1 cultural connection the 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 about
The “Prince of Mathematicians” and other mathematicians and mathematics of the early 19th century.
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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 -3 Fourier Series
Any 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 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 – 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
CD
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 – 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.
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Can your find identities for addition and multiplication? Can your find identities for addition and multiplication? Inverses?
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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.