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Cultural ConnectionThe Industrial Revolution

Student led discussion.

The Nineteenth Century.

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13 – The 19th Century - Liberation of Geometry and Algebra

The student will learn aboutThe “Prince of Mathematicians” and other mathematicians and mathematics of the early 19th century.

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§13-1 The Prince of Mathematics

Student Discussion.

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§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”.

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§13-2 Germain and Somerville

Student Discussion.

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§13 -3 Fourier and Poisson

Student Discussion.

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§13 -3 Fourier SeriesAny function defined on (-π, π) can be represented by:

1nnn

0 nxsinbnxcosa2

a

That is, by a trigonometric series.

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§13- 4 Bolzano

Student Discussion.

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§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.

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§13-5 Cauchy

Student Discussion.

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§13 - 6 Abel and Galois

Student Comment

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§13-7 Jacobi and Dirichlet

Student Discussion.

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§13 – 8 Non-Euclidean Geometry

Student Discussion.

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§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?

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§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?

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§13 – 9 Liberation of Geometry

Student Discussion.

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§13 – 10 Algebraic Structure

Student Discussion.

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§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

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§13 – 10 2x2 matricesMultiplication is not commutative.

0010

1010

0001

0000

0001

1010

Can your find identities for addition and multiplication? Can your find identities for addition and multiplication? Inverses?

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§13 – 11 Liberation of Algebra

Student Discussion.

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§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.

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

Student Discussion.

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§13 – 12 De Morgan Rules

'B'ABA '

'B'ABA '

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§13 – 13 Cayley, Sylvester, and Hermite

Student Discussion.

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§13 – 14 Academies, Societies, and Periodicals

Student Discussion.

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Assignment

Rough draft due on Wednesday.

Read Chapter 14.