PowerPoint Presentation

Senior Seminar ProjectBy Santiago SalazarPythagorean TheoremIn a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the legs.The Pythagorean Propositionby Elisha S. Loomis (1852 - 1940)Four kinds of DemonstrationsThose based upon linear relations the algebraic proofsThose based upon comparison of areas the geometric proofsThose based upon vector operations the quaternionic proofsThose based upon mass and velocity the dynamic proofsPythagoras (ca. 575-495 BCE)Born on the Island Samos, off the coast of modern-day TurkeyThere is no reliable information about himTraveled through Egypt and Mesopotamia, where he probably increased his knowledge of Mathematics, Philosophy, and ReligionAlso traveled to Miletus, where he made advances in geometry under philosophers and mathematicians such as Thales of Miletus, Anaximander, and AnaximenesPythagoras (ca. 575-495 BCE)Moved to Croton (today, Crotone in southern Italy), about 530 BCEThere he founded a society, denominated the Pythagoreans, whose main interests were religion, mathematics, astronomy, and musicThe nature of the universe could be explained by numbers and their ratiosEgyptDuring his trips to Egypt, Pythagoras probably came in contact with the measuring method of the Harpedonapts (rope stretchers)Egyptian used, for architectural purposes, ropes tied with 12 equidistant nodes to create right triangles of lengths 3,4, and 5 unitsThis suggests that the Egyptians had some insights about the special case of the 3-4-5 right triangle well before Pythagoras proved the more general versionHarpedonapts (Rope strechers)

Mesopotamia (Babylonians)Babylonians had some insights about this theorem ca. 1800 BCEThis shows the high level of mathematical knowledge that existed well before the GreeksThey discovered a significant number of Pythagorean triplesThey solved problems that could only be solved with the use of the Pythagorean triples and the Theorem of PythagorasPlimpton 322 (ca. 1800 BCE)

Plimpton 322 (ca. 1800 BCE)A Babylonian clay table containing Pythagorean triplesG.A. Plimpton Collection at Columbia UniversityOne column is missing, which is believed to contain the third number in each Pythagorean triple15 rows and 4 columns

India and ChinaAbout 800 BCE in India, ancient mathematicians solved problems that relate to the Pythagorean relationUses of the Pythagorean Theorem appeared in Nine Chapters on the Mathematical Arts, probably the most influential Chinese mathematical workThe Chinese provided a proof of the Theorem only in the special case of the 3-4-5 triangleDefinitionA Pythagorean triangle is a right-angled triangle whose sides lengths are positive integers.Equivalent ProblemDefinitionsAnalysis of the Problem (part I)Definition 1Definition 2Definition 3Theorem 4Proposition 5Definition 6Analysis of the Problem (part II)Analysis of the Problem (part II)Lemma 7Corollary 8Lemma 9Lemma 10Lemma 11Theorem 12ConsequencesSome Fundamental SolutionsCuriositiesCuriosity 1Curiosity 2Curiosity 3The area of the previously generated Pythagorean triangle is the product of the picked four consecutive numbers DefinitionsCuriosity 4Curiosity 5Curiosity 6Curiosity 7Curiosity 8ReferencesDudley, Underwood. Elementary Number Theory. Second EditionKatz, Victor J. A History of Mathematics An Introduction. Third EditionLoomis, Elisha S. The Pythagorean PropositionPosamentier, Alfred S. The Pythagorean Theorem