Real Numbers

Real numbers are the building blocks of mathematics used in everyday life. They include natural numbers, whole numbers, integers, rational numbers, and irrational numbers. Understanding real numbers is essential for solving problems in algebra, geometry, and other branches of mathematics.

1. The Fundamental Theorem of Arithmetic

Definition:

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

Example 1:

24 can be written as a product of primes:

24 = 2 x 2 x 2 x3 = 2³ x 3

Example 2:

90 can be written as:

90 = 2 x 3 x 3 x 5 = 3² x 5

2. Euclid’s Division Lemma

Statement:

3. HCF and LCM of Two Numbers

1. HCF (Highest Common Factor):

The largest number that divides two or more numbers exactly.

2. LCM (Least Common Multiple):

The smallest number divisible by two or more numbers.

Method 1 – Prime Factorization:

Example 1:

Find HCF and LCM of 12 and 18.

• 12 = 2² × 3

• 18 = 2 × 3²

HCF: Product of common prime factors with least power → 2 × 3 = 6

LCM: Product of all prime factors with highest power → 2² × 3² = 36

Method 2 – Division Method:

Example 2:

HCF of 48 and 60:

60 | 48 → 60 divides 48? No

48 | 60 → divide larger by smaller:

60 ÷ 48 = 1 remainder 12

48 ÷ 12 = 4 remainder 0 → HCF = 12

LCM = (Product of numbers) / HCF = (48 × 60)/12 = 240

Decimal Representation of Real Numbers

1. Terminating decimals → Rational numbers (e.g., 0.25)

2. Non-terminating recurring decimals → Rational numbers (e.g., 0.666…)

3. Non-terminating non-recurring decimals → Irrational numbers (e.g., √2 = 1.414213…)

Euclidean Algorithm for HCF

A faster method to find HCF of large numbers using repeated division.

Example:

HCF of 252 and 105:

• 252 ÷ 105 = 2 remainder 42

• 105 ÷ 42 = 2 remainder 21

• 42 ÷ 21 = 2 remainder 0 → HCF = 21

Summary Table

Exercises for Practice

1. Find the HCF and LCM of 36 and 60 using prime factorization.

2. Express 56 as a product of prime factors.

3. Divide 123 by 7 using Euclid’s lemma.

4. Check whether √50 is rational or irrational.

5. Show that HCF × LCM of 8 and 12 equals the product of the numbers.

This chapter explanation is detailed and sequential, like a textbook, and covers definitions, formulas, examples, properties, and exercises.