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krantz s episodic history mathematics 2006

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Textbook in PDF format An Episodic History of Mathematics delivers a series of snapshots of the history of mathematics from ancient times to the twentieth century. The intent is not to be an encyclopedic history of mathematics, but to give the reader a sense of mathematical culture and history. The book abounds with stories, and personalities play a strong role. The book will introduce readers to some of the genesis of mathematical ideas. Mathematical history is exciting and rewarding, and is a significant slice of the intellectual pie. A good education consists of learning different methods of discourse, and certainly mathematics is one of the most well-developed and important modes of discourse that we have. The focus in this text is on getting involved with mathematics and solving problems. Every chapter ends with a detailed problem set that will provide the student with many avenues for exploration and many new entrees into the subject. Preface The Ancient Greeks Pythagoras Introduction to Pythagorean Ideas Pythagorean Triples Euclid Introduction to Euclid The Ideas of Euclid . Archimedes The Genius of Archimedes Archimedes’s Calculation of the Area of a Circle Zeno’s Paradox and the Concept of Limit The Context of the Paradox? The Life of Zeno of Elea Consideration of the Paradoxes Decimal Notation and Limits Infinite Sums and Limits Finite Geometric Series Some Useful Notation Concluding Remarks The Mystical Mathematics of Hypatia Introduction to Hypatia What is a Conic Section? The Arabs and the Development of Algebra Introductory Remarks The Development of Algebra Al-Khowˆarizmˆ ı and the Basics of Algebra The Life of Al-Khwarizmi The Ideas of Al-Khwarizmi Omar Khayyam and the Resolution of the Cubic The Geometry of the Arabs The Generalized Pythagorean Theorem Inscribing a Square in an Isosceles Triangle A Little Arab Number Theory Cardano, Abel, Galois, and the Solving of Equations Introduction The Story of Cardano First-Order Equations Rudiments of Second-Order Equations Completing the Square The Solution of a Quadratic Equation The Cubic Equation A Particular Equation The General Case Fourth Degree Equations and Beyond The Brief and Tragic Lives of Abel and Galois The Work of Abel and Galois in Context Ren´e Descartes and the Idea of Coordinates Introductory Remarks The Life of Ren´e Descartes The Real Number Line The Cartesian Plane Cartesian Coordinates and Euclidean Geometry Coordinates in Three-Dimensional Space The Invention of Differential Calculus The Life of Fermat Fermat’s Method More Advanced Ideas of Calculus: The Derivative and the Tangent Line Fermat’s Lemma and Maximum/Minimum Problems Complex Numbers and Polynomials A New Number System Progenitors of the Complex Number System Cardano Euler Argand Cauchy Riemann Complex Number Basics . The Fundamental Theorem of Algebra Finding the Roots of a Polynomial Sophie Germain and Fermat’s Last Problem Birth of an Inspired and Unlikely Child Sophie Germain’s Work on Fermat’s Problem Cauchy and the Foundations of Analysis Introduction Why Do We Need the Real Numbers? How to Construct the Real Numbers Properties of the Real Number System Bounded Sequences Maxima and Minima The Intermediate Value Property The Prime Numbers The Sieve of Eratosthenes The Infinitude of the Primes More Prime Thoughts Dirichlet and How to Count The Life of Dirichlet The Pigeonhole Principle Other Types of Counting Riemann and the Geometry of Surfaces Introduction How to Measure the Length of a Curve Riemann’s Method for Measuring Arc Length The Hyperbolic Disc Georg Cantor and the Orders of Infinity Introductory Remarks What is a Number? An Uncountable Set Countable and Uncountable The Existence of Transcendental Numbers The Number Systems The Natural Numbers Introductory Remarks Construction of the Natural Numbers Axiomatic Treatment of the Natural Numbers The Integers Lack of Closure in the Natural Numbers The Integers as a Set of Equivalence Classes Examples of Integer Arithmetic Arithmetic Properties of the Integers The Rational Numbers Lack of Closure in the Integers The Rational Numbers as a Set of Equivalence Classes Examples of Rational Arithmetic Subtraction and Division of Rational Numbers The Real Numbers Lack of Closure in the Rational Numbers Axiomatic Treatment of the Real Numbers The Complex Numbers Intuitive View of the Complex Numbers Definition of the Complex Numbers The Distinguished Complex Numbers 1 and i Algebraic Closure of the Complex Numbers Henri Poincar´e, Child Prodigy Introductory Remarks Rubber Sheet Geometry The Idea of Homotopy The Brouwer Fixed Point Theorem 1The Generalized Ham Sandwich Theorem Classical Ham Sandwiches Generalized Ham Sandwiches Sonya Kovalevskaya and Mechanics The Life of Sonya Kovalevskaya The Scientific Work of Sonya Kovalevskaya Partial Differential Equations A Few Words About Power Series The Mechanics of a Spinning Gyroscope and the Influence of Gravity The Rings of Saturn The Lam´e Equations Bruns’s Theorem Afterward on Sonya Kovalevskaya Emmy Noether and Algebra The Life of Emmy Noether Emmy Noether and Abstract Algebra: Groups Emmy Noether and Abstract Algebra: Rings The Idea of an Ideal Methods of Proof Axiomatics Undefinables Definitions Axioms Theorems, ModusPonendoPonens, and ModusTollens Proof by Induction Mathematical Induction Examples of Inductive Proof Proof by Contradiction Examples of Proof by Contradiction Direct Proof Examples of Direct Proof Other Methods of Proof Examples of Counting Arguments Alan Turing and Cryptography Background on Alan Turing The Turing Machine An Example of a Turing Machine More on the Life of Alan Turing What is Cryptography? Encryption by Way of Affine Transformations Division in Modular Arithmetic Instances of the Affine Transformation Encryption Digraph Transformations References

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