Mathematics as a Human Endeavor

1. Mathematics as a Human Endeavor Ed Dickey University of South Carolina SCCTM Conference / 23 October 2009

2. 1979 SCCTM Conference • 20 October 1979 • “Historical Anecdotes for the Math Class” • Teacher at Spring Valley High School • Taught from experience that injecting history MOTIVATES and HELPS students learn mathematics. • I believe it then and I believe it now

3. Handouts • Invite Letter from Patty Smith, SCCTM President • Handout (on blue mimeograph paper and I STILL have 28!) • Evaluation Form (32 attendees!)

4. Opening Quote “ I have more than an impression- it amounts to a certainty- that algebra is made repellent by the unwillingness or inability of teachers to explain why… There is no sense of history behind the teaching, so the feeling is given that the whole system dropped down ready-made from the skies, to be used by born jugglers.” Jacques Barzun, Teacher in America

5. Help students see mathematics as a human endeavor that evolved through men and women discovering and inventing the many things we study in algebra and school mathematics

6. Help make connections to other cultures

7. Hypatia • 5th Century AD • Library of Alexandria • Commentaries to works of Diophantus and Apollonius • Edited Ptolemy and Euclid • Invented hydrometer

8. Hydrometer To measure specific gravity of liquids

9. Hypatia • Literary legend in Charles Kingsley 1953 novel: Hypatia – or New Foes with an Old Face • Portrayed the scholar as a “helpless, pretentious, erotic heroine” • Murdered by an angry mob of fanatical monks who objected to her being a woman who didn’t know her place

10. Gerolamo Cardano • 1501-1576 • “eccentric” and “difficult” • Gyroscope gimbal • Auto Differential • Combination lock • Imaginary numbers

11. Two-axis gimbal set

12. Cardan Shaft

13. Combination Lock

14. Imaginary Numbers • Contests to Solve Equations • Degree 1 and 2 equations easy • Degree 3 or Cubic • Substitute • Depressed cubic:

15. Solving Cubic Equation • Where in (2), our depressed cubic • Now introduce two new variables in (2) • And get

16. Solving Cubic Equation • Cardano let 3uv + p = 0 in (3), which implies uv = -p/3 so substituting for uv and multiplying by u3 he got • Which is a quadratic in u3 • So using the quadratic formula

17. Solving Cubic Equation • And • Now work back from the substitutions to get x in terms of a, b, c and d. • When you do this, you must accept the existence of imaginary numbers and in the 16th century only an eccentric like Cardano would.

18. Solving the Cubic • Along the backward substitution path you reach: • A place where most would stop but Cardano persisted and this gave him the formula needed to win contests.

19. Cubic Formula for

20. Example

21. .. And FINALLY • Generating 3 values of x • Full explanation at http://en.wikipedia.org/wiki/Cubic_function#Cardano.27s_method

22. TI Nspire CAS

23. WolframAlpha

24. Derive

25. GeoGebra

26. Quartic anyone? • Lodovico Ferrari (at 18!) discovered a quartic formula in 1540. • Cardano published it in Ars Magna (1545) • Many substitutions and “nested depressed cubics”

27. Quartic glimpse

28. Sigh…

29. SIGH….. FINALLY:

30. Example Solve Using Derive…..

31. Derive output (after .5 seconds)

32. So what about 5th degree? • We are now in the mid 1500s • The formula will be AWFUL! • Isn’t there a formula to solve:

33. Enter Niels Abel • 1802-1829 (ouch!) • “Abelian group”

34. Died young • Contracted tuberculosis in Paris at Christmas. • Traveled by sled to visit his fiancee in April but died after a short visit with her on April 6

35. Abel’s Impossibility Theorem • No general solution in radicals to polynomial equations of degree five or higher • Fundamental Theorem of Algebra: every polynomial with real or complex coefficients can be solved with a complex number. • Proved using Galois Theory

36. Evariste Galois • 1811-1832 (20 years old) • Poisson denied a position in the Academy “incomprehensible work” • Fought a duel on behalf off a Mlle du Motel • Stayed up all night writing his papers

37. Example Solve

38. Derive output

39. Symbolism • Cardano used no algebraic symbols • Variables like x were written out as cosi

40. Sample from Tartaglia’s Nova Scientia (1537)

41. Symbolisms • François Viète initiated the use of letters for variables (end of 16th Century) • René Descartes then Isaac Newton moved algebraic symbolism toward today’s conventions

42. Hendrick van Heuraet (1634-1660) Arc length using x and y for

43. Giovanni Saccheri (1667-1733) • “the good little monk” • Actually a Jesuit • “Euclid Freed of Every Flaw” • Demonstrate that denying Euclid Parallel Postulate leads to a contradiction

44. Saccheri Quadrilateral

45. Right Case • Euclid’s 5th Postulate • Given a line and a point not on that line, there is one and only one line through the given point parallel to the given line • Model of PLANE geometry

46. Obtuse Case • Assuming Euclid’s 5th Postulate is false • Equivalent to NO parallel lines • Leads to the conclusion that the sum of the interior angles of a triangle are greater than 180 degrees • To Saccheri this was “absurd” but later it was the basis for Elliptic Geometry • Model of SPHERICAL Geometry

47. Acute Case • Assuming Euclid’s 5th Postulate is false • Equivalent to at least TWO parallel lines • Leads to the conclusion that the sum of the interior angles of a triangle are less than 180 degrees • To Saccheri this too was “absurd” but later it was the basis for Hyperbolic Geometry • Model for RELATIVITY (spacetime Lorentz model)

48. Good Monk or Mathematician? • Was Saccheri a “good monk” allowing the prevalent view (Euclid’s) to define “absurd’ • Or was he a better mathematician allowing the logic of his conclusions to win out • He preserved his status and safety as a monk and avoided conflict with the prevailing Euclidean view point • As time progressed, thinkers challenged the prevailing views…

49. George Cantor • 1845-1918

50. Set Theory and the Infinite • Levels of infinity: countable and uncountable • Continuum Hypothesis (no set whose cardinality is between the integers and the reals) • Gödel showed this cannot be proved or disproved. • Paradoxes… “nowhere dense”