๐ Chapter at a Glance
๐ข Introduction โ The World of Numbers
Numbers are one of the most fundamental ideas in mathematics. We use numbers to count objects, measure quantities, compare values, describe distances and solve everyday problems.
As we progress in mathematics, we encounter different types of numbers. These numbers form a hierarchy known as the number system.
๐ Understanding the Number System
The number system helps us classify numbers according to their properties.
| Number Set | Meaning | Examples |
|---|---|---|
| Natural Numbers | Counting numbers | 1, 2, 3, 4, ... |
| Whole Numbers | Natural numbers including zero | 0, 1, 2, 3, ... |
| Integers | Positive numbers, negative numbers and zero | ..., -2, -1, 0, 1, 2, ... |
| Rational Numbers | Numbers expressible as p/q | 1/2, -3/4, 5 |
| Irrational Numbers | Numbers not expressible as p/q | โ2, โ3, ฯ |
| Real Numbers | Rational and irrational numbers together | All numbers on the number line |
1๏ธโฃ Natural Numbers
Natural numbers are the numbers used for counting.
The set of natural numbers starts with 1.
0๏ธโฃ Whole Numbers
Whole numbers consist of natural numbers together with zero.
โโ Integers
Integers include positive whole numbers, negative whole numbers and zero.
Examples:
- -10
- -5
- 0
- 7
- 15
ยฝ Rational Numbers
A number is called rational if it can be expressed in the form:
where:
- p is an integer.
- q is an integer.
- q โ 0.
๐ Why Every Integer is Rational
Every integer can be written as a fraction whose denominator is 1.
๐ข Rational Numbers and Decimal Representation
The decimal expansion of a rational number is either:
- Terminating
- Non-terminating but recurring
โ Irrational Numbers
An irrational number is a real number that cannot be expressed in the form p/q, where p and q are integers and q โ 0.
Examples include:
The number ฯ is another well-known irrational number.
โ๏ธ Rational vs Irrational Numbers
| Rational Numbers | Irrational Numbers |
|---|---|
| Can be written as p/q | Cannot be written as p/q |
| p and q are integers | No such integer fraction exists |
| Decimal may terminate | Decimal does not terminate |
| Decimal may repeat | Decimal does not repeat periodically |
| Example: 1/2 | Example: โ2 |
๐ Real Numbers
Real numbers consist of both rational and irrational numbers.
Every real number can be represented by a point on the number line.
๐ Real Numbers on the Number Line
The number line provides a geometric representation of real numbers.
Numbers increase as we move towards the right and decrease as we move towards the left.
On a number line:
โ Understanding Square Roots
Some square roots are rational while others are irrational.
Since 5 is an integer, โ25 is rational.
โ2 cannot be expressed as p/q, so it is irrational.
๐ Important Properties of Real Numbers
| Property | Example |
|---|---|
| Closure under addition | 2 + 3 = 5 |
| Closure under multiplication | 2 ร 3 = 6 |
| Commutative addition | a + b = b + a |
| Commutative multiplication | ab = ba |
| Associative addition | (a+b)+c = a+(b+c) |
| Associative multiplication | (ab)c = a(bc) |
| Distributive property | a(b+c) = ab+ac |
๐ง Solved Examples
Classify:
It is in the form p/q, where p and q are integers and q โ 0.
Answer: Rational numberWhich is irrational?
โ2 cannot be represented as p/q.
Answer: โ2 is irrational.Express -7 as a rational number.
7 is an integer and therefore rational.
Answer: Rational numberConsider:
The decimal terminates.
Therefore, it is rational.
The decimal repeats.
Therefore, it represents a rational number.
โญ Important Facts to Remember
- Every natural number is a whole number.
- Every whole number is an integer.
- Every integer is a rational number.
- Every rational number is a real number.
- Irrational numbers are also real numbers.
- Every real number can be represented on the number line.
- Zero is rational.
- The denominator of a rational number cannot be zero.
- โ2, โ3 and โ5 are irrational.
- โ4 = 2, so โ4 is rational.
๐ช Number System Hierarchy
Irrational numbers are also contained within the real numbers but are not rational.
๐ Chapter Practice Questions
โ Practice Question Solutions
Natural numbers are counting numbers:
Whole numbers consist of zero and all natural numbers.
Integers include negative numbers, zero and positive numbers.
A rational number is a number that can be written in the form p/q, where p and q are integers and q โ 0.
An irrational number cannot be expressed in the form p/q where p and q are integers and q โ 0.
Every integer n can be written as:
Therefore, every integer is rational.
4 is an integer and therefore rational.
Answer: Rationalโ7 cannot be expressed as a fraction of two integers.
Answer: IrrationalYes.
Therefore, zero is rational.
Rational numbers can be expressed as p/q, while irrational numbers cannot be expressed in that form.
Examples:
Examples:
0.625 is a terminating decimal.
Therefore, it is rational.
0.121212... is a repeating decimal.
Therefore, it is rational.
Real numbers consist of both rational and irrational numbers.
๐ฏ Class 9 Maths Chapter 3 MCQs
Answer: B. Whole numbers
Answer: C. โ2
Answer: D. โ25
Answer: B. Rational
Answer: D. 0
Answer: B. 0
Answer: A. 0.5
Answer: C. Every rational number is real
Answer: B. Rational
Answer: A. -8
Answer: C. Non-terminating and non-repeating
Answer: C. Rational and irrational numbers
Answer: A. 1/2
Answer: D. โ-1
Answer: A. All whole numbers are integers
๐ง Competency-Based Questions
A student writes:
The student says that โ36 is irrational.
Is the student correct?
Answer: No.
Since 6 is an integer, it is rational.
A number has a decimal expansion that terminates after three decimal places.
Can it be rational?
Answer: Yes. Every terminating decimal represents a rational number.
A student claims that every real number is rational.
Is the statement correct?
Answer: No.
Real numbers include both rational and irrational numbers.
๐ Class 9 Maths Chapter 3 Worksheet
Section A โ Fill in the Blanks
1. The smallest natural number is ______.
2. The smallest whole number is ______.
3. A rational number can be written in the form ______.
4. The denominator of a rational number cannot be ______.
5. โ2 is a/an ______ number.
Section B โ Classify the Numbers
6. 7
7. -4
8. 0
9. 3/5
10. โ11
11. โ81
Section C โ Rational or Irrational?
12. 0.75
13. 0.3333...
14. โ5
15. ฯ
Section D โ Short Answer Questions
16. Why is every integer a rational number?
17. Explain the difference between rational and irrational numbers.
18. Explain why โ25 is rational.
Section E โ Higher Order Thinking
19. Can a number be both an integer and a rational number? Give an example.
20. Are all real numbers rational? Explain.
21. Give two examples each of rational and irrational numbers.
โ Worksheet Answer Key
- 1
- 0
- p/q
- 0
- Irrational
- Natural, whole, integer, rational and real
- Integer, rational and real
- Whole, integer, rational and real
- Rational and real
- Irrational and real
- Rational, integer, whole and real
- Rational
- Rational
- Irrational
- Irrational
- Every integer n can be written as n/1.
- Rational numbers can be expressed as p/q; irrational numbers cannot.
- โ25 = 5, and 5 is an integer.
- Yes. For example, 5 = 5/1.
- No. Real numbers include both rational and irrational numbers.
- Rational: 1/2, 3/4. Irrational: โ2, โ3.
โญ Important Questions for Revision
- Define natural numbers.
- Define whole numbers.
- What are integers?
- Define rational numbers.
- Define irrational numbers.
- What are real numbers?
- Explain why every integer is rational.
- Give examples of irrational numbers.
- Differentiate between rational and irrational numbers.
- Explain the relationship between different number systems.
- Classify given numbers into appropriate number sets.
- Explain terminating and recurring decimals.
- Represent real numbers on a number line.
โก One-Minute Revision
1, 2, 3, ...
0, 1, 2, 3, ...
..., -2, -1, 0, 1, 2, ...
p/q, q โ 0
Non-terminating & non-repeating
Rational + Irrational
5 โ Rational
Irrational
๐ฏ Exam Tips
- Remember the hierarchy: Natural โ Whole โ Integers โ Rational โ Real.
- Every integer is rational because it can be written with denominator 1.
- A rational decimal can terminate or repeat.
- An irrational decimal is non-terminating and non-repeating.
- Do not assume every square root is irrational. For example, โ49 = 7.
- The denominator in p/q must never be zero.
- Real numbers include both rational and irrational numbers.
โ Frequently Asked Questions
What is Chapter 3 of Class 9 Maths?
The chapter is titled The World of Numbers.
What is a rational number?
A rational number can be expressed in the form p/q, where p and q are integers and q is not zero.
What is an irrational number?
An irrational number cannot be expressed as p/q where p and q are integers and q is not zero.
Is zero a rational number?
Yes. Zero can be written as 0/1.
Is โ16 rational?
Yes. โ16 = 4, and 4 is an integer and therefore rational.
Is โ2 irrational?
Yes. โ2 cannot be expressed as a ratio of two integers.
What are real numbers?
Real numbers consist of both rational and irrational numbers and can be represented on the number line.
๐ Class 9 Maths Chapter 3 Study Material
Students searching for Class 9 Maths Chapter 3 notes, The World of Numbers Class 9 notes, Class 9 Maths Chapter 3 2026-27, Class 9 number system notes, rational numbers Class 9, irrational numbers Class 9, real numbers Class 9, Class 9 Maths Chapter 3 MCQs, Class 9 Maths Chapter 3 worksheet, and Class 9 Maths important questions can use this Study Notes Hub page for learning, revision and practice.
This page covers the important ideas related to the world of numbers, including natural numbers, whole numbers, integers, rational numbers, irrational numbers, real numbers, decimal representations, number classification, solved examples, practice questions, MCQs and a student-friendly worksheet.