An episodic history of mathematics : mathematical culture through problem solving / Steven G. Krantz

Author(s): Krantz, Steven G. (Steven George), 1951-
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Subjects: Mathematics--History--Study and teaching (Higher)
Mathematics--Problems, exercises, etc
Mathematics--Study and teaching (Higher)
Mathematicians
Formats: Print
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