1

Cultural ConnectionThe Industrial Revolution

Student led discussion.

The Nineteenth Century.

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

Homework – write 2009 as the sum of at most three triangular numbers.

EUREKA! = Δ + Δ + Δ

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”.

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:

1nnn

0 nxsinbnxcosa2

a

That is, by a trigonometric series.

8

§13- 4 Bolzano

Student Discussion.

9

§13- 4 BolzanoBolzano-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

Easy to show that angles C and D are equal.

A B

CD

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

Is angle D a right angle?

A B

CD

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 2Addition (a + b2) + (c + d2) = ( a + c + (b +d) 2 )

Multiplication (a + b2) (c + d2) = (ac + 2bd + ( bc + ad ) 2 ) )

Is addition commutative?

Is multiplication commutative?

Add (1 + 22) + (3 + 2) =

Multiply (1 + 22) (3 + 2) =

Homework – find the additive identity and the additive inverse of 2 + 52, and the multiplicative identity and the multiplicative inverse of 2 + 52.

Is addition commutative? Associative?

Is multiplication commutative? Associative?

Add (1 + 22) + (3 + 2) = 4 + 3 2

Multiply (1 + 22) (3 + 2) = 7 + 72

19

§13 – 10 2x2 matricesMultiplication is not commutative.

00

10

10

10

00

01

00

00

00

01

10

10

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

Try the following: (2, 3) + (4, 5) =(2, 3) · (4, 5) =

Note: (a, 0) + (b, 0) = (a + b, 0) and (a, 0) · (b, 0) = (ab, 0)

And i 2 = (0, 1) (0, 1) = (-1, 0) = -1

Let (a, b) represent a + bi, then (a, b) + (c, d) = (a + c, b + d) and (a, b) · (c, d) = (ac - bd, ad + bc).

Note: (a, 0) + (b, 0) = (a + b, 0) and (a, 0) · (b, 0) = (ab, 0) the reals are a subset.

22

§13 – 12 Hamilton, Grassmann, Boole, and De Morgan

Student Discussion.

23

§13 – 12 De Morgan Rules

'B'ABA '

'B'ABA '

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.