Lecture 2: Irrational Numbers and Real Numbers

🧮 Class 9 – CBSE Mathematics

Chapter 1: Number Systems

🔢 Lecture 2: Irrational Numbers and Real Numbers

🔹 1. What is an Irrational Number?

A number $s$ is called irrational if:

• It cannot be written in the form $\frac{p}{q}$,
  where:

  • $p$ and $q$ are integers

  • $q \ne 0$

✅ Examples of Irrational Numbers:

• $\sqrt{2}, \sqrt{3}, \sqrt{5}, \sqrt{15}$

• $\pi$ (pi)

• Non-terminating, non-repeating decimals like:
  $0.10110111011110\ldots$

🔹 2. Real Numbers – The Bigger Family

• The collection of all rational and all irrational numbers together is called the set of Real Numbers.

• Denoted by: $\mathbb{R}$

$$\text{Real numbers} = \text{Rational numbers} \cup \text{Irrational numbers}$$

Every real number:

• Can be plotted as a unique point on the number line.

• And every point on the number line represents a unique real number.

Hence, the number line is also called the Real Number Line.

🔹 3. Visualizing Irrational Numbers on the Number Line

📍 Example: Locate $\sqrt{2}$ on the number line

1. Draw a square with side = 1 unit.

2. Use Pythagoras Theorem:

$$OB = \sqrt{1^2 + 1^2} = \sqrt{2}$$

3. Place point O on 0 on the number line.

4. Draw an arc with radius $OB = \sqrt{2}$ and centre at O.

5. The arc cuts the number line at point P.

6. Point P represents $\sqrt{2}$ on the number line.

🔹 4. Summary Table

Type of Number	Example	Can be written as $\frac{p}{q}$?
Rational	$\frac{2}{5}, -3, 0, 4$	Yes
Irrational	$\sqrt{2}, \pi, 0.101101...$	No
Real (includes both)	$\frac{3}{4}, \sqrt{5}, \pi$	Rational or Irrational

📘 Exercise 1.2 – Solutions Outline

Q1: State True or False. Justify.

(i) True – Every irrational number is part of real numbers.
(ii) False – Not every point is of the form $\sqrt{m}$. For example, $\frac{1}{2}$ is a real number but not of the form $\sqrt{m}$.
(iii) False – Rational numbers are also real. Not every real number is irrational.

Q2: Are square roots of all positive integers irrational?

No.
Examples:

• $\sqrt{4} = 2$ → Rational

• $\sqrt{9} = 3$ → Rational
  Only non-perfect squares (like $\sqrt{2}, \sqrt{3}$) are irrational.

Q3: Represent $\sqrt{5}$ on the number line.

✅ Steps (You can draw the figure):

1. Draw a line segment AB = 2 units.

2. At point B, draw BC = 1 unit perpendicular to AB.

3. Use Pythagoras Theorem:

   $$AC = \sqrt{2^2 + 1^2} = \sqrt{5}$$

4. Place AC on the number line with A at 0.

5. Draw an arc of radius $\sqrt{5}$, center at 0.

Point where arc cuts the number line = $\sqrt{5}$.

✅ Tips to Remember

• If the decimal terminates or repeats, it's rational.

• If it goes on forever without a pattern, it’s irrational.

• Perfect squares like $\sqrt{4}, \sqrt{9}, \sqrt{16}$ are rational.

• Non-perfect square roots like $\sqrt{2}, \sqrt{3}$ are irrational.

📝 Worksheet: Irrational and Real Numbers

📚 Chapter 1 – Number System | Class 9 – CBSE

✍️ A. Very Short Answer Type Questions

1. Define an irrational number.

2. Write two examples of irrational numbers.

3. Can $\sqrt{2}$ be expressed in the form $\frac{p}{q}$, where $p, q \in \mathbb{Z}$ and $q \ne 0$?

4. Write one irrational number between 3 and 4.

5. What is meant by a real number?

✅ B. True or False (Justify your answer)

1. Every irrational number is a real number.

2. Every real number is either rational or irrational.

3. $\pi$ is a rational number.

4. $\sqrt{16}$ is an irrational number.

5. Every point on the number line is of the form $\sqrt{m}$, where $m$ is a natural number.

🔁 C. Fill in the Blanks

1. The decimal expansion of an irrational number is ____________ and ____________.

2. Rational numbers and irrational numbers together form the set of ____________ numbers.

3. An example of a non-terminating, non-repeating decimal is ____________.

4. $\sqrt{9} =$ ____________, which is a ____________ number.

5. A real number can be represented by a unique ____________ on the number line.

🎯 D. Multiple Choice Questions (MCQs)

1. Which of the following is an irrational number?
   a) $\frac{3}{5}$
   b) $\sqrt{4}$
   c) $0.25$
   d) $\sqrt{3}$

2. The value of $\sqrt{49}$ is:
   a) 6
   b) 7
   c) 8
   d) 3

3. Which of these numbers is not a real number?
   a) $\pi$
   b) $\sqrt{2}$
   c) $\sqrt{-1}$
   d) $\frac{1}{2}$

4. Decimal expansion of irrational numbers is:
   a) Finite
   b) Terminating
   c) Repeating
   d) Non-terminating, non-repeating

📐 E. Application/Diagram-Based Question

1. Draw and describe how to represent $\sqrt{5}$ on the number line.
   (Use steps involving the Pythagoras Theorem – you may draw it or describe it in your notebook.)

🧠 F. Think and Answer

1. Is the number $0.10110111011110\ldots$ rational or irrational? Explain why.

2. Are all square roots irrational? Give two examples to support your answer.

✨ Bonus Challenge

Find two irrational numbers between 1 and 2 and write them in decimal form (upto 5 digits).

Would you like me to provide the answer key for this worksheet as well?