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An Episodic History of Mathematics by Steven G. Krantz



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An Episodic History of Mathematics written by Steven G. Krantz cover the following topics.

  • 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-Khowˆarizmˆi 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, The 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 443
    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

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