Simple Proof That the Square Root of 2 Is Irrational

Mark Dominus on a proof by Tom M. Apostol published in 2000:

In short, if √2 were rational, we could construct an isosceles right triangle with integer sides. Given one such triangle, it is possible to construct another that is smaller. Repeating the construction, we could construct arbitrarily small integer triangles. But this is impossible since there is a lower limit on how small a triangle can be and still have integer sides.

(Via Michael Tsai.)

Wednesday, 21 February 2007