Class 10 NCERT Maths Chapter 1: Real Numbers

Watch Video explanation: Click Here

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.

• 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