Download presentation

Presentation is loading. Please wait.

Published byClifton Edenfield Modified over 3 years ago

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

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

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

4
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
5 §13-2 Germain and Somerville Student Discussion.

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

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

8
8 §13- 4 Bolzano Student Discussion.

9
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
10 §13-5 Cauchy Student Discussion.

11
11 §13 - 6 Abel and Galois Student Comment

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

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

14
14 §13 – 8 Saccheri Quadrilateral Easy to show that angles C and D are equal. A B C D 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
15 §13 – 8 Lambert Quadrilateral Is angle D a right angle? A B C D Is angle D a right angle? An acute angle?Is angle D a right angle? An acute angle? An obtuse angle?

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

17
17 §13 – 10 Algebraic Structure Student Discussion.

18
18 §13 – 10 a + b 2 Addition (a + b 2) + (c + d 2) = ( a + c + (b +d) 2 ) Multiplication (a + b 2) (c + d 2) = (ac + 2bd + ( bc + ad ) 2 ) ) Is addition commutative? Is multiplication commutative? Add (1 + 2 2) + (3 + 2) = Multiply (1 + 2 2) (3 + 2) = 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. Is addition commutative? Associative? Is multiplication commutative? Associative? Add (1 + 2 2) + (3 + 2) = 4 + 3 2 Multiply (1 + 2 2) (3 + 2) = 7 + 7 2

19
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
20 §13 – 11 Liberation of Algebra Student Discussion.

21
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
22 §13 – 12 Hamilton, Grassmann, Boole, and De Morgan Student Discussion.

23
23 §13 – 12 De Morgan Rules

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

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

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

Similar presentations

Presentation is loading. Please wait....

OK

31.03.2011-ZMQS- 1. 2 31.03.2011 -ZMQS- 3 31.03.2011.

31.03.2011-ZMQS- 1. 2 31.03.2011 -ZMQS- 3 31.03.2011.

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on nature and human development Ppt on security attacks in information security Ppt on health effects of air pollution in delhi Ppt on rainwater harvesting Download ppt on surface chemistry Ppt on geothermal energy advantages and disadvantages Ppt on mathematics for class 9 Ppt on interest rate derivatives Ppt on online image editor Ppt on chapter 12 electricity for kids