TEACHGYAN.COM · CLASS 10 MATHEMATICS

Chapter 1: Real Numbers

Learning goals: Understand prime factorisation, the Fundamental Theorem of Arithmetic, HCF and LCM through prime factors, and proofs that selected numbers are irrational.

1. Revisiting the number system

Natural numbers are 1, 2, 3, …; whole numbers include 0; integers include negative numbers; rational numbers can be written as p/q, where p and q are integers and q ≠ 0. Irrational numbers cannot be expressed as a ratio of two integers. Real numbers consist of all rational and irrational numbers. Examples: 7/4 and −3 are rational; √2 and π are irrational.

2. Prime and composite numbers

A prime number is a natural number greater than 1 with exactly two distinct positive factors: 1 and itself. Examples: 2, 3, 5, 7, 11. The number 2 is the only even prime. A composite number has more than two positive factors, such as 4, 6, 8 and 9. The number 1 is neither prime nor composite.

3. Fundamental Theorem of Arithmetic

Statement: Every composite number can be expressed as a product of primes, and this factorisation is unique apart from the order of the prime factors.

Example: 360 = 2 × 2 × 2 × 3 × 3 × 5 = 2³ × 3² × 5. Whether we divide 360 by 2 first or by 3 first, the final prime factors and their powers remain the same.

4. Finding prime factorisation systematically

Divide the number repeatedly by the smallest possible prime: first 2, then 3, 5, 7 and so on. Continue until the quotient becomes 1. The prime divisors, counted with repetition, form the prime factorisation.

Worked example: Express 156 as a product of primes.

156 = 2 × 78 = 2 × 2 × 39 = 2² × 3 × 13.

Check by multiplication: 4 × 3 × 13 = 156.