Real Numbers Class 10 Full Chapter Notes and Important Questions

Real Numbers is one of the most important chapters in Class 10 Maths because it builds the foundation of the number system and introduces some of the most important ideas in elementary number theory. Even though many students feel this chapter is simple at first, it carries strong conceptual value and regular board weightage. It is also one of those chapters where basic understanding can directly help in MCQs, short questions, proof-based questions, and case-based problems.

This chapter mainly revolves around four major areas: number system basics, the fundamental theorem of arithmetic, prime factorisation, HCF and LCM, and irrational numbers. Most students are already familiar with many of these ideas from earlier classes, but in Class 10 the chapter becomes more structured and exam-oriented.

At Deeksha Vedantu, we always encourage students to treat Real Numbers as a chapter of logic, not just memorisation. If the concepts are clear, the questions become much easier to solve.

Why Real Numbers Is Important in Class 10

Real Numbers is an important chapter because it is concept-based, scoring, and closely connected to many basic ideas used later in mathematics.

Why Students Should Prepare This Chapter Well

Reason	Why it matters
Board relevance	This chapter regularly appears in CBSE-style question papers
Strong conceptual base	It improves mathematical reasoning
Multiple question types	MCQs, 2-mark, 3-mark, proof-based, and case-study questions can come
High scoring potential	Standard methods can fetch accurate marks quickly
Useful formulas and logic	HCF, LCM, irrational proofs, and factorisation are repeated patterns

Chapter Weightage and Key Areas

This chapter belongs to the unit of number system and is an important scoring area in Class 10 Maths.

Chapter Weightage Table

Unit	Chapter	Approximate weightage
Number System	Real Numbers	6 marks

Main Topics Covered in This Chapter

Topic	Main idea
Number system basics	Natural, whole, integers, rational, irrational, and real numbers
Fundamental theorem of arithmetic	Every composite number has a unique prime factorisation
Prime factorisation	Writing a number as a product of prime numbers
HCF and LCM	Finding highest common factor and least common multiple
Irrational numbers	Numbers not expressible in p/q form
Proofs	Showing numbers like √2 are irrational

Understanding the Number System First

Before starting the main chapter concepts, students should understand the structure of the number system clearly.

Number System Hierarchy Table

Set of numbers	Meaning
Natural numbers	Counting numbers like 1, 2, 3, 4, …
Whole numbers	Natural numbers including 0
Integers	Positive numbers, negative numbers, and 0
Rational numbers	Numbers expressible as p/q, where q ≠ 0
Irrational numbers	Numbers not expressible as p/q
Real numbers	Combination of rational and irrational numbers

Real Numbers at a Glance

Part of real numbers	Includes
Rational numbers	Fractions, integers, terminating decimals, repeating decimals
Irrational numbers	√2, √3, √5, π, and non-terminating non-repeating decimals

What Are Rational Numbers

A rational number is any number that can be written in the form p/q, where p and q are integers and q ≠ 0.

Rational Number Table

Example	Why it is rational
1/2	Already in p/q form
-3	Can be written as -3/1
0.5	Can be written as 1/2
0.3333…	Repeating decimal, can be written as a fraction

Decimal Expansion of Rational Numbers

Type of decimal	Rational or not
Terminating decimal	Rational
Non-terminating repeating decimal	Rational

What Are Irrational Numbers

Irrational numbers are the numbers that cannot be written in the form p/q.

Irrational Number Identification Table

Type	Example
Non-terminating, non-repeating decimal	1.412134521…
Square root of an imperfect square	√2, √3, √5
Cube root of an imperfect cube	∛2, ∛7
Famous irrational constant	π

How to Recognise an Irrational Number

Feature	Meaning
Cannot be written as p/q	Irrational
Decimal never ends and never repeats in a fixed pattern	Irrational
Root of a non-perfect square	Usually irrational

Fundamental Theorem of Arithmetic

This is one of the central concepts of the chapter.

Statement of the Theorem

Every composite number can be expressed as a product of prime numbers, and this factorisation is unique except for the order of the prime factors.

Meaning Table

Term	Meaning
Composite number	A number having more than two factors
Prime factorisation	Writing a number only as a product of prime numbers
Unique factorisation	The prime factors remain the same apart from order

Example of Fundamental Theorem of Arithmetic

Number	Prime factorisation
36	2² × 3²
88	2³ × 11
242	2 × 11²

This theorem is the base for solving HCF, LCM, and several factorisation-based questions.

Prime Factorisation Method

Prime factorisation means expressing a number as the product of its prime factors.

Prime Factorisation Table

Number	Prime factorisation
48	2⁴ × 3
72	2³ × 3²
80	2⁴ × 5
90	2 × 3² × 5
144	2⁴ × 3²

Why Prime Factorisation Is Important

Use	Why it helps
HCF	Helps identify common prime factors with least powers
LCM	Helps identify all prime factors with highest powers
Proofs and logic	Builds factor-based reasoning

HCF and LCM: Core Ideas

HCF means Highest Common Factor. LCM means Least Common Multiple.

HCF vs LCM Table

Term	Meaning
HCF	Greatest number that divides the given numbers exactly
LCM	Smallest number that is a multiple of all the given numbers

Rule Using Prime Factorisation

Quantity	Rule
HCF	Take only common prime factors with lowest powers
LCM	Take all prime factors with highest powers

Solved Example 1: Find the HCF and LCM of 48, 72, and 80

Given

Number	Prime factorisation
48	2⁴ × 3
72	2³ × 3²
80	2⁴ × 5

Step 1

For HCF, take only the common prime factor with the least power.

Common factor = 2

Least power of 2 = 2³

Step 2

HCF = 2³ = 8

Step 3

For LCM, take all prime factors with the highest powers.

LCM = 2⁴ × 3² × 5

Step 4

LCM = 16 × 9 × 5 = 720

Answer

Quantity	Value
HCF	8
LCM	720

Special Property of HCF and LCM

This is one of the most important formulas in the chapter.

Formula Table

Formula	Meaning
HCF × LCM = Product of the two numbers	Used for two positive integers

Solved Example 2: Use the Relation Between HCF and LCM

The ratio of two numbers is 3:4 and their HCF is 4. Find their LCM.

Given

Item	Value
Ratio	3:4
HCF	4

Step 1

Let the numbers be 3x and 4x.

Step 2

Since the HCF of 3x and 4x is x, we get:

x = 4

So the numbers are:

12 and 16

Step 3

Use the formula:

HCF × LCM = Product of numbers

4 × LCM = 12 × 16

Step 4

LCM = (12 × 16)/4 = 48

Answer

The LCM of the two numbers is 48.

Important Property: HCF Is Always a Factor of LCM

This is a useful MCQ-level concept.

Concept Table

Statement	Meaning
HCF is always a factor of LCM	If a number cannot divide the LCM, it cannot be the HCF

For example, if the LCM of two numbers is 48, then 3, 16, and 24 can be factors of it, but 15 cannot be the HCF because 15 is not a factor of 48.

Special Case: HCF and LCM of Two Distinct Prime Numbers

This is a very common concept-based question.

Distinct Prime Number Table

Quantity	Result
HCF of two distinct prime numbers	1
LCM of two distinct prime numbers	Product of the two numbers

Example

For 3 and 17:

Quantity	Value
HCF	1
LCM	3 × 17 = 51

Word Problems on HCF and LCM

Many board questions from this chapter are based on word problems.

Common Word Problem Signals Table

Key phrase in question	Usually indicates
Greatest number that divides	HCF
Largest number that divides exactly	HCF
Smallest number divisible by all	LCM
First time they meet again	LCM
Equal grouping or common arrangement	Often HCF

Solved Example 3: Army Band and Contingent Problem

An army contingent of 104 members is to march behind an army band of 96 members. Both are to march in the same number of columns. What is the maximum number of columns possible?

Given

Group	Members
Army contingent	104
Army band	96

Step 1

Find the HCF of 104 and 96.

104 = 2³ × 13

96 = 2⁵ × 3

Step 2

Take the common factor with least power.

HCF = 2³ = 8

Answer

The maximum number of columns is 8.

Terminating Decimal Expansion of Rational Numbers

This is an important theorem-based concept.

If a rational number p/q is in lowest form, then its decimal expansion terminates only if the prime factorisation of q is of the form 2ᵐ × 5ⁿ, where m and n are non-negative integers.

Terminating Decimal Condition Table

Denominator in lowest form	Decimal expansion
Only factors 2 and/or 5	Terminating
Any prime factor other than 2 or 5	Non-terminating repeating

Example Table

Fraction	Denominator factors	Decimal type
1/2	2	Terminating
3/20	2² × 5	Terminating
1/3	3	Non-terminating repeating
7/15	3 × 5	Non-terminating repeating

Irrational Number Proof: Prove That √2 Is Irrational

This is one of the most important proof-based questions in the chapter.

Step 1

Assume that √2 is rational.

So it can be written in the form:

√2 = p/q

where p and q are co-prime integers and q ≠ 0.

Step 2

Square both sides:

2 = p²/q²

So:

p² = 2q²

Step 3

This means p² is divisible by 2.

So p is also divisible by 2.

Let:

p = 2k

Step 4

Substitute in p² = 2q²:

(2k)² = 2q²

4k² = 2q²

q² = 2k²

So q is also divisible by 2.

Step 5

Now both p and q are divisible by 2.

That means p and q have a common factor 2.

But this contradicts our assumption that p and q are co-prime.

Answer

Therefore, our assumption is wrong, and √2 is irrational.

Proof Type: Rational + Irrational Combination

Questions may also ask students to prove that a combination like √2 + √3 is irrational.

Solved Example 4: Prove That √2 + √3 Is Irrational

Step 1

Assume that:

√2 + √3 is rational.

Let:

√2 + √3 = p/q

Step 2

Rearrange:

√3 = p/q – √2

Step 3

Square both sides and simplify.

On simplification, √2 gets expressed in rational form.

But we already know that √2 is irrational.

Step 4

This creates a contradiction.

Answer

Therefore, √2 + √3 is irrational.

Important Board Question Types from Real Numbers

Question Pattern Table

Pattern	What is asked
Fundamental theorem	Statement or application
Prime factorisation	Write the number in product of primes
HCF and LCM	Direct calculation or theorem-based relation
Word problems	Greatest divisor, smallest multiple, or meeting-time problem
Irrational proof	Prove √2 or a similar expression is irrational
Decimal expansion	Check whether decimal will terminate or repeat

Common Mistakes Students Make in This Chapter

Common Mistakes Table

Mistake	Correct idea
Taking highest power in HCF	HCF uses lowest power
Taking only common factors in LCM	LCM uses all factors with highest powers
Forgetting HCF × LCM formula	Use it for two-number problems
Mixing rational and irrational decimal rules	Terminating and repeating decimals are rational
Forgetting co-prime condition in proofs	This is the key contradiction point
Missing factor 2 or 5 rule in denominator	Denominator decides decimal type

Quick Revision Sheet

This section is useful for one-glance revision before tests.

Quick Revision Table

Topic	Formula or key idea
Real numbers	Rational + irrational numbers
Rational number	p/q, where q ≠ 0
Irrational number	Not expressible as p/q
Fundamental theorem of arithmetic	Every composite number has unique prime factorisation
HCF rule	Common factors with lowest powers
LCM rule	All factors with highest powers
Relation	HCF × LCM = Product of two numbers
Terminating decimal condition	Denominator in lowest form should have only 2 and 5 as prime factors
Classic proof	√2 is irrational

Important Practice Questions

Practice Set Table

Question type	Practice prompt
Prime factorisation	Write 180 and 252 in prime factorised form
HCF and LCM	Find the HCF and LCM of 36, 48, and 60
Relation formula	Two numbers have HCF 6 and LCM 144. If one number is 24, find the other
Decimal expansion	Check whether 13/125 and 7/18 have terminating decimal expansions
Irrational proof	Prove that √5 is irrational
Combination proof	Prove that 2 + √3 is irrational

FAQs

Q1. What are real numbers?

Real numbers are all rational and irrational numbers taken together.

Q2. What is the fundamental theorem of arithmetic?

It states that every composite number can be expressed as a product of prime numbers in a unique way, except for the order of the factors.

Q3. How do I find HCF using prime factorisation?

Take only the common prime factors with the lowest powers.

Q4. How do I find LCM using prime factorisation?

Take all the prime factors that appear, with the highest powers.

Q5. What is the relation between HCF and LCM of two numbers?

For two positive integers, HCF × LCM = Product of the two numbers.

Q6. When does a rational number have a terminating decimal expansion?

When its denominator in lowest form has only 2 and/or 5 as prime factors.

Q7. Why is √2 irrational?

Because assuming it is rational leads to a contradiction in the co-prime condition of p and q.

Q8. Is every non-terminating decimal irrational?

No. A non-terminating repeating decimal is rational. Only a non-terminating non-repeating decimal is irrational.

Conclusion

Real Numbers is one of the most concept-rich and scoring chapters in Class 10 Maths because it combines number system understanding, factorisation logic, HCF-LCM methods, decimal expansion rules, and irrational number proofs in one place. Once students understand the chapter carefully, most questions become highly manageable.

The best way to prepare this chapter is to revise the number system hierarchy, learn prime factorisation properly, practise HCF and LCM questions regularly, and master the proof of √2 being irrational. At Deeksha Vedantu, we always remind students that Real Numbers becomes easy when logic stays clear and steps stay structured.