Irrational numbers
- The discovery of irrational numbers is attributed to the ancient Greeks
- Since Pythagoras, it was known that the
diagonal of a unit square has length √ 2 - His followers spent many years trying to prove
it was rational - Hippasus is attributed with proving that √ 2 is irrational, around 500 BCE
- The followers of Pythagoras were shocked by
the discovery - Allegedly, they drowned Hippasus at sea to
suppress this fact from the public
The proof of Hippasus that √2 is not a rational number
- If √ 2 is rational, it can be written as a reduced fraction
p/q, where gcd(p, q) = 1 - From √2 = p/q, squaring both sides, 2 = p2/q 2
- Cross multiplying, p 2 = 2q 2 , so p 2 = p · p is even
- The product of two odd numbers is odd and the product
of two even numbers is even, so p is even, say p = 2a - So p 2 = (2a) 2 = 4a 2 = 2q 2
- Therefore q 2 = 2a 2 , so q 2 is also even
- By the same reasoning, q is even, say q = 2b.
- So p = 2a and q = 2b, which means gcd(p, q) ≥ 2, which contradicts our assumption that p/q was in reduced form.
Summary
- The proof of Hippasus follows a pattern commonly used in mathematical reasoning
- To show that a fact P holds, assume not(P) and derive a contradiction
- Using a similar strategy, can show that for any natural number n that is not a perfect square, √ n is irrational
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