The Real Number System
Real numbers include all rational and irrational numbers.
Euclid's Division Lemma
For positive integers a and b:
a = bq + r, where 0 ≤ r < b
Example: Divide 455 by 42
- 455 = 42 × 10 + 35
- So q = 10, r = 35
Euclid's Division Algorithm (for HCF)
To find HCF of 96 and 404: 1. 404 = 96 × 4 + 20 2. 96 = 20 × 4 + 16 3. 20 = 16 × 1 + 4 4. 16 = 4 × 4 + 0
HCF(404, 96) = 4
Fundamental Theorem of Arithmetic
Every composite number can be expressed as a unique product of prime numbers.
Example: 12 = 2 × 2 × 3 = 2² × 3
HCF and LCM Using Prime Factorisation
For numbers 12 and 18:
- 12 = 2² × 3
- 18 = 2 × 3²
- HCF = 2¹ × 3¹ = 6 (lowest powers)
- LCM = 2² × 3² = 36 (highest powers)
Verify: HCF × LCM = 6 × 36 = 216 = 12 × 18 ✓
Irrational Numbers
A number is irrational if it cannot be written as p/q (where p, q are integers, q ≠ 0).
Examples: √2, √3, √5, π
Proving √2 is Irrational
Assume √2 = p/q (lowest terms, no common factors).
Then 2 = p²/q² → p² = 2q² → p² is even → p is even
Let p = 2m, then 4m² = 2q² → q² = 2m² → q is even
But then p and q have a common factor 2 — contradiction!
Therefore √2 is irrational. ∎
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