What are Irrational Numbers?
Irrational numbers are real numbers that cannot be expressed as p/q (where p and q are integers, q ≠ 0). Their decimal expansions are non-terminating and non-repeating.
Examples of Irrational Numbers:
- √2, √3, √5, √7, ... (square roots of non-perfect squares)
- π (3.1415926535...)
- e (2.718281828...)
- Golden ratio φ (1.618033...)
Proving √2 is Irrational (Classic Proof):
- Assume √2 = p/q in simplest form (p, q coprime integers)
- Square both sides: 2 = p²/q² → p² = 2q²
- This means p² is even → p is even → p = 2k
- Substitute: (2k)² = 2q² → 4k² = 2q² → q² = 2k²
- This means q² is even → q is even
- But then p and q are both even → contradicting "simplest form"
- Therefore, √2 cannot be rational
Locating √2 on the Number Line:
Construct a right triangle with legs of length 1. The hypotenuse = √2.