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

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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.

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

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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”.

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5 §13-2 Germain and Somerville Student Discussion.

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6 §13 -3 Fourier and Poisson Student Discussion.

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

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8 §13- 4 Bolzano Student Discussion.

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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.

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10 §13-5 Cauchy Student Discussion.

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11 §13 - 6 Abel and Galois Student Comment

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12 §13-7 Jacobi and Dirichlet Student Discussion.

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

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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?

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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?

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16 §13 – 9 Liberation of Geometry Student Discussion.

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17 §13 – 10 Algebraic Structure Student Discussion.

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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

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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?

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20 §13 – 11 Liberation of Algebra Student Discussion.

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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.

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

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23 §13 – 12 De Morgan Rules

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

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

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

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