Hans H>Bill Gosper: "Years ago, Conway said, perhaps on this list, that he was working on a book of Extreme Proofs. His version of this one was to cut (or fold) a square of paper along the diagonal..." Michael Reid: "So it's possible that this proof is several millenia old." According to Miller & Montague, Stanley Tennenbaum discovered Conway's 'origami' proof in the 1950's: http://web.williams.edu/go/math/sjmiller/public_html/math/papers/irrationali... According to < http://www.qedcat.com/proofs/geometricproof.html >, it may go back even further, mentioning a 'numerical' version in 'A Course of Modern Analysis' by Whittaker & Watson (1927). As I mentioned to Mike Hirschhorn, I think Conway's[?] origami is novel because you don't actually need to compute the quantities b-a and 2a-b. You can just say they're obviously differences of integers forming a smaller isosceles right triangle. --Bill On Tue, Nov 20, 2012 at 10:11 PM, Mike Hirschhorn <m.hirschhorn@unsw.edu.au>wrote:

Dear Bill,

I do not know who first thought of the proof that if b^2=2a^2 then (2a-b)^2=2(b-a)^2, leading to a contradiction, though I found it many years ago, and it was accepted as a filler in the American Mathematical Monthly till I found it in an article entitled ``Irrationality without number theory'' in the Monthly of the previous year, under the previous editor, and withdrew it.

If you can find that article, you may find a reference to earlier work which includes this proof.

I found it by reversing the recurrence that gives the convergents to the continued fraction for sqrt{2}.

Best wishes,

Mike.

________________________________________ From: mathfuneavesdroppers@googlegroups.com [ mathfuneavesdroppers@googlegroups.com] on behalf of Bill Gosper [ billgosper@gmail.com] Sent: Wednesday, 21 November 2012 3:08 PM To: math-fun@mailman.xmission.com Subject: Re: [math-fun] Best proof that sqrt(2) is irrational?

Hans Havermann>

Dan Asimov:

Does anyone know who first thought of the following proof that sqrt(2) is irrational?

It looks like Tom M. Apostol's proof, published as "Irrationality of The Square Root of Two - A Geometric Proof" in the American Mathematical Monthly, November 2000, pp. 841–842.

Years ago, Conway said, perhaps on this list, that he was working on a book of Extreme Proofs. His version of this one was to cut (or fold) a square of paper along the diagonal, making an isosceles right triangle. Label the sides a,a,b. Then bisect-fold one of the 45˚ angles.

This creates an obvious right-angled b-a,b-a,2a-b. --rwg