Important Questions Class 9 Maths Chapter 3 The World of Numbers 2026-2027

The World of Numbers explains how numbers grew from counting numbers to real numbers. It covers natural numbers, zero, integers, rational numbers, irrational numbers, decimal expansions and the real number line.

Important questions class 9 maths chapter 3 help students practise proofs, calculations, number line questions, decimal conversion and rational number problems. This chapter needs concept clarity because later algebra, coordinate geometry and mensuration all depend on real numbers.

Class 9 Maths Chapter 3 begins with a simple human need: counting. It then moves from natural numbers to zero, negative numbers, rational numbers, irrational numbers and real numbers. The chapter also connects mathematics with Indian contributions such as Śhūnya and Brahmagupta’s rules.

Students should not revise this chapter only through definitions. Exam questions test whether students can classify numbers, prove irrationality, convert decimals into fractions, locate numbers on a number line and find rational numbers between two values.

Key Takeaways from Class 9 Maths Chapter 3

Important Questions Class 9 Maths Chapter 3 with Answers

These questions cover the full chapter. Students should practise both direct answers and stepwise solutions.

Important Questions Class 9 Maths Chapter 3: Basic Concepts

Q1. What are natural numbers?
Natural numbers are counting numbers.

They are written as:

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

Natural numbers began from the human need to count objects, animals and quantities.

Q2. Are natural numbers closed under addition and subtraction?
Natural numbers are closed under addition.

For example:

5 + 3 = 8

8 is also a natural number.

Natural numbers are not closed under subtraction.

For example:

3 − 5 = −2

−2 is not a natural number.

Q3. What is zero?
Zero is the number that represents nothing or absence of quantity.

Brahmagupta defined zero as the result of subtracting a number from itself.

For example:

7 − 7 = 0

Zero made place value, arithmetic and negative numbers easier to express.

Q4. What are integers?
Integers include negative numbers, zero and positive numbers.

They are written as:

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

Integers help represent real-life situations such as debts, losses, temperatures below zero and sea levels below ground.

Q5. 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.

Examples include 3/5, −7/9, 4, 0 and −12.

Integers are also rational numbers because they can be written with denominator 1.

Class 9 Maths Chapter 3 Question Answer on Integers

Integers are important because they connect zero, positive numbers and negative numbers. Students should understand the debt and fortune model clearly.

Q1. Calculate: (−12) × 5
(−12) × 5 = −60.

A negative number multiplied by a positive number gives a negative number.

Q2. Calculate: (−8) × (−7)
(−8) × (−7) = 56.

The product of two negative numbers is positive.

Q3. The temperature in Ladakh is 4°C at noon. It drops by 15°C at midnight. Find the midnight temperature.
Midnight temperature = 4°C − 15°C

= −11°C

So, the midnight temperature is −11°C.

Q4. A trader takes a loan of ₹850, earns ₹1,200, and then loses ₹450. Find his final standing.
Loan = −850

Profit = +1200

Loss = −450

Final standing = −850 + 1200 − 450

= 350 − 450

= −100

The trader has a final debt of ₹100.

Q5. Why is subtracting a negative number the same as adding a positive number?
Subtracting a negative number means removing a debt.

For example:

10 − (−5) = 10 + 5 = 15

If ₹5 of debt gets removed, the final value increases by ₹5.

Rational Numbers Class 9 Important Questions

Rational numbers class 9 questions appear in operations, proof-based questions and number line problems. Students should remember the p/q form and q ≠ 0 condition.

Q1. Prove that 2/3 and 4/6 are equal rational numbers.
Two rational numbers a/b and c/d are equal if:

ad = bc

For 2/3 and 4/6:

2 × 6 = 12

3 × 4 = 12

Since both products are equal, 2/3 = 4/6.

Q2. Find the sum: 2/5 + 3/10
LCM of 5 and 10 = 10

2/5 = 4/10

So,

2/5 + 3/10 = 4/10 + 3/10

= 7/10

Q3. Find the difference: 5/6 − 1/4
LCM of 6 and 4 = 12

5/6 = 10/12

1/4 = 3/12

So,

5/6 − 1/4 = 10/12 − 3/12

= 7/12

Q4. Find the product: 2/3 × 3/10
2/3 × 3/10 = 6/30

= 1/5

Q5. Find the quotient: 2/3 ÷ 3/10
2/3 ÷ 3/10 = 2/3 × 10/3

= 20/9

So, the quotient is 20/9.

Important Questions Class 9 Maths Chapter 3 on Number Line

Number line questions test whether students understand the size and position of numbers. Rational numbers can lie between integers.

Q1. How do you represent 3/4 on a number line?
To represent 3/4, divide the interval between 0 and 1 into four equal parts.

Start from 0 and move three parts to the right. That point represents 3/4.

Q2. How do you represent −5/4 on a number line?
−5/4 lies to the left of 0.

Since −5/4 = −1¼, it lies between −1 and −2.

Divide the interval between −1 and −2 into four equal parts. Mark one part left of −1.

Q3. What is the absolute value of −7/3?
The absolute value gives distance from 0.

|−7/3| = 7/3

Distance is always non-negative.

Q4. Find the distance between −4 and 3 on the number line.
Distance = |a − b|

= |3 − (−4)|

= |3 + 4|

= 7

The distance is 7 units.

Q5. Find three rational numbers between 1 and 2.
Three rational numbers between 1 and 2 are:

5/4, 3/2 and 7/4

All three lie greater than 1 and less than 2.

Irrational Numbers Class 9 Important Questions

Irrational numbers class 9 questions test classification, decimal expansion and proof skills. Students should remember that non-terminating, non-repeating decimals are irrational.

Q1. What are irrational numbers?
Irrational numbers cannot be expressed in the form p/q, where p and q are integers and q ≠ 0.

Examples include √2, √3, √5 and π.

Their decimal expansions are non-terminating and non-repeating.

Q2. Is √81 rational or irrational?
√81 = 9.

9 is an integer.

So, √81 is a rational number.

Q3. Is √12 rational or irrational?
√12 = √(4 × 3)

= 2√3

Since √3 is irrational, 2√3 is also irrational.

So, √12 is irrational.

Q4. Is 0.33333... rational or irrational?
0.33333... is rational.

It is a non-terminating repeating decimal.

0.33333... = 1/3

Q5. Is 1.01001000100001... rational or irrational?
This number is irrational.

It does not terminate and does not repeat one fixed block of digits.

Prove Root 2 Is Irrational Class 9

This proof is one of the most important proof-based questions from Class 9 Maths Chapter 3. Write each contradiction step clearly.

Prove Root 2 Is Irrational Class 9: Stepwise Proof

Q1. Prove that √2 is irrational.
Assume that √2 is rational.

Then it can be written as p/q, where p and q are co-prime integers and q ≠ 0.

√2 = p/q

Squaring both sides:

2 = p²/q²

So,

p² = 2q²

This means p² is even. Therefore, p is also even.

Let p = 2k.

Substitute in p² = 2q²:

(2k)² = 2q²

4k² = 2q²

q² = 2k²

This means q² is even. Therefore, q is also even.

So, both p and q are even. This means they have a common factor 2.

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

Therefore, √2 is irrational.

Q2. Prove that √5 is irrational.
Assume that √5 is rational.

Then:

√5 = p/q

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

Squaring both sides:

5 = p²/q²

So,

p² = 5q²

This means p² is divisible by 5. Therefore, p is divisible by 5.

Let p = 5k.

Substitute:

(5k)² = 5q²

25k² = 5q²

q² = 5k²

This means q² is divisible by 5. Therefore, q is also divisible by 5.

So, p and q have a common factor 5.

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

Therefore, √5 is irrational.

Decimal Expansion Class 9 Important Questions

Decimal expansion class 9 questions are scoring when students know the rule. Check the denominator in lowest form before deciding the decimal type.

Q1. When does a rational number have a terminating decimal expansion?
A rational number p/q has a terminating decimal expansion if q has only 2 or 5 as prime factors.

The rational number must be in lowest form.

Example:

7/20 has denominator 20 = 2² × 5

So, it has a terminating decimal expansion.

Q2. Without division, decide whether 4/15 has a terminating decimal expansion.
4/15 is in lowest form.

15 = 3 × 5

The denominator has 3 as a prime factor.

So, 4/15 has a non-terminating repeating decimal expansion.

Q3. Without division, decide whether 13/250 has a terminating decimal expansion.
250 = 2 × 5³.

The denominator has only 2 and 5 as prime factors.

So, 13/250 has a terminating decimal expansion.

Q4. Convert 0.35 into p/q form.
0.35 = 35/100

= 7/20

So, 0.35 = 7/20.

Q5. Convert 0.6 repeating into p/q form.
Let x = 0.666...

Then,

10x = 6.666...

Subtract:

10x − x = 6.666... − 0.666...

9x = 6

x = 6/9

x = 2/3

So, 0.6 repeating = 2/3.

Q6. Convert 0.45 repeating into p/q form.
Let x = 0.454545...

Then,

100x = 45.454545...

Subtract:

100x − x = 45.454545... − 0.454545...

99x = 45

x = 45/99

x = 5/11

So, 0.45 repeating = 5/11.

Q7. Convert 0.16, where 6 repeats, into p/q form.
Let x = 0.1666...

Since 1 is non-repeating:

10x = 1.666...

Now multiply again by 10:

100x = 16.666...

Subtract:

100x − 10x = 16.666... − 1.666...

90x = 15

x = 15/90

x = 1/6

So, 0.1666... = 1/6.

Real Numbers Class 9 Important Questions

Real numbers class 9 questions test whether students can connect rational and irrational numbers to the number line. Every point on the number line represents a real number.

Q1. What are real numbers?
Real numbers are all rational and irrational numbers together.

They fill the complete number line.

Examples include −3, 0, 5/7, √2, π and 4.25.

Q2. How are rational numbers different from irrational numbers?
Rational numbers can be written in the form p/q.

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

Rational decimals terminate or repeat. Irrational decimals never terminate and never repeat.

Q3. Are all integers rational numbers?
Yes, all integers are rational numbers.

For example:

−5 = −5/1

0 = 0/1

9 = 9/1

So, every integer can be written in p/q form.

Q4. Are all rational numbers real numbers?
Yes, all rational numbers are real numbers.

Real numbers include both rational and irrational numbers.

Q5. Are all real numbers rational numbers?
No, all real numbers are not rational numbers.

Numbers such as √2, √3 and π are real but irrational.

Class 9 Maths Chapter 3 MCQs with Answers

Class 9 Maths Chapter 3 MCQs help students check classification rules quickly. These are useful before solving long proof or decimal questions.

Q1. Which of the following is a natural number?
(a) 0
(b) −4
(c) 7
(d) 2/3

Answer: (c) 7
Natural numbers are counting numbers starting from 1.

Q2. Which of the following is irrational?
(a) 4/5
(b) √9
(c) √7
(d) 0.25

Answer: (c) √7
√7 cannot be written as p/q.

Q3. Which decimal expansion represents a rational number?
(a) 1.414213... non-repeating
(b) 0.272727...
(c) √5
(d) π

Answer: (b) 0.272727...
A repeating decimal is rational.

Q4. Which denominator gives a terminating decimal?
(a) 18
(b) 45
(c) 40
(d) 21

Answer: (c) 40
40 = 2³ × 5, so it gives a terminating decimal.

Q5. Which statement is true?
(a) Every real number is rational
(b) Every irrational number is real
(c) Every rational number is natural
(d) Every integer is irrational

Answer: (b) Every irrational number is real
Real numbers include both rational and irrational numbers.

Class 9 Maths Chapter 3 Extra Questions with Answers

Class 9 Maths Chapter 3 extra questions help students practise calculation, density of rational numbers and mixed concept application.

Q1. Find six rational numbers between 3 and 4.
Write 3 and 4 with denominator 10:

3 = 30/10

4 = 40/10

Six rational numbers between them are:

31/10, 32/10, 33/10, 34/10, 35/10, 36/10

Q2. Find five rational numbers between 2/5 and 3/5.
Convert to a larger common denominator.

2/5 = 20/50

3/5 = 30/50

Five rational numbers are:

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

Q3. Find five rational numbers between 1/6 and 2/5.
LCM of 6 and 5 = 30

1/6 = 5/30

2/5 = 12/30

Five rational numbers are:

6/30, 7/30, 8/30, 9/30, 10/30

Q4. Find x if x/3 + x/5 = 16/15.
x/3 + x/5 = 16/15

LCM of 3 and 5 = 15

5x/15 + 3x/15 = 16/15

8x/15 = 16/15

8x = 16

x = 2

Q5. If a and b are non-zero rational numbers and a/b + 1 = 0, determine whether ab is positive or negative.
a/b + 1 = 0

a/b = −1

So, a = −b

This means a and b have opposite signs.

Therefore, ab is negative.

Q6. Find the rational number between 7/12 and 5/6 using average method.
A rational number between a and b is:

(a + b)/2

= (7/12 + 5/6)/2

= (7/12 + 10/12)/2

= (17/12)/2

= 17/24

So, 17/24 lies between 7/12 and 5/6.

HOTS Questions on The World of Numbers Class 9

HOTS questions test whether students can apply number properties in unfamiliar situations. Write each step clearly.

Q1. Is 0.999... equal to 1?
Yes, 0.999... is equal to 1.

Let x = 0.999...

Then 10x = 9.999...

Subtract:

10x − x = 9.999... − 0.999...

9x = 9

x = 1

So, 0.999... = 1.

Q2. Can there be a rational number and an irrational number between 1 and 2?
Yes, both types of numbers lie between 1 and 2.

For example, 3/2 is rational.

√2 is irrational and lies between 1 and 2.

Q3. Is the sum of two irrational numbers always irrational?
No, the sum of two irrational numbers is not always irrational.

For example:

√2 + (−√2) = 0

0 is rational.

Q4. Is the product of two irrational numbers always irrational?
No, the product of two irrational numbers is not always irrational.

For example:

√2 × √2 = 2

2 is rational.

Q5. Why are rational numbers called dense?
Rational numbers are called dense because between any two rational numbers, there are infinitely many rational numbers.

For example, between 1 and 2, we can find 3/2, 5/4, 7/4 and many more.