Author(s): | Krantz, Steven G. (Steven George), 1951- |
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Location: |
Retrieving Holdings Information |

Subjects: | Mathematics--History--Study and teaching (Higher) Mathematics--Problems, exercises, etc Mathematics--Study and teaching (Higher) Mathematicians |

Formats: | |

Material Type: | Books |

Language: | English |

Audience: | Unspecified |

Published: | [Washington, DC] : Mathematical Association of America, c2010 |

Series: | MAA textbooks MAA textbooks |

LC Classification: | Q, QA |

Physical Description: | xiii, 381 p. : ill. ; 27 cm |

Table of Contents: | Preface 1. The Ancient Greeks and the Foundations of Mathematics 1 1.1. Pythagoras 1 1.2. Euclid 6 1.3. Archimedes 13 Exercises 22 2. Zeno's Paradox and the Concept of Limit 25 2.1. The Context of the Paradox 25 2.2. The Life of Zeno of Elea 26 2.3. Consideration of the Paradoxes 30 2.4. Decimal Notation and Limits 33 2.5. Infinite Sums and Limits 34 2.6. Finite Geometric Series 36 2.7. Some Useful Notation 38 2.8. Concluding Remarks 39 Exercises 39 3. The Mystical Mathematics of Hypatia 43 3.1. Introduction to Hypatia 43 3.2. What is a Conic Section? 47 Exercises 50 4. The Islamic World and the Development of Algebra 55 4.1. Introductory Remarks 55 4.2. The Development of Algebra 55 4.3. The Geometry of the Arabs 64 4.4. A Little Arab Number Theory 67 Exercises 70 5. Cardano, Abel, Galois, and the Solving of Equations 73 5.1. Introduction 73 5.2. The Story of Cardano 74 5.3. First-Order Equations 77 5.4. Rudiments of Second-Order Equations 78 5.5. Completing the Square 79 5.6. The Solution of a Quadratic Equation 80 5.7. The Cubic Equation 83 5.8. Fourth-Degree Equations and Beyond 86 5.9. The Work of Abel and Galois in Context 91 Exercises 92 6. Rene Descartes and the Idea of Coordinates 95 6.0. Introductory Remarks 95 6.1. The Life of Rene Descartes 96 6.2. The Real Number Line 98 6.3. The Cartesian Plane 100 6.4. The Use of Cartesian Coordinates to Study Euclidean Geometry 102 6.5. Coordinates in Three-Dimensional Space 104 Exercies 107 7. Pierre de Fermat and the Invention of Differential Calculus 109 7.1. The Life of Fermat 109 7.2. Fermat's Method 111 7.3. More Advanced Ideas of Calculus: The Derivative and the Tangent Line 113 7.4. Fermat's Lemma and Maximum/Minimum Problems 117 Exercises 123 8. The Great Isaac Newton 125 8.1. Introduction to Newton 125 8.2. The Idea of the Integral 130 8.3. Calculation of the Integral 132 8.4. The Fundamental Theorem of Calculus 135 8.5. Some Preliminary Calculations 137 8.6. Some Examples 140 Exercises 149 9. The Complex Numbers and the Fundamental Theorem of Algebra 151 9.1. A New Number System 151 9.2. Progenitors of the Complex Number System 151 9.3. Complex Number Basics 156 9.4. The Fundamental Theorem of Algebra 161 9.5. Finding the Roots of a Polynomial 165 Exercises 166 10. Carl Friedrich Gauss: The Prince of Mathematics 169 10.1. Gauss the Man 169 10.2. The Binomial Theorem 173 10.3. The Chinese Remainder Theorem 184 10.4. A Constructive Means for Finding the Solution x 186 10.5. Quadratic Reciprocity and the Gaussian Integers 186 10.6. The Gaussian Integers 189 Exercises 192 11. Sophie Germain and the Attack on Fermat's Last Problem 195 11.1. Birth of an Inspired and Unlikely Child 195 11.2. Sophie German's Work on Fermat's Problem 200 Exercises 204 12. Cauchy and the Foundations of Analysis 207 12.1. Introduction 207 12.2. Why Do We Need the Real Numbers? 210 12.3. How to Construct the Real Numbers 211 12.4. Properties of the Real Number System 215 Exercises 221 13. The Prime Numbers 223 13.1. The Sieve of Eratosthenes 223 13.2. The Infinitude of the Primes 225 13.3. More Prime Thoughts 226 13.4. The Concept of Relatively Prime 231 Exercises 233 14. Dirichlet and How to Count 237 14.1. The Life of Dirichlet 237 14.2. The Pigeonhole Principle 239 14.3. Ramsey Theory 242 Exercises 244 15. Bernhard Riemann and the Geometry of Surfaces 247 15.0. Introduction 247 15.1. How to Measure the Length of a Curve 249 15.2. Riemann's Method for Measuring Arc Length 251 15.3. The Hyperbolic Disc 253 15.4. The Use of the Integral 256 Exercises 258 16. Georg Cantor and the Orders of Infinity 261 16.1. Introductory Remarks 261 16.2. What is a Number? 264 16.3. The Existence of Transcendental Numbers 271 Exercises 273 17. The Number Systems 275 17.1. The Natural Numbers 276 17.2. The Integers 278 17.3. The Rational Numbers 280 17.4. The Real Numbers 281 17.5. The Complex Numbers 284 Exercises 285 18. Henri Poincare, Child Phenomenon 289 18.1. Introductory Remarks 289 18.2. Rubber Sheet Geometry 292 18.3. The Idea of Homotopy 293 18.4. The Brouwer Fixed Point Theorem 294 18.5. The Generalized Ham Sandwich Theorem 299 Exercises 302 19. Sonya Kovalevskaya and the Mathematics of Mechanics 305 19.1. The Life of Sonya Kovalevskaya 305 19.2. The Scientific Work of Sonya Kovalevskaya 309 19.3. Afterward on Sonya Kovalevskaya 314 Exercises 315 20. Emmy Nother and Algebra 319 20.1. The Life of Emmy Noether 319 20.2. Emmy Noether and Abstract Algebra: Groups 322 20.3. Emmy Noether and Abstract Algebra: Rings 325 Exercises 328 21. Methods of Proof 331 21.1. Axiomatics 333 21.2. Proof by Induction 334 21.3. Proof by Contradiction 337 21.4. Direct Proof 339 21.5. Other Methods of Proof 341 Exercises 343 22. Alan Turing and Cryptography 345 22.0. Background on Alan Turing 345 22.1. The Turing Machine 346 22.2. More on the Life of Alan Turing 347 22.3. What is Cryptography? 349 22.4. Encryption by Way of Affine Transformations 353 22.5. Digraph Transformations 358 Exercises 362 Bibliography 365 Index 371 About the Author 381 |

Alternate Titles: | Online version: Krantz, Steven G. (Steven George), 1951- Episodic history of mathematics. (OCoLC)777921241 [Washington, DC] : Mathematical Association of America, c2010 |

Additional Authors: | Mathematical Association of America |

Notes: | ISBN: 0883857669 ISBN: 9780883857663 Includes bibliographical references (p. 365-369) and index Contents: The ancient Greeks and the foundations of mathematics -- Zeno's paradox and the concept of limit -- The mystical mathematics of Hypatia -- The Islamic world and the development of algebra -- Cardano, Abel, Galois, and the solving of equations -- René Descartes and the idea of coordinates -- Pierre de Fermat and the invention of differential calculus -- The great Isaac Newton -- The complex numbers and the fundamental theorem of algebra -- Carl Friedrich Gauss: the prince of mathematics -- Sophie Germain and the attack on Fermat's last problem -- Cauchy and the foundations of analysis -- The prime numbers -- Dirichlet and how to count -- Bernhard Riemann and the geometry of surfaces -- Georg Cantor and the orders of infinity -- The number systems -- Henri Poincaré, child phenomenon -- Sonya Kovalevskaya and the mathematics of mechanics -- Emmy Noether and algebra -- Methods of proof -- Alan Turing and cryptography Summary: "An Episodic History of Mathematics delivers a series of snapshots of mathematics and mathematicians from ancient times to the twentieth century. Giving readers a sense of mathematical culture and history, the book also acquaints readers with the nature and techniques of mathematics via exercises. It introduces the genesis of key mathematical concepts. For example, while Krantz does not get into the intricate mathematical details of Andrew Wiles's proof of Fermat's Last Theorem, he does describe some of the streams of thought that posed the problem and led to its solution. The focus in this text, moreover, is on doing - getting involved with the mathematics and solving problems. Every chapter ends with a detailed problem set that will provide students with avenues for exploration and entry into the subject. It recounts the history of mathematics; offers broad coverage of the various schools of mathematical thought to give readers a wider understanding of mathematics; and includes exercises to help readers engage with the text and gain a deeper understanding of the material."--Publisher's description |

OCLC Number: | 501976977 |

ISBN/ISSN: | 0883857669 9780883857663 |