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?

 

📝 Worksheet: Rational Numbers – Class 9 CBSE

✳️ Section A: Definitions (1 mark each)

1. Define a rational number.

2. What is meant by co-prime numbers?

3. Why is the denominator of a rational number never zero?

4. What is the origin of the word “rational”?

5. Write the symbol used to represent the set of rational numbers.

✳️ Section B: Fill in the Blanks (1 mark each)

1. A rational number is a number of the form __________, where p and q are integers and q ≠ 0.

2. All whole numbers are __________ numbers.

3. The rational numbers $\frac{3}{5}$ and $\frac{6}{10}$ are __________ rational numbers.

4. The decimal expansion of a rational number is either __________ or __________.

5. The number 0 can be written as $\frac{0}{1}, \frac{0}{2}, \frac{0}{-5}$ where __________.

✳️ Section C: True or False (Write T/F and give reasons) (2 marks each)

1. Every whole number is a natural number.

2. Every rational number is an integer.

3. Zero is a rational number.

4. Every integer is a rational number.

5. Every natural number is a rational number.

✳️ Section D: Reasoning Questions (2–3 marks each)

1. Is 0 a rational number? Explain with three different ways of writing it as a rational number.

2. Is $\frac{3}{0}$ a rational number? Why or why not?

3. How are equivalent rational numbers formed? Give an example.

4. Is $\frac{-8}{12}$ in simplest form? If not, simplify it.

✳️ Section E: Find Rational Numbers Between (2–3 marks each)

1. Find five rational numbers between 1 and 2.

2. Find six rational numbers between 3 and 4.

3. Find five rational numbers between $\frac{3}{5}$ and $\frac{4}{5}$.

4. Find three rational numbers between $\frac{-1}{2}$ and $\frac{1}{2}$.

5. Find two rational numbers between -1 and 0.

✳️ Section F: Challenge Question (4 marks)

Q: If a rational number lies between $\frac{2}{7}$ and $\frac{5}{7}$, how many such numbers can be found? Show at least six examples and explain your method.

✅ Answer Key: Rational Numbers – Class 9 CBSE

✳️ Section A: Definitions

1. A rational number is a number that can be written in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q\mathrm{\ne }0.$

2. Co-prime numbers are two numbers that have no common factor other than 1.

3. The denominator of a rational number is never zero because division by zero is undefined.

4. The word “rational” comes from "ratio", because rational numbers are expressed as a ratio of two integers.

5. The set of rational numbers is denoted by Q.

✳️ Section B: Fill in the Blanks

1. $\frac{p}{q}$

2. Rational

3. Equivalent

4. Terminating or Non-terminating Repeating

5. $q \ne 0$
   

✳️ Section C: True or False with Reasons

1. False – Every whole number is not a natural number. For example, 0 is a whole number but not a natural number.

2. False – Not every rational number is an integer. For example, $\frac{1}{2}$ is a rational number but not an integer.

3. True – 0 is a rational number because it can be written as $\frac{0}{1}, \frac{0}{2}, \frac{0}{-5}$, etc., where the denominator is not zero.

4. True – Every integer is a rational number because any integer $a$ can be written as $\frac{a}{1}$.

5. True – Every natural number can be written in the form $\frac{p}{q}$, like $\frac{3}{1}$, so it is a rational number.

✳️ Section D: Reasoning Questions

1. Yes, 0 is a rational number because:

   $$0 = \frac{0}{1} = \frac{0}{2} = \frac{0}{-5} \quad (q \ne 0)$$
   

2. No, $\frac{3}{0}$ is not a rational number because division by zero is undefined.

3. Equivalent rational numbers are formed by multiplying or dividing both numerator and denominator by the same non-zero number.
   Example: $\frac{2}{3} = \frac{4}{6} = \frac{6}{9}$

4. $\frac{-8}{12}$ is not in simplest form.
   Divide numerator and denominator by 4: $\frac{-2}{3}$ is the simplest form.

✳️ Section E: Find Rational Numbers Between

1. Between 1 and 2:

Multiply both by 10 to get like denominators:

$$\frac{10}{10}, \frac{11}{10}, \frac{12}{10}, \frac{13}{10}, \frac{14}{10}, \frac{15}{10}, \frac{16}{10}, \frac{17}{10}, \frac{18}{10}, \frac{19}{10}, \frac{20}{10}$$

So, five rational numbers are:

$$\frac{11}{10}, \frac{12}{10}, \frac{13}{10}, \frac{14}{10}, \frac{15}{10}$$

2. Between 3 and 4:

$$\frac{31}{10}, \frac{32}{10}, \frac{33}{10}, \frac{34}{10}, \frac{35}{10}, \frac{36}{10}$$

(Equivalent to 3.1 to 3.6)

3. Between $\frac{3}{5}$ and $\frac{4}{5}$:

Convert to denominator 25:

$$\frac{15}{25}, \frac{16}{25}, \frac{17}{25}, \frac{18}{25}, \frac{19}{25}, \frac{20}{25}$$

So, five numbers are:

$$\frac{16}{25}, \frac{17}{25}, \frac{18}{25}, \frac{19}{25}, \frac{20}{25}$$

4. Between $\frac{-1}{2}$ and $\frac{1}{2}$:

$$\frac{-2}{5}, \frac{-1}{5}, 0, \frac{1}{5}$$

(Any three from these: $\frac{-1}{5}, 0, \frac{1}{5}$)

5. Between -1 and 0:

$$\frac{-9}{10}, \frac{-4}{5}$$

✳️ Section F: Challenge Question

Between $\frac{2}{7}$ and $\frac{5}{7}$:

Choose common denominator (e.g., 70):

$$\frac{20}{70}, \frac{21}{70}, \frac{22}{70}, \frac{23}{70}, \frac{24}{70}, \frac{25}{70}, \ldots, \frac{49}{70}, \frac{50}{70}$$

So we can find many rational numbers, such as:

$$\frac{21}{70}, \frac{22}{70}, \frac{23}{70}, \frac{24}{70}, \frac{25}{70}, \frac{26}{70}$$

(At least 6 shown above.)

 

✍️ Class 9 Mathematics – Chapter 1: Number Systems

📘 Topic: Rational Numbers

🔷 Definition: Rational Numbers

A number is called a rational number if it can be written in the form:

$$r = \frac{p}{q}$$

Where:

• p and q are integers

• q ≠ 0

📌 The word ‘rational’ comes from ‘ratio’, and the symbol Q for rational numbers comes from ‘quotient’.

❗ Why q ≠ 0?

If the denominator q is zero, the expression becomes undefined in mathematics. For example, $\frac{3}{0}$ is not defined.

✳️ Co-prime Form of Rational Numbers

When we represent a rational number $\frac{p}{q}$, we usually write it in simplest form, where p and q have no common factors other than 1. That means they are co-prime.

🔄 Equivalent Rational Numbers

Two rational numbers are equivalent if they represent the same value, even if they have different numerators and denominators.

✴️ For example:

$$\frac{2}{3} = \frac{4}{6} = \frac{6}{9}$$

✅ True/False Statements – Reasoning Based

1. Are the following statements true or false?

(i) Every whole number is a natural number.
❌ False.
→ 0 is a whole number but not a natural number.
Natural numbers start from 1, but whole numbers start from 0.

(ii) Every integer is a rational number.
✅ True.
→ Every integer a can be written as $\frac{a}{1}$, which is a rational number.

(iii) Every rational number is an integer.
❌ False.
→ $\frac{3}{4}$ is a rational number but not an integer.

🚀 How to Find Rational Numbers Between Two Given Numbers

Let’s say we want to find rational numbers between two numbers, like between 1 and 2.

🌟 General Step-by-Step Procedure:

Step 1: Write both numbers with the same denominator.

1 = $\frac{1}{1}$ and 2 = $\frac{2}{1}$

Multiply numerator and denominator of both with 10:

$$1 = \frac{10}{10},\quad 2 = \frac{20}{10}$$

Step 2: Find numbers between the numerators:

Between 10 and 20, we have: 11, 12, 13, 14, 15...

So, rational numbers between 1 and 2 are:

$$\frac{11}{10}, \frac{12}{10}, \frac{13}{10}, \frac{14}{10}, \frac{15}{10}$$

Step 3: Simplify if needed.

📚 Exercise 1.1 – Solved Answers

Q1: Is zero a rational number?

✅ Yes, 0 is a rational number.

It can be written as:

$$0 = \frac{0}{1},\ \frac{0}{2},\ \frac{0}{-5} \quad \text{(where } q \ne 0 \text{)}$$

Q2: Find six rational numbers between 3 and 4.

$$3 = \frac{30}{10},\quad 4 = \frac{40}{10}$$

Numbers between 30 and 40: 31, 32, 33, 34, 35, 36

So, six rational numbers:

$$\frac{31}{10},\frac{32}{10},\frac{33}{10},\frac{34}{10},\frac{35}{10},\frac{36}{\mathrm{10}}$$

Q3: Find five rational numbers between $\frac{3}{5}$ and $\frac{4}{5}$

Multiply numerator and denominator by 10:

$$\frac{3}{5} = \frac{30}{50},\quad \frac{4}{5} = \frac{40}{50}$$

Between 30 and 40: 31, 32, 33, 34, 35

So five rational numbers:

$$\frac{31}{50}, \frac{32}{50}, \frac{33}{50}, \frac{34}{50}, \frac{35}{50}$$

Q4: State whether the following statements are true or false. Give reasons.

(i) Every natural number is a whole number.
✅ True
→ Natural numbers start from 1, and all natural numbers are included in whole numbers.

(ii) Every integer is a whole number.
❌ False
→ Integers include negative numbers like -1, -2, etc., which are not whole numbers.

(iii) Every rational number is a whole number.
❌ False
→ Rational numbers like $\frac{3}{5}$ are not whole numbers.

📌 Summary Chart

Set	Examples	Contains
Natural Numbers (N)	1, 2, 3, 4, ...	Counting numbers
Whole Numbers (W)	0, 1, 2, 3, ...	Natural + 0
Integers (Z)	..., -3, -2, -1, 0, 1, 2, ...	Positive and negative whole numbers
Rational Numbers (Q)	$\frac{3}{4}, -\frac{7}{2}, 0, 5$	Numbers written as $\frac{p}{q}$, q ≠ 0

CBSE| HISTORY|CLASS 7| Thinking About Time and Historical Periods

 

Thinking About Time and Historical Periods

History is not simply the story of events recorded according to dates and years. For historians, time is not just the passing of minutes, days, and centuries, but a framework to understand how societies, ideas, and institutions transform. The way time is studied and divided into historical periods plays a key role in how we understand our past.

🔍 1. Understanding Time Beyond the Clock and Calendar

• Historians look at time as a reflection of changes in:

  • Social structures

  • Economic organisation

  • Belief systems and ideas

• They study how certain practices, institutions or customs emerged, developed, or faded with time.

📚 2. The British Periodisation of Indian History

• In the mid-19th century, British historians divided Indian history into three simplistic religious periods:

  • Hindu period (ancient)

  • Muslim period (medieval)

  • British period (modern)

• This periodisation focused only on the religion of rulers, ignoring:

  • Cultural richness

  • Economic transitions

  • Social diversity

• It falsely assumed that no other significant development occurred except a change in religion.

❌ Why This Periodisation Is Problematic

• It over-simplifies history by ignoring the complexities of:

  • Trade and commerce

  • Caste, class, and occupation

  • Art, architecture, and literature

• It dismisses the diversity of India's people and cultures.

• Most modern historians now reject this method of periodisation.

🧠 3. The Modern Approach to Studying Historical Time

• Today, historians look at economic, social, cultural and political developments to define historical periods.

• They seek to understand:

  • The growth of agrarian and peasant societies

  • The formation of regional and imperial states

  • The spread of religions like Hinduism and Islam

  • The arrival of European traders and companies

🏰 4. The “Medieval” Period in Indian History

• The time between 700 CE and 1750 CE is often called “medieval” India.

• However, this term too has limitations:

  • It is borrowed from European history and may not fully apply to Indian conditions.

  • It often gets contrasted with “modern”, implying that the medieval era lacked progress.

• In truth, this period witnessed tremendous change:

  • Flourishing regional cultures and kingdoms

  • Technological and artistic advancements

  • Prosperous economies that attracted Europeans

🔁 5. Continuity and Change

• Despite transformations, some customs, beliefs, and institutions remained the same.

• Students are encouraged to:

  • Observe how the present reflects the past

  • Identify both changes and continuities

  • Compare historical periods with previous knowledge from earlier grades

🧭 Conclusion

Time in history is not static. It is dynamic and multidimensional. The thousand years between 700 and 1750 CE were filled with diversity, transformation, prosperity, and interaction among people, cultures, and ideas. Historians aim to look beyond religion and dates to understand the deeper forces shaping societies. As students of history, we are encouraged to look for patterns, processes, and the real people and stories behind historical change.

CBSE| HISTORY|CLASS 7| Old and New Religions (700–1750 CE)

 

Old and New Religions (700–1750 CE)

The thousand-year period between 700 and 1750 CE was a time of dynamic religious transformation in the Indian subcontinent. This era saw the evolution of existing religious traditions, the rise of new devotional movements, and the arrival of new faiths that reshaped society, belief systems, and power structures.

🧠 1. Nature of Religious Belief

• Religion was not just a personal experience but deeply collective in nature.

• It was closely tied to social and economic life, often reflecting the structure and changes within local communities.

• As societies transformed, so did their religious beliefs and practices.

🛕 2. Evolution within Hinduism

This period saw significant developments in the traditions now called Hinduism:

🔸 New Deities and Temples

• Emergence of the worship of new deities.

• Royal patronage led to the construction of large temples, which became both spiritual and economic centres.

• These temples often reflected the power and prestige of the rulers.

🔸 Rise of the Brahmanas

• Brahmanas (priests) gained prominence through their knowledge of Sanskrit texts.

• They were respected and patronised by rulers, becoming dominant social figures.

• Their role as ritual specialists connected religious authority with political power.

🙏 3. The Bhakti Movement

• A major religious development of this era was the Bhakti tradition.

• Bhakti emphasized a personal connection with a loving deity.

• Key features:

  • Devotion (bhakti) could be expressed by anyone, regardless of caste.

  • Rejected elaborate rituals and the intermediary role of priests.

  • Stressed love, surrender, and equality before God.

• Bhakti created space for inclusive spirituality and reshaped regional religious expressions (more in Chapter 6).

🕌 4. Arrival and Spread of Islam

• Islam was introduced in India by merchants and migrants in the 7th century.

• Its teachings were based on the Quran, which Muslims believe to be the word of God (Allah).

• Core beliefs:

  • Monotheism – belief in one God.

  • Equality before Allah, irrespective of caste or class.

  • Emphasis on mercy, love, and justice.

🔸 Patronage and Spread

• Many rulers became patrons of Islam and supported the ulama (religious scholars and jurists).

• Islam spread through:

  • Trade and migration

  • Sufi saints

  • Political integration under Muslim rulers

🧕 5. Diversity Within Islam

• Like Hinduism, Islam too was not monolithic; it evolved in various ways:

🔹 Shia and Sunni Sects

• Shia Muslims: Believed Ali (Prophet Muhammad’s son-in-law) was the rightful leader.

• Sunni Muslims: Accepted the early caliphs (Khalifas) as rightful leaders.

🔹 Schools of Islamic Law

• Different legal schools interpreted Sharia (Islamic law) differently:

  • In India, Hanafi and Shafi’i schools were most prominent.

🔹 Sufism

• A mystic tradition within Islam that emphasized love, devotion, and union with God.

• Sufi saints played a major role in spreading Islam through compassion and simplicity, often attracting non-Muslim followers too.

📚 Conclusion

Between 700 and 1750 CE, the Indian subcontinent experienced profound religious changes:

• Hinduism evolved with new devotional paths and ritual authority.

• The Bhakti movement offered an inclusive, emotional alternative to traditional rituals.

• Islam, with its message of equality and monotheism, became a major force, adapting itself to the Indian context in diverse forms.

This period was thus marked by religious diversity, adaptation, and dialogue, shaping the spiritual map of India as we know it today.