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Course Lectures and Notes
INSTRUCTION
Professor: Alex Kontorovich
Office: Fine Hall 1210
Office Hours: Tuesdays 2-3 pm. Or by appointment.
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| WEEK |
LECTURE 1 |
LECTURE 2 |
NOTES/READINGS |
Week 1
9/03, 9/05 |
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Lecture 1:
Historical Overview
Early notions of Number/Shape
Babylonian clay tablets
Sexagesimal numerals
YBC 7289, Plimpton 322
Lecture 2:
Rhynd Papyrus
Moscow Papyrus
Converse to Pythagorean Theorem
Vinculum
Egyptian (unit) fractions
Erdos-Straus Conjecture
Thales's Theorem
Pythagorean Axiom of Commensurability
Play around on your own at: geogebra.org
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Week 2
9/10, 9/12 |
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Lecture 3:
Pythagorean music theory
sound as a pressure wave
resonant frequencies
the Number Theory of the overtone series
microtones
Pythagorean comma
Theorem: Pianos don't exist
equal temperament tuning
Axiom of Commensurability
Hippasus, irrationality of sqrt 2
Play around on your own at: Imaginary's Scale Lab
Lecture 4:
Pythagorean number theory
Mathematical Induction
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Week 3
9/17, 9/19 |
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Lecture 5:
Prime Factorization
Straightedge and Compass Constructions
Quadrature (of the Rectangle)
Lecture 6:
Quadrature of the Rectangle, Square,
And All Rectilinear Shapes
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Week 4
9/24, 9/26 |
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Read Dunham Chapter 1
Lecture 7:
Hippocrates's Quadrature of the Lune,
Squaring the Circle,
Impossible Construction Problems from Antiquity
Lecture 8:
Impossible Constructions, Squaring the Circle,
Angle trisection
Doubling the Cube
Pavla Rosenstein's Presentation on Mesopotamian / Babylonian Cultures and Matheamtics
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Week 5
10/01, 10/03 |
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Lecture 9:
Algebraic Numbers,
Constructible Numbers,
Transcendental Numbers,
Carl Friedrich Gauss,
Pierre Wantzel,
Charles Hermite,
Ferdinand von Lindemann,
Pi is Transcendental, and hence,
you can't Square the Circle.
Alexander the Great,
Alexandria,
Euclid,
The Elements.
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Week 6
10/08, 10/10 |
REVIEW |
MIDTERM |
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Week 7
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FALL |
RECESS |
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Week 8
10/22, 10/24 |
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Read Dunham Chapter 2
Lecture 10:
Introduction to Euclid's Elements
The Elements (Original Text)
The Elements (Byrne's Version)
Undefined Terms
The Five Postulates
Spherical Geometry
Lecture 11:
Book 1. Prop 1: Construct Equilateral Triangle (Mistake!).
Prop 2: Collapsing Compass as good as Rigid.
Prop 46: Construct Square.
Prop 47: Windmill proof of Pythagorean Theorem.
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Week 9
10/29, 10/31 |
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Lecture 12:
End of Book I, I.47, I.48
Alternative Postulates to P5
Playfair's Axiom,
Angles in a Triangle,
Gauss,
Bolyai,
Lobachevsky,
Tom Lehrer song: Lobachevsky,
Riemann,
Beltrami,
Synthetic Geometry vs Models
Lecture 13:
Beltrami model (really Poincare model) of hyperbolic space,
Spot It game, NY Times article on finite projective planes,
MC Escher, Circle Limits,
The Parallel Postulate Cannot be Proved from the others,
Books II, III, and IV of the Elements,
Construction of the Regular n-gon,
Gauss's construction of the 17-gon,
Fermat Primes
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Week 10
11/05, 11/07 |
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Read Dunham Chapter 3
Lecture 14:
Quiz Review,
Euclid's Elements Book V,
Eudoxus of Cnidus,
Arguments of Proportionality replacing Axiom of Commensurability
Hyperbolic Crochet,
Long Lost Mayan City Found,
Lecture 15:
Book VI: Similarity (AAA),
Radian,
Books VII-IX: Number Theory
Unit,
"Whole Number",
Divisor,
Perfect Numbers,
Mersenne Primes,
Euclid-Euler Theorem on Even Perfect Numbers,
Euclidean Algorithm
(both for finding a factor, or proving two numbers coprime),
Infinitude of the Primes,
Unique Factorization into Primes,
Goldbach conjecture,
Odd Perfect Number Conjecture,
World's Largest (Mersenne) Prime Found!,
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Week 11
11/12, 11/14 |
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Read Dunham Chapters 4 and 7
Lecture 16:
Book X : (in)commensurability,
Books XI-XIII : Solid Geometry,
Pyramids,
Prisms,
Volume,
Construction and Classification of Platonic ("Regular") Solids,
Socrates,
Plato,
The Academy,
Aristotle,
Archimedes of Syracuse,
Levers,
Weapons of War,
Eureka
Watch: Archimedes Palimpsest Video
Watch: Surface Area Video
Lecture 17:
Read: Archimedes and Ubiquity of Pi,
Pi in the Bible, Kings 7:23,
Pi as a length,
Pi as an area,
Parabola from Focus and Directrix,
Conic Sections,
Apollonius of Perga,
Parabola (and Ellipse) as a Conic Section,
Archimedes: Quadrature of the Parabola,
Zeno's Paradoxes,
Pascal's Triangle,
Newton's Binomial Theorem,
Madhava of Sangamagrama
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Week 12
11/19, 11/21 |
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Read Dunham, Chapters 5 and 6
Lecture 18:
Trigonometry,
Solar / Lunar Eclipses,
Aristarchus,
On the Sizes and Distances,
Eratosthenes,
On the Measure of the Earth,
Calendars:
(Solar /
Lunar /
Lunisolar
in
Christianity / Islam / Judaism, resp.),
Copernicus,
Galileo,
Kepler,
Brahe,
Newton,
Hubble,
Gersonides ~1300 CE
Lecture 19:
Algebra,
Heron of Alexandria,
Diophantus of Alexandria,
Hypatia of Alexandria,
Emperor Constantine,
Library of Alexandria,
Aryabhata,
Brahmagupta,
Virahanka,
Bhaskara II,
al-Khwarizmi (of Baghdad) "algorithm",
al-Jabr ("Algebra"),
Crusades,
Fibonacci (Leonardo of Pisa),
Hindu-Arabic Numberals,
Luca Pacioli,
Summa di Arithmetica,
Solving Linear and Quadratic Equations,
Scipione del Ferro,
Depressed Cubic Equations,
Antonio Fior,
Nicolo "Tartaglia" (The Stammerer) Fontana,
Gerolamo Cardano,
Lodovico Ferrari,
Solution to Cubic and Quartic Equations,
Ars Magna (1545),
Paolo Ruffini,
Niels Henrik Abel,
Evariste Galois,
Insolvability of Quintic and Higher Equations in Radicals,
Numberphile video for one of the last uses of Roman Numerals vs Hindu-Arabic Numberals
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Week 13
NO CLASS |
FRIDAY SCHEDULE |
HAPPY THANKSGIVING! |
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Week 14
12/03, 12/05 |
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Read Dunham, Chapters 9 and 11
Lecture 20:
Newton,
Principia ,
Leibniz,
Calculus Priority Dispute,
Johann Bernouilli,
Jacob Bernouilli,
Marquis de l'Hopital,
Leonhard Euler,
Euler's Identity (proof of God's existence),
Divergence of the Harmonic Series,
Basel Problem,
Apery's Constant,
Riemann Hypothesis
Lecture 21:
Euler,
Gauss,
Gottingen,
Cauchy,
Fourier,
Sophie Germain,
Dirichlet,
Riemann,
Cantor,
Weierstrass,
Hilbert,
Borel,
Lebesgue,
Weyl,
Courant,
von Neumann,
Noether,
Einstein,
Godel,
Turing,
Heat Equation,
d'Alembert,
Daniel Bernouilli,
Wave Equation,
Hilbert Hotel,
Countable/Uncountable Infinity,
Cantor Diagonalization,
Countability of Algebraic Numbers,
Uncountability of Reals,
Transcendence (of e and pi),
Ridicule of Cantor's work,
Russell,
Russell's Paradox,
Hilbert's Program,
Godel's Incompleteness Theorem,
Turing Machine,
Halting Problem,
More on Godel's incompleteness/ Turing's Halting
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