts the assumption that they are coprime.

Therefore:

√3 is irrational.

Proof That √5 Is Irrational

Assume:

√5 = p/q

Here, p and q are coprime integers and q ≠ 0.

Squaring gives:

p² = 5q²

Therefore, 5 divides p.

Let:

p = 5k

Substitute:

25k² = 5q²

q² = 5k²

Therefore, 5 also divides q.

Both p and q are divisible by 5, contradicting the coprime condition.

Therefore:

√5 is irrational.

Reusable Pattern for √r

For a prime number r:

1. Assume √r = p/q in lowest terms.
2. Square to obtain p² = rq².
3. Conclude that r divides p.
4. Write p = rk and substitute.
5. Conclude that r divides q.
6. Contradict the assumption that p and q are coprime.

Why Does the Method Not Prove That √4 Is Irrational?

The method does not prove that √4 is irrational because √4 = 2, and 4 is not prime.

A false argument may claim:

If 4 divides p², then 4 divides p.

This is not always true. If p = 2, then 4 divides p² because p² = 4, but 4 does not divide p.

The valid theorem used in the prescribed proofs concerns a prime divisor. Four is composite.

How Do You Prove Other Expressions Are Irrational?

An expression containing a known irrational number can often be proved irrational by assuming the entire expression is rational and isolating the irrational part.

Rational Plus Irrational

If r is rational and x is irrational, then r + x is irrational.

If r + x were rational, subtracting r would make x rational. This contradicts the assumption that x is irrational.

Non-Zero Rational Times Irrational

If r is a non-zero rational number and x is irrational, then rx is irrational.

If rx were rational, dividing by r would make x rational.

Prove That 3 + 2√5 Is Irrational

Assume:

3 + 2√5 = r

Here, r is rational.

Then:

2√5 = r − 3

√5 = (r − 3)/2

The right-hand side is rational. This would make √5 rational, which is a contradiction.

Therefore:

3 + 2√5 is irrational.

Prove That 7√5 Is Irrational

Assume:

7√5 = r

Here, r is rational.

Dividing by 7:

√5 = r/7

This would make √5 rational, producing a contradiction.

Therefore:

7√5 is irrational.

Prove That 6 + √2 Is Irrational

Assume:

6 + √2 = r

Here, r is rational.

Then:

√2 = r − 6

The right-hand side is rational. This contradicts the known irrationality of √2.

Therefore:

6 + √2 is irrational.

Operations Involving Irrational Numbers

Real Numbers Class 10 Solved Examples

These examples cover classification, prime factorisation, HCF and LCM reasoning, and algebraic irrationality proofs.

Example 1: Classify the Numbers

Example 2: Find the Prime Factorisation of 756

756 = 2 × 378

756 = 2 × 2 × 189

756 = 2² × 3 × 63

756 = 2² × 3 × 3 × 21

756 = 2² × 3³ × 7

Therefore:

756 = 2² × 3³ × 7

Example 3: Find HCF and LCM

Find the HCF and LCM of 90 and 168.

90 = 2 × 3² × 5

168 = 2³ × 3 × 7

For the HCF, select the lowest powers of common primes:

HCF = 2 × 3 = 6

For the LCM, select the highest powers of all primes:

LCM = 2³ × 3² × 5 × 7

LCM = 2,520

Check:

6 × 2,520 = 15,120

90 × 168 = 15,120

Example 4: Detect an Invalid Proof

A student writes:

“√6 = p/q, so p² = 6q². Therefore, 6 divides p.”

The conclusion has not yet been justified because 6 is composite.

The student must use the prime factors 2 and 3 separately. Since both 2 and 3 divide p², both divide p. Only then can the student conclude that 6 divides p.

Example 5: Disprove a Universal Claim

Claim: The sum of two irrational numbers is always irrational.

Counterexample:

√2 + (−√2) = 0

Both terms are irrational, but zero is rational. Therefore, the claim is false.

Real Numbers Class 10 Competency-Based Questions

Competency-based questions test whether students can interpret, apply and justify mathematical ideas rather than only recall definitions.

CBSE’s 2026–27 secondary curriculum states that approximately 50% of questions are competency-focused. These can include case-based, source-based, integrated and data-interpretation questions.

Multiple-Choice Questions

Which number is irrational?

0.125

B. 7/9

C. √49

D. √11

Answer: D. √11 is irrational.

The prime factorisation of 360 is:

A. 2² × 3² × 5

B. 2³ × 3² × 5

C. 2³ × 3 × 5²

D. 2² × 3³ × 5

Answer: B. 360 = 2³ × 3² × 5.

If HCF(a, b) = 12, LCM(a, b) = 420 and a = 60, find b.

b = (12 × 420)/60

b = 84

Answer: 84.

Assertion–Reason Question

Assertion: 3 + √2 is irrational.

Reason: The sum of a rational number and an irrational number is irrational.

Answer: Both statements are true, and the reason correctly explains the assertion.

Error-Analysis Question

A student writes:

“Let √5 = p/q. We get p² = 5q², so both are divisible by 5. This is a contradiction.”

Two steps are missing:

1. The student must state that p and q are coprime and q ≠ 0.
2. The student must prove that 5 divides p, substitute p = 5k, and then prove that 5 divides q.

Case-Based Question

A school has 96 Mathematics Basic worksheets and 144 Mathematics Standard worksheets. The worksheets must be packed into the greatest possible number of identical sets.

96 = 2⁵ × 3

144 = 2⁴ × 3²

HCF = 2⁴ × 3

HCF = 48

The school can make 48 identical sets.

Each set contains:

96 ÷ 48 = 2 Basic worksheets

144 ÷ 48 = 3 Standard worksheets

Mathematics Basic vs Standard Practice

Mathematics Basic and Standard share the central Real Numbers concepts, but students should practise at the level appropriate to their registered paper.

Mathematics Basic Practice

1. Write the prime factorisation of 225.
2. Find the HCF of 54 and 90.
3. State the Fundamental Theorem of Arithmetic.
4. Identify whether √3 is rational or irrational.
5. Complete the missing steps in a proof that √2 is irrational.

Mathematics Standard Practice

1. If HCF(a, b) = 8, LCM(a, b) = 336 and one number is 48, find the other number.
2. Prove that 5 + 3√2 is irrational.
3. Identify the first invalid step in a proposed proof that √9 is irrational.
4. Decide whether two numbers with HCF 18 and LCM 280 can exist.
5. Prove that √7 is irrational.

For Question 4, the proposed pair cannot exist because the HCF must divide the LCM, but 18 does not divide 280.

Common Mistakes in Real Numbers Class 10

The most common errors involve outdated syllabus material, incomplete prime factorisation and missing logical steps in irrationality proofs.

How to Revise Real Numbers in 30 Minutes

A 30-minute revision should cover the current syllabus, one complete proof and a short self-test.

1. Minutes 0–5: Review the current syllabus.
2. Minutes 5–10: Recall definitions and the theorem.
3. Minutes 10–15: Complete one prime-factorisation question.
4. Minutes 15–22: Write one irrationality proof without using notes.
5. Minutes 22–27: Solve one application and one reasoning question.
6. Minutes 27–30: Check mistakes and conclusions.

How to Revise Real Numbers in One Day

A one-day revision should combine concept review, written proof practice and self-testing.

1. Review the classification of real numbers and the Fundamental Theorem of Arithmetic.
2. Complete three prime-factorisation questions and four HCF–LCM questions.
3. Write the complete proofs for √2, √3 and √5.
4. Complete one MCQ set, one error-analysis question and one case-based problem.
5. Attempt the diagnostic test below without using notes.

Are NCERT Questions Enough for Real Numbers?

NCERT provides the essential concepts and exercises, but students should also practise current-syllabus competency and reasoning questions.

Use this sequence:

1. Read the current NCERT chapter.
2. Complete its examples and exercises.
3. Practise proof completion and error analysis.
4. Attempt current CBSE sample-paper questions.
5. Use older previous-year questions only after checking their syllabus status.

Real Numbers Class 10 Quick Summary

Real Numbers Chapter 1 for CBSE 2026–27 centres on unique prime factorisation and algebraic proofs of irrationality.

• Rational and irrational numbers together form the real numbers.
• A rational number can be written as p/q, where q ≠ 0.
• Every composite number has a unique prime factorisation apart from factor order.
• HCF uses the lowest powers of common prime factors.
• LCM uses the highest powers of all required prime factors.
• For two positive integers, HCF × LCM equals their product.
• An irrationality proof assumes a lowest-term rational representation.
• The contradiction shows that the numerator and denominator share a prime factor.
• The prescribed core examples are √2, √3 and √5.
• Euclid’s Division Lemma and decimal-expansion tests are older topics for this session unless a school specifically assigns them.

Test Your Understanding

This ten-question diagnostic checks the central Real Numbers concepts.

1. Is every integer rational?
2. Is every irrational number real?
3. Write the prime factorisation of 180.
4. Find the HCF of 60 and 84.
5. Find the LCM of 60 and 84.
6. Why must p/q be in lowest terms in an irrationality proof?
7. Is √25 irrational?
8. Is 2√2 rational or irrational?
9. Can two irrational numbers have a rational product?
10. State the contradiction used to prove that √3 is irrational.

Answers

1. Yes. An integer n can be written as n/1.
2. Yes.
3. 180 = 2² × 3² × 5.
4. HCF(60, 84) = 12.
5. LCM(60, 84) = 420.
6. The proof must contradict the assumption that the numerator and denominator have no common factor.
7. No. √25 = 5.
8. Irrational.
9. Yes. For example, √2 × √2 = 2.
10. Both the numerator and denominator become divisible by 3 even though they were assumed coprime.

Resources: 

• NCERT Solutions for Class 10 Mathematics
• CBSE Class 10 Maths Curriculum 2026–27