Class 10 NCERT Maths Chapter 1: Real Numbers

The journey of numbers continues in Class 10 with Chapter 1 – Real Numbers. This chapter deepens your understanding of the number system by revisiting prime factorization and exploring irrational numbers in more detail.

What Are Real Numbers?

Real numbers include all rational and irrational numbers.

• Rational numbers: Numbers that can be written in the form p/q where q ≠ 0

• Irrational numbers: Numbers that cannot be expressed as p/q, like √2, π, etc.

📚 Topics Covered in Chapter 1

1. Fundamental Theorem of Arithmetic (FTA)

Statement: Every composite number can be expressed as a product of prime numbers and this factorization is unique, except for the order of the prime factors.

Example: \( 84 = 2 \times 2 \times 3 \times 7 = 2^2 \times 3 \times 7 \)

Use: Helps in solving HCF, LCM and perfect square-related problems.

2. Proofs of Irrationality

You will learn how to prove certain numbers are irrational using algebraic techniques.

Irrational Numbers to Prove:

• √2
• √3
• √5
• Expressions like 3 + 2√5

Method: Assume the number is rational, write it as p/q in lowest terms, then square both sides. If it leads to a contradiction, the number is irrational.

Important Concepts and Skills

• Understand and describe real numbers and their properties
• Extend knowledge of powers, exponents, and radicals
• Apply the Fundamental Theorem of Arithmetic in real-life contexts
• Prove irrationality of square roots and expressions like 3 + 2√5

Examples:

Example 1: Prime Factorization

Find the prime factorization of 150:
150 = 2 × 3 × 5 × 5 = 2 × 3 × 5²

Example 2: Prove √2 is Irrational

Assume √2 = p/q (in lowest terms).
Squaring: 2 = p²/q² → p² = 2q² ⇒ p is even ⇒ p = 2k
Substituting leads to q also being even ⇒ contradiction.
Hence, √2 is irrational.

Example 3: Prove 3 + 2√5 is Irrational

Assume 3 + 2√5 = p/q.
Isolate √5 and square both sides. Eventually leads to contradiction.
So, 3 + 2√5 is irrational.

Summary Table:

Concept	Description
Real Numbers	Set of all rational and irrational numbers
Fundamental Theorem of Arithmetic	Every composite number has a unique prime factorization
Irrational Number	Cannot be expressed as p/q
Proving Irrationality	Use contradiction to show numbers like √2 are irrational