CBSE Class 9 Maths Revision Notes Chapter 3 The World of Numbers

The number system expands from natural numbers to integers, rational numbers, irrational numbers and finally real numbers. Decimal expansions and number-line representation help distinguish and understand different types of numbers.

The chapter The World of Numbers explains how number systems developed to meet different mathematical needs. Counting gave rise to natural numbers, zero expanded them into whole numbers, and negative quantities led to integers.

These CBSE Class 9 Maths Revision Notes Chapter 3 also cover rational numbers, irrational numbers, real numbers and decimal expansions. Students can revise important properties, number-line methods and common proofs for the 2026–27 academic year.

Key Takeaways

• Rational number: Any number expressible as p/q, where p and q are integers and q ≠ 0.
• Irrational number: A number that cannot be expressed as p/q.
• Real numbers: The union of rational and irrational numbers.
• Decimal test: Rational numbers have terminating or recurring decimals, while irrational numbers have non-terminating, non-recurring decimals.

Access Class 9 Maths Chapter 3 The World of Numbers Notes in 30 Minutes

Revise the chapter in three parts:

First 10 minutes: Natural numbers, zero, integers and rational numbers
Next 10 minutes: Number-line representation, absolute value and density
Final 10 minutes: Irrational numbers, real numbers and decimal expansions

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Number Systems in Class 9 Maths Chapter 3 Notes

Different number systems developed as mathematical needs increased. Each new set contains or extends an earlier set.

Natural Numbers

Natural numbers are the counting numbers.

N = {1, 2, 3, 4, ...}

They are used to count objects.

Examples:

• 5 books
• 12 students
• 100 trees

Natural numbers do not include zero or negative numbers.

Whole Numbers

Whole numbers include zero along with natural numbers.

W = {0, 1, 2, 3, 4, ...}

Every natural number is a whole number, but zero is not a natural number under this definition.

Integers

Integers include:

• Positive numbers
• Negative numbers
• Zero

Z = {..., −3, −2, −1, 0, 1, 2, 3, ...}

Integers are useful for representing temperature, debt, height below sea level and financial loss.

Rational Numbers

A rational number can be expressed as:

p/q

where p and q are integers and q ≠ 0.

Examples:

3/4, −5/7, 8, 0, 1.25

An integer is rational because it can be written with denominator 1.

For example:

5 = 5/1

−9 = −9/1

Irrational Numbers

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

Examples:

√2, √3, √5, π

Their decimal expansions are:

• Non-terminating
• Non-recurring

Real Numbers

Real numbers include all rational and irrational numbers.

R = Q ∪ I

Every point on the number line represents a real number.

Number-System Hierarchy

The inclusion can be written as:

N ⊂ W ⊂ Z ⊂ Q ⊂ R

Irrational numbers are also part of R but are separate from Q.

Zero and Integers in CBSE Class 9 Maths Chapter 3 Notes

Zero became an important mathematical number when formal rules were developed for using it in arithmetic.

Meaning of Zero

Zero represents the absence of quantity.

It also acts as the point separating positive and negative numbers on the number line.

Brahmagupta’s Rules for Zero

For any number a:

a + 0 = a

a − 0 = a

a × 0 = 0

a − a = 0

Division by zero is not defined.

Positive and Negative Integers

Positive integers lie to the right of zero.

Negative integers lie to the left of zero.

Example:

−5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5

Brahmagupta described positive numbers as fortunes and negative numbers as debts.

Rules for Adding Integers

Rules for Multiplying Integers

Examples:

(−3) × 4 = −12

(−3) × (−4) = 12

Subtracting a Negative Number

Subtracting a negative number is the same as adding a positive number.

a − (−b) = a + b

Example:

10 − (−5) = 15

Rational Numbers in Class 9 Maths Revision Notes Chapter 3

Rational numbers include integers and fractions. They can be positive, negative or zero.

Standard Form of a Rational Number

A rational number p/q is in standard form when:

• q ≠ 0
• p and q have no common factor other than 1
• The denominator is usually taken as positive

Example:

12/30 = 2/5

Here, 2/5 is the standard form.

Equivalent Rational Numbers

Equivalent rational numbers have the same value.

For example:

1/2 = 2/4 = 3/6 = 50/100

Multiplying or dividing the numerator and denominator by the same non-zero integer gives an equivalent rational number.

Equality of Rational Numbers

Two rational numbers a/b and c/d are equal if:

ad = bc

Example:

2/3 and 4/6

2 × 6 = 12

3 × 4 = 12

Therefore:

2/3 = 4/6

Addition of Rational Numbers

For equal denominators:

a/b + c/b = (a + c)/b

Example:

2/7 + 3/7 = 5/7

For different denominators, first find a common denominator.

Example:

2/5 + 3/10

= 4/10 + 3/10

= 7/10

Subtraction of Rational Numbers

a/b − c/b = (a − c)/b

Example:

5/6 − 1/4

LCM of 6 and 4 = 12

= 10/12 − 3/12

= 7/12

Multiplication of Rational Numbers

a/b × c/d = ac/bd

Example:

2/3 × 3/10

= 6/30

= 1/5

Division of Rational Numbers

a/b ÷ c/d = a/b × d/c

where c ≠ 0.

Example:

2/3 ÷ 3/10

= 2/3 × 10/3

= 20/9

Properties of Rational Numbers

Closure Property

Rational numbers are closed under:

• Addition
• Subtraction
• Multiplication
• Division, except division by zero

This means that the result remains a rational number.

Distributive Property

For rational numbers p, q and r:

p(q + r) = pq + pr

Example:

1/2 × (3/4 + 8/3)

= 1/2 × 3/4 + 1/2 × 8/3

Number Line in The World of Numbers Notes

Integers and rational numbers can be represented on a number line.

Numbers increase towards the right and decrease towards the left.

Representing Integers

Mark zero as the origin.

• Positive integers lie to the right.
• Negative integers lie to the left.
• Equal intervals represent equal units.

Representing a Positive Rational Number

To represent p/q:

1. Identify the two integers between which the number lies.
2. Divide the unit interval into q equal parts.
3. Move p parts from zero towards the right.

Example:

To represent 3/4, divide the interval from 0 to 1 into four equal parts and mark the third part.

Representing a Negative Rational Number

For a negative rational number, move towards the left.

Example:

−3/4 lies between −1 and 0.

Divide that interval into four equal parts and mark three parts to the left of zero.

Representing an Improper Fraction

Example:

9/4 = 2 1/4

It lies between 2 and 3.

Divide the interval between 2 and 3 into four equal parts and mark the first part after 2.

Absolute Value in Class 9 Maths Chapter 3 Notes

The absolute value of a number is its distance from zero on the number line.

It is written as:

|x|

Examples:

|5/3| = 5/3

|−5/3| = 5/3

|0| = 0

Absolute value is always non-negative.

|x| ≥ 0

Distance Between Two Rational Numbers

The distance between two numbers a and b is:

|a − b|

Example:

Distance between −4 and 3:

|−4 − 3|

= |−7|

= 7

Density of Rational Numbers

Rational numbers are dense.

This means that between any two rational numbers, there are infinitely many rational numbers.

Finding One Rational Number Between Two Numbers

For rational numbers a and b, one rational number between them is:

(a + b)/2

Example:

Between 1 and 3/2:

(1 + 3/2)/2

= (5/2)/2

= 5/4

Therefore, 5/4 lies between 1 and 3/2.

Finding Several Rational Numbers

To find several rational numbers between two fractions:

1. Convert them to equivalent fractions with a larger common denominator.
2. Select numerators between the two given numerators.

Example:

Between 2/5 and 3/5:

2/5 = 20/50

3/5 = 30/50

Some rational numbers between them are:

21/50, 22/50, 23/50, 24/50, 25/50

Irrational Numbers in Class 9 Chapter 3 Maths Notes

Some lengths and values cannot be expressed as ratios of integers.

These are called irrational numbers.

Definition of an Irrational Number

A number is irrational if it cannot be written as p/q, where p and q are integers and q ≠ 0.

Examples:

√2, √3, √5, √10, π

Decimal Expansion of Irrational Numbers

An irrational number has a decimal expansion that:

• Never terminates
• Never repeats in a fixed block

Examples:

√2 = 1.4142135623...

π = 3.1415926535...

Proof of Irrationality of Square Root 2

The proof uses contradiction.

Step 1: Assume √2 Is Rational

Suppose:

√2 = p/q

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

Step 2: Square Both Sides

2 = p²/q²

Therefore:

p² = 2q²

Step 3: Show That p Is Even

Since p² is divisible by 2, p must be even.

Let:

p = 2k

Step 4: Substitute p = 2k

p² = 2q²

4k² = 2q²

Therefore:

q² = 2k²

So q is also even.

Step 5: Reach a Contradiction

Both p and q are even, so they have a common factor 2.

This contradicts the assumption that p and q are co-prime.

Therefore:

√2 is irrational.

Construction of Square Root 2 on the Number Line

To locate √2:

1. Mark O at 0 and A at 1 on the number line.
2. Draw AB perpendicular to OA with AB = 1 unit.
3. Join O to B.
4. By the Pythagoras theorem:

OB² = OA² + AB²

= 1² + 1²

= 2

Therefore:

OB = √2

1. With O as centre and OB as radius, draw an arc cutting the number line at P.
2. Point P represents √2.

The same method can be extended to construct √3, √5 and other square-root lengths.

Pi as an Irrational Number

π represents the ratio of the circumference of a circle to its diameter.

Its decimal expansion is:

π = 3.1415926535...

It is non-terminating and non-recurring.

Therefore, π is irrational.

Fractions such as 22/7 and 3.1416 are approximations, not exact values of π.

Decimal Expansions in Class 9 Maths Chapter 3 Notes

Decimal expansions help identify whether a number is rational or irrational.

Terminating Decimal

A terminating decimal ends after a finite number of digits.

Examples:

3/8 = 0.375

7/20 = 0.35

13/250 = 0.052

Non-Terminating Recurring Decimal

A recurring decimal continues forever but repeats a fixed digit or block.

Examples:

1/3 = 0.333...

5/11 = 0.454545...

1/7 = 0.142857142857...

These numbers are rational.

Non-Terminating Non-Recurring Decimal

Such a decimal continues forever without repeating a fixed pattern.

Examples:

√2 = 1.414213...

π = 3.141592...

These numbers are irrational.

Decimal-Type Comparison

Test for Terminating Decimal Expansion

Let p/q be a rational number in lowest terms.

Its decimal expansion terminates if the prime factorisation of q contains only:

• 2
• 5
• Both 2 and 5

Therefore:

q = 2^m × 5^n

for non-negative integers m and n.

Examples

7/20

20 = 2² × 5

Therefore, the decimal terminates.

4/15

15 = 3 × 5

The denominator contains 3.

Therefore, the decimal is non-terminating recurring.

13/250

250 = 2 × 5³

Therefore, the decimal terminates.

Converting Terminating Decimals into p/q Form

Example:

0.35

= 35/100

= 7/20

Example:

2.125

= 2125/1000

= 17/8

Converting Pure Recurring Decimals into p/q Form

A pure recurring decimal starts repeating immediately after the decimal point.

Example: Convert 0.666... into p/q

Let:

x = 0.666...

Multiply by 10:

10x = 6.666...

Subtract:

10x − x = 6.666... − 0.666...

9x = 6

x = 6/9

x = 2/3

Example: Convert 0.454545... into p/q

Let:

x = 0.454545...

Since two digits repeat, multiply by 100:

100x = 45.454545...

Subtract:

100x − x = 45

99x = 45

x = 45/99

x = 5/11

Converting General Recurring Decimals into p/q Form

A general recurring decimal has some non-repeating digits followed by repeating digits.

Example: Convert 0.1666... into p/q

Let:

x = 0.1666...

Multiply by 10:

10x = 1.666...

Multiply again by 10:

100x = 16.666...

Subtract:

100x − 10x = 16.666... − 1.666...

90x = 15

x = 15/90

x = 1/6

General Method

Cyclic Numbers

Some recurring blocks show repeating cyclic arrangements.

For example:

1/7 = 0.142857142857...

The repeating block is 142857.

Multiplying it by 1 to 6 gives cyclic rearrangements:

142857 × 1 = 142857
142857 × 2 = 285714
142857 × 3 = 428571
142857 × 4 = 571428
142857 × 5 = 714285
142857 × 6 = 857142

The same digits repeat in a different order.

Real Numbers and the Number Line

Rational and irrational numbers together form the set of real numbers.

Every real number has a unique position on the number line.

Examples include:

• −3
• 0
• 2/5
• √2
• π
• 4.75

Rational and Irrational Numbers

Important Rules in Class 9 Maths Chapter 3

Quick Revision of The World of Numbers Class 9 Maths Chapter 3

• Natural numbers begin from 1.
• Whole numbers include zero.
• Integers include positive and negative numbers.
• Rational numbers can be written as p/q.
• The denominator of a rational number cannot be zero.
• Irrational numbers cannot be written as p/q.
• Real numbers include rational and irrational numbers.
• Rational numbers are dense on the number line.
• Absolute value represents distance from zero.
• √2 and π are irrational.
• Rational decimals terminate or recur.
• Irrational decimals neither terminate nor recur.
• A rational decimal terminates when its denominator has only factors 2 and 5.
• Recurring decimals can be converted into fractions using algebra.

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