Maths Class 9 NCERT Lesson 3 - The World of Numbers (Color)

📝 Section 1 — Classroom Notes

🌎 Real-World Hook: Think about your school report card. Your marks are natural numbers (1 to 100). Your bank balance can be zero or negative (debt) — so banks use integers. A pizza cut into 8 pieces means each piece is 1/8 — a rational number. The diagonal of a square tile is √2 metres — an irrational number. All these together form the real number system that describes everything in our world. Numbers are not just maths — they are the language of the universe!

🔢 Natural Numbers

Natural numbers are for counting.
They start from 1 and go on forever.
Written as: N = {1, 2, 3, 4, 5, ...}
You can always add two natural numbers.
The answer is always a natural number too.
Example: 3 + 5 = 8 (still natural).

⭕ Zero (Shunya)

Zero means nothing, or empty.
It was invented in India by Brahmagupta.
Any number + 0 = same number.
Any number - 0 = same number.
Any number x 0 = 0.
When we add zero to natural numbers, we get whole numbers.
Whole numbers: W = {0, 1, 2, 3, ...}

🔄 Integers

Integers include positive, negative, and zero.
Written as: Z = {..., -3, -2, -1, 0, 1, 2, 3, ...}
Positive = fortune (money you have).
Negative = debt (money you owe).
Positive x positive = positive.
Negative x negative = positive.
Positive x negative = negative.

🍕 Rational Numbers

A rational number can be written as p/q.
p and q are integers. q is not zero.
Examples: 1/2, -3/4, 5 (which is 5/1).
Symbol: Q (for quotient).
All integers are rational numbers too!
Example: 7 = 7/1, -3 = -3/1.

📏 Number Line

A number line is a straight line of numbers.
Zero is in the middle (the origin).
Positive numbers go to the right.
Negative numbers go to the left.
Fractions sit between whole numbers.
Example: 1/2 is between 0 and 1.

♾️ Density of Rational Numbers

Between any 2 rational numbers, there is always another.
You find it by taking the average.
Average = add both, then divide by 2.
Average of a and b = (a + b) / 2.
Example: between 1 and 2, we find 3/2.
There are infinite rational numbers between any two!

🌀 Irrational Numbers

Irrational numbers cannot be written as p/q.
Their decimal never ends and never repeats.
Examples: √2 = 1.41421356... (goes on forever).
π (pi) = 3.14159265... (goes on forever).
√2 was discovered when measuring a square's diagonal.
Side = 1, diagonal = √(1² + 1²) = √2.

🌍 Real Numbers

Real numbers = rational + irrational together.
Symbol: R.
They fill the entire number line with no gaps.
Every measurement in the real world is a real number.

🔟 Decimal Types

Terminating decimal = it stops.
Example: 3/8 = 0.375 (it ends).
Repeating decimal = digits repeat forever.
Example: 1/3 = 0.333... (3 repeats).
Both terminating and repeating = rational.
Non-repeating, non-ending = irrational.

🔄 Convert Repeating Decimal to p/q

Let x = the repeating decimal.
Multiply by 10 (or 100) to shift digits.
Subtract to remove the repeating part.
Solve for x.
Example: x = 0.666..., 10x = 6.666...
10x - x = 6, so 9x = 6, x = 6/9 = 2/3.

🎨 Section 2 — Easy Illustrations

Picture 1: The Number Family Tree

All numbers fit inside the Real Numbers family. Rational numbers include integers and fractions. Irrational numbers stand apart.

Picture 2: The Number Line

The number line shows integers, fractions (purple dots), and irrational numbers (red dot) all together.

Picture 3: Why √2 Exists

The diagonal of a unit square has length √2, which is an irrational number.

Picture 4: Decimal Types

Three types of decimals. Only the third type (non-repeating, non-ending) is irrational.

Picture 5: Density of Rationals

Take the average of any two rational numbers to find one between them. You can do this forever!

✏️ Section 3 — Worked Examples

Example 1: Classify Numbers

Question: Is √9 rational or irrational? What about √5?

1 Find √9. Ask: what number x itself = 9?

2 3 x 3 = 9, so √9 = 3.

3 3 is a whole number = 3/1. So rational.

4 Now try √5. No whole number gives 5.

5 √5 = 2.2360679... (never ends, never repeats).

√9 = 3 (rational). √5 = 2.236... (irrational).

Example 2: Convert Repeating Decimal to p/q

Question: Write 0.666... as a fraction p/q.

1 Let x = 0.666...

2 Multiply both sides by 10: 10x = 6.666...

3 Subtract: 10x - x = 6.666... - 0.666...

4 Simplify: 9x = 6, so x = 6/9 = 2/3.

0.666... = 2/3

Example 3: Find a Rational Number Between Two Numbers

Question: Find a rational number between 1/4 and 1/2.

1 Add the two numbers: 1/4 + 1/2.

2 Make the same bottom: 1/4 + 2/4 = 3/4.

3 Divide by 2: (3/4) / 2 = 3/8.

4 Check: 1/4 = 0.25, 3/8 = 0.375, 1/2 = 0.5. Yes!

3/8 is between 1/4 and 1/2.

📊 Section 4 — Cheat Sheet Poster

Topic	What it Means	Key Fact
Natural Numbers (N)	Counting numbers	1, 2, 3, 4, 5...
Whole Numbers (W)	Natural + zero	0, 1, 2, 3, 4...
Integers (Z)	Positive + negative + zero	...-2, -1, 0, 1, 2...
Rational (Q)	Can be written as p/q	1/2, -3/4, 7, 0.5
Irrational	Cannot be p/q	√2, π, √3
Real Numbers (R)	Rational + Irrational	Everything on the line!
Zero Rules	Brahmagupta's laws	a+0=a, ax0=0
Negative x Negative	Gives positive	(-3)x(-4) = +12
Terminating Decimal	It ends (rational)	3/8 = 0.375
Repeating Decimal	It loops (rational)	1/3 = 0.333...
Non-repeating Decimal	Forever, no pattern (irrational)	√2 = 1.41421...
Density	Infinite rationals between any two	Average = (a+b)/2

📄 Section 5 — Take-Home Handout

Print-friendly. Use Arial or OpenDyslexic font when printing.

🔢 Natural Numbers and Integers

Natural numbers: 1, 2, 3, 4, 5... (for counting).
Add zero = whole numbers: 0, 1, 2, 3...
Add negatives = integers: ...-2, -1, 0, 1, 2...
Positive = fortune. Negative = debt.

Example: (-3) x (-4) = ?

1 Negative x negative = positive. Answer: +12.

🍕 Rational Numbers (p/q)

Any number written as p/q (q not 0).
Includes all integers and fractions.
Decimal is either terminating or repeating.

Example: 3/8 = 0.375 (terminating). 1/3 = 0.333... (repeating).

🌀 Irrational Numbers

Cannot be written as p/q.
Decimal never ends and never repeats.
Examples: √2 = 1.414..., π = 3.14159...

Example: √9 = 3 (rational). √5 = 2.236... (irrational).

🌍 Real Numbers

Real numbers = rational + irrational.
Symbol: R.
They fill the entire number line.
N ⊂ Z ⊂ Q ⊂ R (each fits inside the next).

🔄 Convert Repeating Decimal to Fraction

Let x = the decimal. Multiply by 10 (or 100).
Subtract. Solve for x.

Example: 0.666... = ?

1 x = 0.666..., 10x = 6.666...

2 10x - x = 6, so 9x = 6, x = 2/3.

My notes / drawings:

📝 Practice Questions

1. Classify each number (write: Natural, Integer, Rational, or Irrational):

(a) 0.75 → __________

(b) √7 → __________

(d) 22/7 → __________

2. Convert 0.333... into a fraction p/q. Show your working:

Work space

3. Find a rational number exactly between 1/3 and 1/2. Show your working:

Work space

✅ Detailed Solutions — Practice Questions

✓ Question 1: Classify the Numbers

a 0.75 = 75/100 = 3/4. Can be written as p/q. → Rational number. (Also terminating decimal.)

b √7 = 2.6457... (never ends, never repeats). Cannot be written as p/q. → Irrational number.

c −5 is a negative whole number. → Integer (and also rational, since −5 = −5/1).

d 22/7 is written as p/q form (p=22, q=7). → Rational number. (Note: 22/7 is an approximation of π, but the fraction 22/7 itself is rational.)

🏆 Answers: (a) Rational (b) Irrational (c) Integer / Rational (d) Rational

✓ Question 2: Convert 0.333... to p/q

1 Let x = 0.333...

2 Multiply both sides by 10: 10x = 3.333...

3 Subtract the first equation from the second: 10x − x = 3.333... − 0.333...

4 Simplify: 9x = 3.

5 Divide: x = 3 ÷ 9 = 1/3.

🏆 Answer: 0.333... = 1/3.

✓ Question 3: Rational Number between 1/3 and 1/2

1 Method: The number halfway between two numbers = (a + b) ÷ 2.

2 Add the fractions: 1/3 + 1/2. Make same denominator: 2/6 + 3/6 = 5/6.

3 Divide by 2: (5/6) ÷ 2 = 5/12.

4 Check: 1/3 = 4/12, 5/12 = 5/12, 1/2 = 6/12. Yes, 5/12 is between them! ✓

🏆 Answer: 5/12 is a rational number between 1/3 and 1/2.

💡 There are infinitely many correct answers — any fraction between 1/3 and 1/2 works, e.g., 3/8 = 0.375 is also correct.

🃏 Section 6 — Revision Flashcards

Cut these out or use them to quiz yourself!

FRONT - Card 1

BACK

Counting numbers: 1, 2, 3, 4, 5...

They go on forever. Symbol: N.

FRONT - Card 2

BACK

a + 0 = a (no change)

a - 0 = a (no change)

a x 0 = 0 (always zero)

FRONT - Card 3

BACK

Positive + negative + zero.

...-3, -2, -1, 0, 1, 2, 3...

Symbol: Z

FRONT - Card 4

BACK

A number written as p/q.

p, q are integers. q is not 0.

Symbol: Q. e.g. 1/2, -3/4, 7

FRONT - Card 5

BACK

Cannot be written as p/q.

Decimal never ends, never repeats.

e.g. √2, π, √3

FRONT - Card 6

BACK

Terminating: stops. 0.375

Repeating: loops. 0.333...

Both are rational numbers.

FRONT - Card 7

BACK

Take the average!

(a + b) / 2

Between 1 and 2: (1+2)/2 = 3/2

FRONT - Card 8

BACK

Rational + Irrational = Real.

Symbol: R.

They fill the entire number line!

🔢 Maths Class 9 NCERT Lesson 3 The World of Numbers Special Edition for Easy Learning

📋 What is Inside?

📝 Section 1 — Classroom Notes

🔢 Natural Numbers

⭕ Zero (Shunya)

🔄 Integers

🍕 Rational Numbers

📏 Number Line

♾️ Density of Rational Numbers

🌀 Irrational Numbers

🌍 Real Numbers

🔟 Decimal Types

🔄 Convert Repeating Decimal to p/q

🎨 Section 2 — Easy Illustrations

Picture 1: The Number Family Tree

Picture 2: The Number Line

Picture 3: Why √2 Exists

Picture 4: Decimal Types

Picture 5: Density of Rationals

✏️ Section 3 — Worked Examples

Example 1: Classify Numbers

Example 2: Convert Repeating Decimal to p/q

Example 3: Find a Rational Number Between Two Numbers

📊 Section 4 — Cheat Sheet Poster

📄 Section 5 — Take-Home Handout

🔢 Natural Numbers and Integers

🍕 Rational Numbers (p/q)

🌀 Irrational Numbers

🌍 Real Numbers

🔄 Convert Repeating Decimal to Fraction

📝 Practice Questions

✅ Detailed Solutions — Practice Questions

✓ Question 1: Classify the Numbers

✓ Question 2: Convert 0.333... to p/q

✓ Question 3: Rational Number between 1/3 and 1/2

🃏 Section 6 — Revision Flashcards

🧠 Section 7 — Mind Map