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The Pythagorean Theorem in different Civilizations Christine Larson & Ashley Alex • What Civilizations? – Greeks, Chinese, Indians • How was it approached? • What techniques were used to prove it? • How did it effect the rest of Mathematics? Some Greek Methods… •Contributed in the respect of Geometry and formal proof •Euclid’s Elements • abandoned algebra (slogan), focused more on geometry •Led to geometric algebra, Thales boat and shore problem, heights of structures, ratios, distances related to astronomy such as measuring the distance to the sun, distance between the moon and sun and used to solve equations with two unknowns Some Indian Methods… • Observed in the Sulba Sutras; also used a cutting and rotating method (very similar to Chinese proof) • Used as an explanation of how to construct squares from original figures, find heights by use of shadows • Greater shift towards solving systems, linear congruencies, trigonometry http://www.mathsisfun.com/pythagoras.html Some Chinese Methods… • Made a convincing argument without using an axioms (both Zhao Shuang and Liu Hui) • Focused more on estimation, solving equations, and Chinese Remainder Theorem • Helped solve problems with similar triangles, measure certain depths or other measurements (like homework problem) Claim: c2 = (a-b)2 + 2ab Cut and Rotate method… Chinese Indian algebraic Convincing arguments; “theorein:” to look at cut and rotate concrete examples “empirical” proof a2 + b2 = c2 Astronomical applications Chapter 9 similar to Diophantus’s Arithmetica Shared a geometric focus Developed constructions for “completing the square” axiomatic proof system; organized Deductive reasoning no numbers Greeks • Collection of 84 proofs… http://www.cut-the-knot.org/pythagoras/index.shtml • Wizard of Oz http://www.youtube.com/watch?v=DUCZXn9RZ9s • Chinese http://www.davis-inc.com/pythagor/proof2.html