Proof of Impossibility - The Existence of Irrational Numbers: Pythagoras Proof

The Existence of Irrational Numbers: Pythagoras Proof

Although not usually considered an "impossibility proof", the proof by Pythagoras or his students that the square-root of 2 cannot be expressed as the ratio of two integers (counting numbers) has had a profound effect on mathematics: it bifurcated "the numbers" into two non-overlapping collections—the rational numbers and the irrational numbers. This bifurcation was used by Cantor in his diagonal method, which in turn was used by Turing in his proof that the Entscheidungsproblem (the decision problem of Hilbert) is undecidable.

ca 500 B.C. "It is unknown when, or by whom, the 'theorem of Pythagoras' was discovered. 'The discovery', says Heath, 'can hardly have been made by Pythagoras himself, but it was certainly made in his school.' Pythagoras lived about 570-490. Democritus, born about 470, wrote 'on irrational lines and solids'...

Proofs followed for various square roots of the primes up to 17. "There is a famous passage in Plato's Theaetetus in which it is stated that Teodorus (Plato's teacher) proved the irrationality of

'taking all the separate cases up to the root of 17 square feet..." (Hardy and Wright, p. 42).

A more general proof now exists that:

The mth root of an integer N is irrational, unless N is the mth power of an integer n" (Hardy and Wright, p. 40).