1. Introduction to Real Numbers

In earlier classes, students learned about natural numbers, whole numbers, integers, rational numbers, and irrational numbers. Together, rational and irrational numbers form the set of real numbers.

In this chapter, we study important properties of integers that help us:

• Understand divisibility

• Find HCF and LCM

• Factor numbers into primes

• Prove numbers like √2 and √3 are irrational

• Determine whether a rational number has a terminating decimal expansion

2. Euclid’s Division Algorithm

What is it?

For any two positive integers a and b, we can write:

$$a=bq+r$$

Where:

• q = quotient

• r = remainder

• 0 ≤ r < b

This is simply the mathematical form of long division.

Example 1

Divide 45 by 7.

45 = 7 × 6 + 3

Here:

• Quotient = 6

• Remainder = 3

Since remainder is less than divisor, the condition is satisfied.

Example 2

Divide 250 by 12.

250 = 12 × 20 + 10

Additional Practice

1. Write 133 in the form a = bq + r when divided by 9.

Answer:
133 = 9 × 14 + 7

3. Fundamental Theorem of Arithmetic

Statement

Every composite number can be expressed as a product of prime numbers. This factorisation is unique, except for the order of factors.

This means a number can only be broken into primes in one specific way.

Prime Factorisation

Prime factorisation means expressing a number as a product of prime numbers.

Example 1

Factorise 360.

360 = 2 × 2 × 2 × 3 × 3 × 5
= 2³ × 3² × 5

Example 2

Factorise 945.

945 ÷ 3 = 315
315 ÷ 3 = 105
105 ÷ 3 = 35
35 = 5 × 7

So,

945 = 3³ × 5 × 7

Example 3

Factorise 7429.

7429 ÷ 17 = 437
437 ÷ 19 = 23

So,

7429 = 17 × 19 × 23

4. HCF and LCM Using Prime Factorisation

Steps

1. Find prime factors of each number.

2. For HCF: choose smallest power of common factors.

3. For LCM: choose highest power of all factors.

Example 1

Find HCF and LCM of 18 and 24.

18 = 2 × 3²
24 = 2³ × 3

HCF = 2¹ × 3¹ = 6
LCM = 2³ × 3² = 72

Verification:
6 × 72 = 432
18 × 24 = 432

Example 2

Find HCF and LCM of 12, 20, 30.

12 = 2² × 3
20 = 2² × 5
30 = 2 × 3 × 5

HCF = 2¹
LCM = 2² × 3 × 5 = 60

Example 3 (Application)

Two bells ring at intervals of 20 minutes and 30 minutes. If they ring together now, after how many minutes will they ring together again?

20 = 2² × 5
30 = 2 × 3 × 5

LCM = 2² × 3 × 5 = 60

They will ring together again after 60 minutes.

5. Relationship Between HCF and LCM

For two positive integers a and b:

HCF × LCM = a × b

This property works only for two numbers, not necessarily for three.

6. Irrational Numbers

A number is irrational if it cannot be written as p/q, where p and q are integers and q ≠ 0.

Examples:

• √2

• √3

• √5

• π

7. Important Theorem

If a prime number divides a², then it also divides a.

This concept is used in proofs of irrational numbers.

8. Proof That √2 Is Irrational (Simplified)

Assume √2 = a/b in simplest form.

Squaring both sides:

2b² = a²

So 2 divides a², which means 2 divides a.

Let a = 2c.

Substitute:

2b² = 4c²
b² = 2c²

Now 2 divides b as well.

So both a and b have common factor 2, which contradicts the assumption that they are coprime.

Therefore, √2 is irrational.

9. Proof That √3 Is Irrational (Outline)

Assume √3 = a/b.

Then:

3b² = a²

This implies 3 divides a.

Let a = 3c.

Then 3 divides b.

Contradiction.

Therefore, √3 is irrational.

10. Decimal Expansion of Rational Numbers

For a rational number p/q in lowest form:

• If q has only 2 and/or 5 as prime factors → decimal is terminating.

• If q has other prime factors → decimal is non-terminating repeating.

Example 1

1/40

40 = 2³ × 5

Only 2 and 5 present → terminating decimal.

Example 2

3/28

28 = 2² × 7

7 is present → non-terminating repeating decimal.

11. Additional Solved Problems

Problem 1

Check whether 6ⁿ can end with 0.

6 = 2 × 3

No factor 5 present.
So 6ⁿ cannot end with 0.

Problem 2

Find LCM of 306 and 657 if HCF = 9.

LCM = (306 × 657) ÷ 9
LCM = 22338

Problem 3

Prove √5 is irrational (short explanation).

Assume √5 = a/b.
5b² = a²
So 5 divides a.
Let a = 5c.
Then 5 divides b.
Contradiction.
Hence √5 is irrational.

12. Practice Questions (With Answers)

1. Factorise 156.
   Answer: 2² × 3 × 13

2. Find HCF of 84 and 90.
   Answer: 6

3. Determine if 7/125 is terminating.
   Answer: Yes (125 = 5³)

4. Determine if 5/18 is terminating.
   Answer: No (18 contains 3)