Channel: Mathologer

Category: Education

Tags: root 6irrational numberdavid montaguestanley tennenbaumtennenbaumproofroot 2infinite descentroot 3john conwayroot 5proof by contradiction

Description: NEW (Christmas 2019). Two ways to support Mathologer Mathologer Patreon: patreon.com/mathologer Mathologer PayPal: paypal.me/mathologer (see the Patreon page for details) Get ready for some brand new and very pretty visual proofs of the fact that root 2, root 3, root 5 and root 6 are irrational numbers. Root 2 being irrational also translates into the fact that the equation x^2+x^2=y^2 has no solutions in positive integers, root 3 being irrational translates into the fact that the equation x^2+x^2+x^2=y^2 has no solutions in positive integers, etc. What I find very attractive about these proofs is that the destructive core of these proofs by contradiction lead a second secret constructive life, giving birth to infinitely many nearest miss solutions of our impossible equations like for example 15^2+15^2+15^2=26^2-1. Here is the paper by Steven J. Miller and David Montague which features the basic root 3 and pentagonal root 5 choreographies. arxiv.org/abs/0909.4913 Footnotes: -our nearest miss solutions like, for example, 15^2+15^2+15^2=26^2-1 correspond to the solutions of the equation y^2 - n x^2 = 1 with n=2, 3, 5 and 6. This is the famous Pell's equation, which happens to have solutions for all integers n that are not squares. -there is also a second type of nearest miss solutions like 4^2+4^2+4^2=7^2+1 (a plus instead of a minus at the end). Starting with one of these our choreographies also generate all other such nearest misses. -the original Tennenbaum square choreography and the first puzzle root three choreography generate both types of nearest misses from any nearest miss solution. -The close approximations to the various roots corresponding to our nearest miss solutions are partial fractions of the continued fraction expansion of the roots. -lots more things to be said here but we are getting close to the word limit for descriptions and so I better stop :) Thank you very much to Marty for all his nitpicking of the script for this video and Danil for his ongoing Russian support. Today's t-shirt is the amazing square root t-shirt (google "square root tshirt"). Note that the tree looks like a square root sign AND that the roots of the tree are really square. Enjoy!