Chapter 1: Real Numbers
Exploring the building blocks of mathematics: Prime Factorisation & Irrationals.
🌳 Fundamental Theorem
Every composite number can be expressed as a product of primes.
This prime factorisation is absolutely unique, apart from the order in which the prime factors occur.
32760 = 2×2×2×3×3×5×7×13
= 2³ × 3² × 5 × 7 × 13
⚙️ HCF & LCM Rules
Using Prime Factorisation Method.
Product of the smallest power of each common prime factor.
Product of the greatest power of each prime factor involved.
HCF (a,b) × LCM (a,b) = a × b
The Golden Verification Formula.
📐 Irrational Roots
Let p be a prime number. If p divides a², then p divides a, where a is a positive integer.
To prove √2 is irrational, we initially assume the opposite (that it is rational and equals a/b where a,b are coprime). We then prove this leads to a mathematical contradiction.
⭐ Board Exam Super-Hacks
Hack 1: Proving numbers end in zero
If a number like 4ⁿ or 6ⁿ were to end in 0, its prime factorisation MUST contain the prime 5. Since 4ⁿ = (2²)ⁿ, it only has 2 as a prime factor. Thus, it can never end in zero.
Hack 2: The Secret 3-Number Formula
For three numbers (p, q, r), HCF × LCM ≠ p × q × r. Use this explicit formula instead:
LCM(p,q,r) = [p · q · r · HCF(p,q,r)] / [HCF(p,q) · HCF(q,r) · HCF(p,r)]
⚠️ Common Exam Traps
Trap 1: The Co-prime Step
When proving √3 is irrational, students forget to state: "Let a and b be coprime". Without this statement, your final contradiction (that they share a common factor 3) makes no sense!
Trap 2: Rational + Irrational
Remember: The sum or difference of a rational and an irrational number is ALWAYS irrational (e.g. 5 - √3). Don't try to calculate it, just write it algebraically.