Proof by Contradiction
Rational Approximations
Convergents of continued fraction [1; 2, 2, 2, ...]
The Legend
Hippasus of Metapontum, a 5th-century BCE Pythagorean, allegedly discovered that √2 cannot be expressed as a ratio of integers—shattering the Pythagorean belief that all numbers are rational. Legend says his fellow Pythagoreans were so outraged by this heretical finding that they drowned him at sea, though historians debate the veracity of this tale.
The Mathematics
Theorem: √2 is irrational (cannot be written as p/q where p, q are integers with no common factors).
Proof by Contradiction:
1. Assume √2 = p/q in lowest terms (gcd(p,q)=1)
2. Then 2 = p²/q², so p² = 2q²
3. Thus p² is even, which means p is even. Write p=2m
4. Then (2m)² = 2q², so 4m² = 2q², thus q² = 2m²
5. So q² is even, meaning q is even
6. But if p and q are both even, they share factor 2, contradicting gcd(p,q)=1
7. Therefore √2 cannot be rational ∎
Continued Fraction: √2 = [1; 2, 2, 2, ...] = 1 + 1/(2 + 1/(2 + 1/(2 + ...)))
Convergents: 1/1, 3/2, 7/5, 17/12, 41/29, 99/70, ... Each gives progressively better rational approximations.