Download Chapter 13 (The early nineteenth century).

Cultural Connection
The Industrial Revolution
The Nineteenth Century.
Student led discussion.
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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.
2
§13-1 The Prince of Mathematics
Student Discussion.
3
§13-1 Carl Fredrich Gauss
3 yr.
10 yr.
18 yr.
19 yr.
20 yr.
Error in father’s bookkeeping.
Σ 1 + 2 + . . . + 100 = 5050.
17 sided polygon.
Every positive integer is the sum of at
most three triangular numbers.
Dissertation –proof of “Fundamental
Theorem of Algebra”.
Homework – write 2009 as the sum of at most
three triangular numbers.
EUREKA! = Δ + Δ + Δ
4
§13-2 Germain and Somerville
Student Discussion.
5
§13 -3 Fourier and Poisson
Student Discussion.
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§13 -3 Fourier Series
Any function defined on (-π, π) can be represented
by:

a0
  an cos nx  bn sin nx 
2 n1
That is, by a trigonometric series.
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§13- 4 Bolzano
Student Discussion.
8
§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.
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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.
12
§13 – 8 Non-Euclidean Geometry
Student Discussion.
13
§13 – 8 Saccheri Quadrilateral
D
A
C
B
Easy to show that angles C and D are equal. Are
they right angles? Acute angles? Obtuse angles?
14
§13 – 8 Lambert Quadrilateral
C
A
D
B
Is angle D a right angle? An acute angle? An obtuse
angle?
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§13 – 9 Liberation of Geometry
Student Discussion.
16
§13 – 10 Algebraic Structure
Student Discussion.
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§13 – 10 a + b 2
Addition
Add
(a + b2) + (c + d2) = ( a + c + (b +d) 2 )
(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) = 7 + 72
Is addition commutative? Associative?
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.
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§13 – 10 2x2 matrices
Multiplication is not commutative.
1 0 0 1 0 1
0 0  0 1  0 0

 
 

0 1 1 0 0 0
0 1  0 0  0 0

 
 

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
Let (a, b) represent a + bi, then
(a, b) + (c, d) = (a + c, b + d) and
(a, b) · (c, d) = (ac - bd, ad + bc).
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) the reals are a subset.
And i 2 = (0, 1) (0, 1) = (-1, 0) = -1
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§13 – 12 Hamilton, Grassmann,
Boole, and De Morgan
Student Discussion.
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§13 – 12 De Morgan Rules
A  B
'
 A'  B'
A  B
 A'  B'
'
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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.
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