Class 9 Maths Chapter 3 The World of Numbers Solutions

Class 9 Maths Chapter 3 The World of Numbers Solutions helps students understand the basics of the number system in a simple and clear manner. In this chapter, students learn about natural numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers with solved examples and practice questions. These solutions are useful for homework, revision, and exam preparation according to the latest NCERT syllabus.

Students can download official textbooks and study materials from the NCERT website:
https://ncert.nic.in/

For CBSE curriculum updates and learning resources, visit:
https://cbseacademic.nic.in/

Students can also access digital learning resources from:
https://diksha.gov.in/

Topics Covered in Class 9 Maths Chapter 3 The World of Numbers Solutions

• Natural Numbers

• Whole Numbers

• Integers

• Rational Numbers

• Irrational Numbers

• Real Numbers

• Decimal Expansion

• Representation of Numbers on Number Line

• Laws of Exponents

NCERT Solutions for Class 9 Maths Chapter 3 Natural Numbers

Natural numbers are counting numbers that begin from 1.

Examples:

1, 2, 3, 4, 5…

Properties of Natural Numbers

Closure Property

Natural numbers are closed under addition and multiplication.

Example:

4 + 3 = 7

5 × 2 = 10

Natural numbers are not closed under subtraction because subtraction may produce negative numbers.

Example:

3 − 7 = -4

Class 9 Maths Chapter 3 Whole Numbers

Whole numbers include zero along with natural numbers.

Examples:

0, 1, 2, 3, 4, 5…

Important Facts About Whole Numbers

• Zero is the smallest whole number.

• Whole numbers do not include negative numbers.

• Whole numbers can be represented on the number line.

NCERT Solutions for Class 9 Maths Chapter 3 Integers

Integers include positive numbers, negative numbers, and zero.

Examples:

… -5, -4, -3, -2, -1, 0, 1, 2, 3 …

Addition of Integers

Same Sign Integers

Add the numbers and keep the same sign.

Example:

(-7) + (-2) = -9

Different Sign Integers

Subtract the smaller number from the larger number and keep the sign of the larger number.

Example:

9 + (-4) = 5

Rational Numbers in Class 9 Maths Chapter 3

A rational number can be written in the form:

p/q where q ≠ 0

Examples of rational numbers:

1/2, 3/5, -7/9

Properties of Rational Numbers

Rational numbers are closed under:

• Addition

• Subtraction

• Multiplication

Decimal Expansion of Rational Numbers

Rational numbers may have:

• Terminating decimals

• Recurring decimals

Examples:

1/4 = 0.25

2/3 = 0.666…

Irrational Numbers in The World of Numbers Solutions

Numbers that cannot be written in fractional form are called irrational numbers.

Examples:

√2, √5, π

Properties of Irrational Numbers

• Their decimal expansion is non-terminating.

• Their decimal expansion is non-repeating.

Proof that √2 is Irrational

Assume that √2 is rational and can be written as p/q where p and q are coprime integers.

Squaring both sides:

2q² = p²

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

Let:

p = 2k

Substituting:

2q² = 4k²

q² = 2k²

Thus, q is also even, which contradicts the assumption that p and q are coprime.

Hence, √2 is irrational.

Real Numbers in Class 9 Maths Chapter 3

Real numbers include both rational and irrational numbers.

Examples:

-8, 0, 4/7, √3, π

Representation of Real Numbers on Number Line

Every real number can be represented on the number line.

• Rational numbers occupy exact positions.

• Irrational numbers are represented using geometric methods.

Important Formulas in Class 9 Maths Chapter 3 The World of Numbers Solutions

Laws of Exponents

am × an = am+n

am / an = am−n

(am)n = amn

Solved Examples from Class 9 Maths Chapter 3 The World of Numbers Solutions

Example 1

Find:

2/7 + 3/7

Solution

= (2 + 3)/7

= 5/7

Example 2

Find the decimal expansion of:

5/8

Solution

5 ÷ 8 = 0.625

Therefore, it is a terminating decimal.

Example 3

State whether √11 is rational or irrational.

Solution

√11 cannot be written in p/q form.

Hence, √11 is irrational.

Practice Questions for Class 9 Maths Chapter 3

Short Answer Questions

1. Define rational numbers.

2. Give examples of irrational numbers.

3. Is zero a whole number?

4. Write the decimal form of 3/8.

5. State whether √13 is rational or irrational.

Long Answer Questions

1. Prove that √3 is irrational.

2. Explain closure property for integers.

3. Represent real numbers on the number line.

4. Explain decimal expansion of rational numbers.

FAQs – Class 9 Maths Chapter 3 The World of Numbers Solutions

What is a rational number?

A rational number is any number written in the form p/q where q ≠ 0.

What are irrational numbers?

Irrational numbers cannot be expressed as fractions and have non-terminating non-repeating decimal expansions.

Is π a real number?

Yes, π is an irrational real number.

Why is the number system important?

The number system forms the foundation of mathematics and helps students understand algebra, geometry, and advanced calculations.

Conclusion

Class 9 Maths Chapter 3 The World of Numbers Solutions provides complete explanations of natural numbers, integers, rational numbers, irrational numbers, and real numbers with examples and practice questions. These solutions help students improve conceptual understanding and prepare effectively for school exams.

Students should regularly practice all examples and exercises to strengthen their mathematical skills.

Suggested Internal Links

• Class 9 Maths Chapter 1 Solutions

• Class 9 Maths Chapter 2 Solutions

• NCERT Solutions for Class 9 Science

• CBSE Class 9 Study Material

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Class 9 Maths Chapter 3 The World of Numbers Solutions

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Rational and irrational numbers in Class 9 Maths Chapter 3

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Real numbers representation on number line Class 9 Maths