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An Episodic History of Mathematics by Steven G. Krantz
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About this book :-
An Episodic History of Mathematics written by Steven Krantz
The focus in this text is on doing—getting involved with the math- ematics and solving problems. This book is unabashedly mathematical: The history is primarily a device for feeding the reader some doses of mathematical meat. In the course of reading this book, the neophyte will become involved with mathematics by working on the same prob- lems that Zeno and Pythagoras and Descartes and Fermat and Riemann worked on. This is a book to be read with pencil and paper in hand, and a calculator or computer close by. The student will want to experiment, to try things, to become a part of the mathematical process. This history is also an opportunity to have some fun. Most of the mathematicians treated here were complex individuals who led colorful lives. They are interesting to us as people as well as scientists. There are wonderful stories and anecdotes to relate about Pythagoras and Galois and Cantor and Poincar ́e, and we do not hesitate to indulge ourselves in a little whimsy and gossip. This device helps to bring the subject to life, and will retain reader interest.
Book Detail :-
Title: An Episodic History of Mathematics
Author(s): Steven Krantz
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Book Contents :-
An Episodic History of Mathematics written by Steven 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
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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