Mindless link propagation: Mark-Jason Dominus reports on a recent-ish proof that the square root of 2 is irrational. It's equivalent to the following simple algebraic proof: if a/b is the "simplest" integer ratio equal to sqrt(2) then consider (2b-a)/(a-b), which a little manipulation shows is also equal to sqrt(2) but has smaller numerator and denominator, contradiction.
MJD quite rightly observes:
The Greeks being who they were, their essentially arithmetical argument was phrased in terms of geometry, with all the numbers and arithmetic represented by operations on line segments. The Tom Apostol proof is much more in the style of the Greeks than is the one that the Greeks actually found!
Speaking of irrationality, I made a classic programmers' mistake this weekend. We needed some shelving for CDs, and it occurred to me that we could take two identical bog-standard pine bookcases (we have two pairs, both somewhat underused because we bought a whole lot of new shelving when we moved house) and use both sets of shelves in one bookcase to get them spaced correctly for CDs. Quite right. But I'd thought this would result in half-height shelves all the way up, whereas in fact there's one shelf too few for that.