Conway: gosper.org/homeplate.html On 2018-05-29 12:56, Dan Asimov wrote:

My favorite irrationality proof is this which is almost instant if you allow pictures:

Suppose there exist integers p, q with

(p/q)^2 = 2.

Then p^2 = 2 q^2 implies there is a (p, p, q) right triangle by Pythagoras.

Draw this (p, p, q) triangle with the hypotenuse as base: Call it ABC with A = right angle.

Draw the circle with center at B and radius = p. This cuts the hypotenuse at the point D into lengths p and q-p.

Extend a perpendicular from the hypotenuse at D up to point E on side AC. This cuts side AC into pieces of sides q - p and p - (q - p) = 2p - q.

Then ADE is a 45-45-90 right triangle with integer sides that are smaller than the original. Which implies an infinite regress of smaller positive integers, contradiction.

—Dan

Mike Stay wrote: ----- My favorite irrationality proof is one I heard from John Baez. Suppose the cube root of two were not irrational; then there would be two positive integers p, q such that p/q = ∛2. Multiplying both sides by q and cubing, we get p³ = 2q³ = q³ + q³, which has no solutions in the positive integers by Fermat's Last Theorem! -----