spendius wrote:He was as cool as a Harlem Globetrotter in his day.
Irrationals man. That's where it's at. Ya dig?
Irrationals belong to Hippasus. Pythagoras preached rationals. Legend has it that Pythagoras had Hippasus drown when he showed that the square root of 2 wasn't rational.
Hippasus proof was by contradiction (reductio ad absurdum).
Quote:If √2 is rational it has the form m/n for integers m, n not both even. Then m² = 2n² whence m is even, say m = 2p. Thus 4p² = 2n² so 2p² = n² whence n is also even, a contradiction.
He also showed by construction that the square foot of two could be found on the number line. Dedekind and Cantor showed that the cardinality of the set irrational numbers was larger than the set of rational numbers (there are an infinite number of irrational numbers between any two rational numbers)--wrap your head about that one.
Rap