1. Senior Seminar Project By Santiago Salazar

2. Pythagorean Theorem In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the legs.

3. The Pythagorean Proposition by Elisha S. Loomis (1852 - 1940)

4. Four kinds of Demonstrations Those based upon linear relations –the algebraic proofs Those based upon comparison of areas –the geometric proofs Those based upon vector operations –the quaternionic proofs Those based upon mass and velocity –the dynamic proofs

5. Pythagoras (ca. 575-495 BCE) Born on the Island Samos, off the coast of modern-day Turkey There is no reliable information about him Traveled through Egypt and Mesopotamia, where he probably increased his knowledge of Mathematics, Philosophy, and Religion Also traveled to Miletus, where he made advances in geometry under philosophers and mathematicians such as Thales of Miletus, Anaximander, and Anaximenes

6. Pythagoras (ca. 575-495 BCE) Moved to Croton (today, Crotone in southern Italy), about 530 BCE There he founded a society, denominated the Pythagoreans, whose main interests were religion, mathematics, astronomy, and music The nature of the universe could be explained by numbers and their ratios

7. Egypt During 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 units This 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 version

8. Harpedonapts (Rope strechers)

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

10. Plimpton 322 (ca. 1800 BCE)

11. A Babylonian clay table containing Pythagorean triples G.A. Plimpton Collection at Columbia University One column is missing, which is believed to contain the third number in each Pythagorean triple 15 rows and 4 columns

12. India and China About 800 BCE in India, ancient mathematicians solved problems that relate to the Pythagorean relation Uses of the Pythagorean Theorem appeared in “Nine Chapters on the Mathematical Arts,” probably the most influential Chinese mathematical work The Chinese provided a proof of the Theorem only in the special case of the 3-4-5 triangle

13. Definition A Pythagorean triangle is a right-angled triangle whose side’s lengths are positive integers.

14. Equivalent Problem

15. Definitions

16. Analysis of the Problem (part I)

17. Definition 1

18. Definition 2

19. Definition 3

20. Theorem 4

21. Proposition 5

22. Definition 6

23. Analysis of the Problem (part II)

25. Lemma 7

26. Corollary 8

27. Lemma 9

28. Lemma 10

29. Lemma 11

30. Theorem 12

31. Consequences

32. Some Fundamental Solutions 2143516925 321251314425169 418151764225289 432472557649625 52202129400441841 54409411600811681 6112353714412251369 7228455378420252809

33. Curiosities

34. Curiosity 1

35. Curiosity 2

36. Curiosity 3 The area of the previously generated Pythagorean triangle is the product of the picked four consecutive numbers

37. Definitions

38. Curiosity 4

39. Curiosity 5

40. Curiosity 6

41. Curiosity 7

42. Curiosity 8

43. References Dudley, Underwood. Elementary Number Theory. Second Edition Katz, Victor J. A History of Mathematics An Introduction. Third Edition Loomis, Elisha S. The Pythagorean Proposition Posamentier, Alfred S. The Pythagorean Theorem