✦ NCF 2023 ALIGNED • VISUAL LEARNING SERIES ✦
Chapter 1: Real Numbers
Exploring the building blocks of mathematics: Prime Factorisation & Irrationals.
Curated by Akash Srivastva
The set of Real Numbers consists of all rational numbers (of the form p/q where p, q are integers and q ≠ 0) and all irrational numbers.
In this chapter we study the Fundamental Theorem of Arithmetic and use it to prove the irrationality of √2, √3, √5 and to understand the nature of HCF, LCM and decimal expansions.
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Fundamental Theorem of Arithmetic
Theorem 1.1
Every composite number can be expressed (factorised) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur.
Detailed Example (from NCERT):
32760 = 2 × 2 × 2 × 3 × 3 × 5 × 7 × 13
= 2³ × 3² × 5 × 7 × 13
32760 = 2 × 2 × 2 × 3 × 3 × 5 × 7 × 13
= 2³ × 3² × 5 × 7 × 13
Important Application:
Can 4ⁿ end with the digit 0 for any natural number n?
No. Because 4ⁿ = (2²)ⁿ = 2²ⁿ. The only prime factor is 2. For a number to end with 0 it must be divisible by both 2 and 5. By uniqueness, the prime 5 can never appear.
Can 4ⁿ end with the digit 0 for any natural number n?
No. Because 4ⁿ = (2²)ⁿ = 2²ⁿ. The only prime factor is 2. For a number to end with 0 it must be divisible by both 2 and 5. By uniqueness, the prime 5 can never appear.
Proofs of Irrationality
Theorems 1.2 & 1.3
Theorem 1.2: Let p be a prime number. If p divides a², then p divides a, where a is a positive integer.
Complete Proof that √2 is irrational:
1. Assume √2 = a/b (co-prime).
2. 2b² = a² → 2 divides a² → 2 divides a.
3. a = 2c → b² = 2c² → 2 divides b.
4. Common factor 2 → contradiction.
5. Hence √2 is irrational.
1. Assume √2 = a/b (co-prime).
2. 2b² = a² → 2 divides a² → 2 divides a.
3. a = 2c → b² = 2c² → 2 divides b.
4. Common factor 2 → contradiction.
5. Hence √2 is irrational.
Same method proves √3, √5 and √p (p prime) are irrational. Also: 5−√3, 3√2, 1/√2, 7√5, 6+√2 are irrational.
HCF & LCM Applications
Prime Factorisation Method
HCF = Product of the smallest power of each common prime factor.
LCM = Product of the greatest power of every prime factor involved.
Example: 6 = 2¹×3¹ | 20 = 2²×5¹
HCF = 2 LCM = 60
HCF(a,b) × LCM(a,b) = a × b
HCF = 2 LCM = 60
HCF(a,b) × LCM(a,b) = a × b
Note: This relation holds only for two numbers. For three or more numbers it does not hold in general.
Rational & Irrational Properties
Arithmetic Combinations
Example 1: 5 − √3 is irrational
Assume it is rational → √3 becomes rational → contradiction.
Assume it is rational → √3 becomes rational → contradiction.
Example 2: 3√2 is irrational
Assume it is rational → √2 becomes rational → contradiction.
Assume it is rational → √2 becomes rational → contradiction.
Rules:
• Rational ± Irrational = Irrational
• Non-zero Rational × Irrational = Irrational
• Rational ÷ Irrational = Irrational
• Rational ± Irrational = Irrational
• Non-zero Rational × Irrational = Irrational
• Rational ÷ Irrational = Irrational