Cultural Connection

1. Cultural Connection The Industrial Revolution The Nineteenth Century. Student led discussion.

2. 13 – The 19th Century - Liberation of Geometry and Algebra The student will learn about The “Prince of Mathematicians” and other mathematicians and mathematics of the early 19th century.

3. §13-1 The Prince of Mathematics Student Discussion.

4. §13-1 Carl Fredrich Gauss 3 yr. Error in father’s bookkeeping. 10 yr. Σ 1 + 2 + . . . + 100 = 5050. 18 yr. 17 sided polygon. 19 yr. Every positive integer is the sum of at most three triangular numbers. 20 yr. Dissertation –proof of “Fundamental Theorem of Algebra”. Homework – write 2009 as the sum of at most three triangular numbers. EUREKA! = Δ + Δ + Δ

5. §13-2 Germain and Somerville Student Discussion.

6. §13 -3 Fourier and Poisson Student Discussion.

7. §13 -3 Fourier Series Any function defined on (-π, π) can be represented by: That is, by a trigonometric series.

8. §13- 4 Bolzano Student Discussion.

9. §13- 4 Bolzano Bolzano-Weirstrass Theorem – Every bounded infinite set of points contains at least one accumulation point. Intermediate Value Theorem – for f (x) real and continuous on an open interval R and f (a) = α and f (b) = β, then f takes on any value γ lying between α and β at at least one point c in R between a and b.

10. §13-5 Cauchy Student Discussion.

11. §13 - 6 Abel and Galois Student Comment

12. §13-7 Jacobi and Dirichlet Student Discussion.

13. §13 – 8 Non-Euclidean Geometry Student Discussion.

14. §13 – 8 Saccheri Quadrilateral C D B A Easy to show that angles C and D are equal. Easy to show that angles C and D are equal. Are they right angles? Easy to show that angles C and D are equal. Are they right angles? Acute angles? Easy to show that angles C and D are equal. Are they right angles? Acute angles? Obtuse angles?

15. §13 – 8 Lambert Quadrilateral D C B A Is angle D a right angle? Is angle D a right angle? An acute angle? Is angle D a right angle? An acute angle? An obtuse angle?

16. §13 – 9 Liberation of Geometry Student Discussion.

17. §13 – 10 Algebraic Structure Student Discussion.

18. §13 – 10 a + b 2 Addition (a + b2) + (c + d2) = ( a + c + (b +d) 2 ) Add (1 + 22) + (3 + 2) = Add (1 + 22) + (3 + 2) = 4 + 3 2 Multiplication (a + b2) (c + d2) = (ac + 2bd + ( bc + ad ) 2 ) ) Multiply (1 + 22) (3 + 2) = Multiply (1 + 22) (3 + 2) = 7 + 72 Is addition commutative? Is addition commutative? Associative? Is multiplication commutative? Is multiplication commutative? Associative? Homework – find the additive identity and the additive inverse of 2 + 52, and the multiplicative identity and the multiplicative inverse of 2 + 52.

19. §13 – 10 2x2 matrices Multiplication is not commutative. Can your find identities for addition and multiplication? Can your find identities for addition and multiplication? Inverses?

20. §13 – 11 Liberation of Algebra Student Discussion.

21. §13 – 11 Complex Numbers Note: Let (a, b) represent a + bi, then Note: (a, 0) + (b, 0) = (a + b, 0) and (a, b) + (c, d) = (a + c, b + d) and (a, 0) + (b, 0) = (a + b, 0) and (a, 0) · (b, 0) = (ab, 0) (a, b) · (c, d) = (ac - bd, ad + bc). (a, 0) · (b, 0) = (ab, 0) the reals are a subset. Try the following: (2, 3) + (4, 5) = (2, 3) · (4, 5) = And i 2 = (0, 1) (0, 1) = (-1, 0) = -1

22. §13 – 12 Hamilton, Grassmann, Boole, and De Morgan Student Discussion.

23. §13 – 12 De Morgan Rules

24. §13 – 13 Cayley, Sylvester, and Hermite Student Discussion.

25. §13 – 14 Academies, Societies, and Periodicals Student Discussion.

26. Assignment Rough draft due on Wednesday. Read Chapter 14.