Class 10 Mathematics – Real Numbers | Interactive Visual Revision Notes

📝 Section 1 — Classroom Notes

🔢 Topic 1: What Are Real Numbers?

🌎 Real World Connection: When you check a bank balance, it might be −₹500 (debt) or +₹1,000 (savings) — both are real numbers. When cutting a pizza into 4 equal slices, each slice is 1/4 (rational). When measuring the diagonal of a square tile, you get √2 times the side length (irrational). Every measurement in daily life — temperature, speed, distance — is a real number.

Real numbers include all numbers we use.
They sit on a number line.
They include rational and irrational numbers.
Examples: 3, −5, 0.75, √2, π
We write the set as R.

🔢 Topic 2: Types of Numbers

Natural numbers (N): 1, 2, 3, 4, 5 ...
Whole numbers (W): 0, 1, 2, 3, 4 ...
Integers (Z): ...−3, −2, −1, 0, 1, 2, 3...
Every natural number is a whole number.
Every whole number is an integer.
Every integer is a rational number.

🔢 Topic 3: Rational Numbers

A rational number = p ÷ q form.
Here p and q are integers.
q must not be zero.
Examples: 3/4, −7/2, 5, 0.6
Decimals that end are rational.
Decimals that repeat are rational.
0.333... = 1/3 (repeating = rational).

🔢 Topic 4: Irrational Numbers

An irrational number cannot be p/q.
Its decimal never ends and never repeats.
Examples: √2, √3, √5, π
√2 = 1.41421356... (goes on forever).
We can prove they are irrational.
We use a method called contradiction.

🔢 Topic 5: Prime and Composite Numbers

A prime number has only 2 factors.
Its factors are 1 and itself.
Examples: 2, 3, 5, 7, 11, 13, 17 ...
A composite number has more than 2 factors.
Examples: 4, 6, 8, 9, 10, 12 ...
The number 1 is neither prime nor composite.

🔢 Topic 6: Fundamental Theorem of Arithmetic

Every composite number = product of primes.
This way of writing is unique.
Only the order of primes can change.
Example: 36 = 2 × 2 × 3 × 3 = 2² × 3²
Example: 75 = 3 × 5 × 5 = 3 × 5²
This is also called prime factorisation.

Every composite number = p₁ᵃ × p₂ᵇ × p₃ᶜ × ...

🔢 Topic 7: Finding HCF

HCF = Highest Common Factor.
First, write each number as primes.
Pick the smallest power of shared primes.
Multiply those together = HCF.
Example: 18 = 2 × 3² and 45 = 3² × 5
Common prime: 3. Smallest power: 3² = 9.
So HCF(18, 45) = 9.

🔢 Topic 8: Finding LCM

LCM = Lowest Common Multiple.
First, write each number as primes.
Pick the greatest power of all primes.
Multiply those together = LCM.
Example: 18 = 2 × 3² and 45 = 3² × 5
Greatest powers: 2¹, 3², 5¹
LCM = 2 × 9 × 5 = 90.

🔢 Topic 9: HCF × LCM Rule

For any two numbers a and b:
HCF(a, b) × LCM(a, b) = a × b
Check: HCF(18, 45) = 9, LCM = 90.
9 × 90 = 810.
18 × 45 = 810. They match!
This rule works for two numbers only.

HCF(a, b) × LCM(a, b) = a × b

🔢 Topic 10: Decimal Expansion

If denominator = 2ᵃ × 5ᵇ only, decimal ends.
We call this a terminating decimal.
Example: 7/8 = 7/(2³) = 0.875 (ends).
If denominator has other primes, it repeats.
We call this non-terminating repeating.
Example: 1/6 = 0.1666... (repeats).
Check the denominator after simplifying!

🎨 Section 2 — Easy Illustrations

Picture 1: The Number System Family

Every smaller box fits inside the bigger one. N is inside W, W inside Z, and so on.

Picture 2: Factor Tree for 360

Click "Play" to see the factor tree grow step by step. Green circles = prime factors.

Picture 3: HCF and LCM Venn Diagram

HCF = shared factors. LCM = all factors with highest powers.

Picture 4: Rational vs Irrational on Number Line

Both rational and irrational numbers live on the same number line.

Picture 5: Terminating vs Non-Terminating Decimals

Check the denominator: only 2s and 5s = ends. Anything else = repeats forever.

✏️ Section 3 — Worked Examples

Example 1: Find HCF and LCM Using Prime Factorisation

Question: Find HCF and LCM of 24 and 54.

1 Write 24 as primes: 24 = 2 × 2 × 2 × 3 = 2³ × 3.

2 Write 54 as primes: 54 = 2 × 3 × 3 × 3 = 2 × 3³.

3 HCF = smallest powers of shared primes.

4 Shared primes: 2 and 3. Smallest: 2¹ and 3¹.

5 HCF = 2 × 3 = 6.

6 LCM = greatest powers of all primes.

7 Greatest: 2³ and 3³. LCM = 8 × 27 = 216.

HCF = 6, LCM = 216

Check: 6 × 216 = 1296 = 24 × 54 ✓

Example 2: Prove √5 is Irrational

Question: Prove that √5 is irrational.

1 Assume √5 is rational (opposite of what we want).

2 Then √5 = a/b, where a and b share no common factor.

3 Square both sides: 5 = a²/b², so 5b² = a².

4 This means 5 divides a², so 5 divides a.

5 Write a = 5c. Then 5b² = 25c², so b² = 5c².

6 So 5 divides b too. But we said a and b share no factor!

7 This is a contradiction. Our assumption was wrong.

√5 is irrational. (Proved by contradiction.)

Example 3: Is the Decimal Terminating?

Question: Does 21/40 have a terminating decimal?

1 Check if 21/40 is in simplest form. HCF(21,40) = 1. Yes!

2 Write denominator as primes: 40 = 2³ × 5.

3 Check: only 2s and 5s? Yes!

4 So the decimal terminates.

5 21/40 = 21 × 25 / (40 × 25) = 525/1000 = 0.525

Yes! 21/40 = 0.525 (terminating)

Rule: denominator = 2ᵃ × 5ᵇ only → terminates

📊 Section 4 — Cheat Sheet

Term	What It Means	Key Fact
Natural Numbers	Counting numbers	1, 2, 3, 4 ...
Whole Numbers	Natural + zero	0, 1, 2, 3 ...
Integers	Whole + negatives	...−2, −1, 0, 1, 2...
Rational	Can be written as p/q	q ≠ 0
Irrational	Cannot be p/q	√2, π
Prime Number	Only 2 factors	2, 3, 5, 7, 11...
Composite	More than 2 factors	4, 6, 8, 9...
FTA	Every composite = primes	Unique factorisation
HCF	Highest Common Factor	Smallest shared powers
LCM	Lowest Common Multiple	Greatest all powers
Terminating	Decimal ends	Denom = 2ᵃ × 5ᵇ
Non-terminating	Decimal repeats	Other primes in denom

📄 Section 5 — Take-Home Handout

📝 A. Fill in the Blanks

1. Every composite number can be written as a product of _________.
2. HCF stands for _________ Common _________.
3. The number 1 is neither _________ nor _________.
4. √3 is an _________ number.
5. If the denominator has only 2s and 5s, the decimal _________.

✓ B. True or False

1. Every natural number is a real number. _______
2. √9 is an irrational number. _______
3. HCF(a,b) × LCM(a,b) = a × b for any two numbers. _______
4. The number 0 is a natural number. _______
5. 7/12 has a terminating decimal. _______

✏️ C. Solve These

1. Find the HCF and LCM of 36 and 84.

Show your work here

2. Check if 9/125 has a terminating decimal. Find the decimal.

Show your work here

🔗 D. Match the Columns

Column A	Column B
1. √11	a. Terminating decimal
2. 3/8	b. Composite number
3. 51	c. Irrational number
4. HCF(15, 25)	d. 5

🎨 E. Draw the Factor Tree

Draw the factor tree for 180 and write its prime factorisation.

Draw your factor tree here

✅ Detailed Solutions

A. Fill in the Blanks — Answers

1 Every composite number can be written as a product of primes.

2 HCF stands for Highest Common Factor.

3 The number 1 is neither prime nor composite.

4 √3 is an irrational number.

5 If the denominator has only 2s and 5s, the decimal terminates.

B. True or False — Answers

1 TRUE — Every natural number is a real number (naturals ⊂ reals).

2 FALSE — √9 = 3, which is rational, not irrational.

3 TRUE — HCF(a,b) × LCM(a,b) = a × b is a standard result.

4 FALSE — 0 is a whole number, not a natural number.

5 FALSE — 7/12: denominator 12 = 2² × 3 has factor 3, so non-terminating.

C1. Find HCF and LCM of 36 and 84

1 Factorise 36: 2 × 2 × 3 × 3 = 2² × 3²

2 Factorise 84: 2 × 2 × 3 × 7 = 2² × 3 × 7

3 HCF = smallest powers of shared primes = 2² × 3 = 12

4 LCM = greatest powers of all primes = 2² × 3² × 7 = 252

HCF = 12 | LCM = 252

C2. Is 9/125 a terminating decimal?

1 Factorise denominator: 125 = 5³ → only powers of 5, so terminating.

2 Multiply numerator and denominator by 2³ = 8:

3 9/125 = (9 × 8)/(125 × 8) = 72/1000 = 0.072

9/125 = 0.072 (terminating decimal)

D. Match the Columns — Answers

1 √11 → c (Irrational: cannot be written as p/q)

2 3/8 → a (Terminating: 8 = 2³, so 3/8 = 0.375)

3 51 → b (Composite: 51 = 3 × 17)

4 HCF(15, 25) → d (15 = 3×5, 25 = 5², HCF = 5)

E. Factor Tree for 180

1 180 = 2 × 90 = 2 × 2 × 45 = 2 × 2 × 3 × 15 = 2 × 2 × 3 × 3 × 5

2 Prime factorisation: 180 = 2² × 3² × 5

180 = 2² × 3² × 5

🃏 Section 6 — Revision Flashcards

👉

🔢

What is the Fundamental Theorem of Arithmetic?

Every composite number can be written as a product of primes in a unique way.

🔎

How do you find HCF using prime factorisation?

Write both numbers as primes. Pick the smallest power of each shared prime. Multiply them.

📈

How do you find LCM using prime factorisation?

Write both numbers as primes. Pick the greatest power of every prime. Multiply them all.

🔗

What is the HCF × LCM rule?

HCF(a,b) × LCM(a,b) = a × b

Works for two numbers only.

❓

What is an irrational number?

A number that cannot be written as p/q. Its decimal never ends and never repeats.

Examples: √2, √3, π

💡

When does a decimal terminate?

When the denominator (in simplest form) has only 2s and 5s as prime factors.

🔧

How do you prove a number is irrational?

Use proof by contradiction. Assume it is rational, then show this leads to an impossible result.

🎯

Is the number 1 prime or composite?

Neither!

1 is not prime (needs exactly 2 factors) and not composite (needs more than 2).

Class 10 Mathematics — Real Numbers

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🔢 Class 10 Mathematics — Real Numbers Interactive Visual Revision Notes Aligned with the Class 10 Mathematics syllabus. Independently created study material. Not affiliated with or endorsed by NCERT.

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