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Excursions in the History of Mathematics / by Israel Kleiner
Resource type: Ressourcentyp: Buch (Online)Book (Online)Language: English Series: SpringerLink BücherPublisher: Boston : Springer Science+Business Media, LLC, 2012Description: Online-Ressource (XXI, 347p. 36 illus, digital)ISBN:- 9780817682682
- 510.9
- QA21-27
Contents:
Summary: A. Number Theory -- 1. Highlights in the History of Number Theory: 1700 BC - 2008 -- 2. Fermat: The Founder of Modern Number Theory -- 3. Fermat's Last Theorem: From Fermat to Wiles -- B. Calculus/Analysis -- 4. A History of the Infinitely Small and the Infinitely Large in Calculus, with Remarks for the Teacher -- 5. A Brief History of the Function Concept -- 6. More on the History of Functions, Including Remarks on Teaching -- C. Proof -- 7. Highlights in the Practice of Proof: 1600 BC - 2009 -- 8. Paradoxes: What are they Good for? -- 9. Principle of Continuity: 16th - 19th centuries -- 10. Proof: A Many-Splendored Thing -- D. Courses Inspired by History -- 11. Numbers as a Source of Mathematical Ideas -- 12. History of Complex Numbers, with a Moral for Teachers -- 13. A History-of-Mathematics Course for Teachers, Based on Great Quotations -- 14. Famous Problems in Mathematics -- E. Brief Biographies of Selected Mathematicians -- 15. The Biographies -- Index.Summary: This book comprises five parts. The first three contain ten historical essays on important topics: number theory, calculus/analysis, and proof, respectively. Part four deals with several historically oriented courses, and Part five provides biographies of five mathematicians who played major roles in the historical events described in the first four parts of the work. Each of the first three parts—on number theory, calculus/analysis, and proof—begins with a survey of the respective subject and is followed in more depth by specialized themes. Among the specialized themes are: Fermat as the founder of modern number theory, Fermat’s Last Theorem from Fermat to Wiles, the history of the function concept, paradoxes, the principle of continuity, and an historical perspective on recent debates about proof. The fourth part contains essays describing mathematics courses inspired by history. The essays deal with numbers as a source of ideas in teaching, with famous problems, and with the stories behind various "great" quotations. The last part gives an account of five mathematicians—Dedekind, Euler, Gauss, Hilbert, and Weierstrass—whose lives and work we hope readers will find inspiring. Key features of the work include: * A preface describing in some detail the author's ideas on teaching mathematics courses, in particular, the role of history in such courses; * Explicit comments and suggestions for teachers on how history can affect the teaching of mathematics; * A description of a course in the history of mathematics taught in an In-Service Master's Program for high school teachers; * Inclusion of issues in the philosophy of mathematics; * An extensive list of relevant references at the end of each chapter. Excursions in the History of Mathematics was written with several goals in mind: to arouse mathematics teachers’ interest in the history of their subject; to encourage mathematics teachers with at least some knowledge of the history of mathematics to offer courses with a strong historical component; and to provide an historical perspective on a number of basic topics taught in mathematics courses.PPN: PPN: 1651095361Package identifier: Produktsigel: ZDB-2-SEB | ZDB-2-SXMS | ZDB-2-SMA
""Excursions in the History of Mathematics""; ""Preface""; ""Permissions""; ""Contents""; ""Part A Number Theory""; ""Chapter 1 Highlights in the History of Number Theory: 1700 BC�2008 ""; ""1.1 Early Roots to Fermat""; ""1.2 Fermat""; ""1.2.1 Fermat's Little Theorem""; ""1.2.2 Sums of Two Squares""; ""1.2.3 Fermat's Last Theorem""; ""1.2.4 Bachet's Equation""; ""1.2.5 Pell's Equation""; ""1.2.6 Fermat Numbers""; ""1.3 Euler""; ""1.3.1 Analytic Number Theory""; ""1.3.2 Diophantine Equations""; ""1.3.3 Partitions""; ""1.3.4 The Quadratic Reciprocity Law""; ""1.4 Lagrange""
""1.4.1 Pell's Equation""""1.4.2 Sums of Four Squares""; ""1.4.3 Binary Quadratic Forms""; ""1.5 Legendre""; ""1.6 Gauss' Disquisitiones Arithmeticae""; ""1.6.1 Introduction""; ""1.6.2 Quadratic Reciprocity""; ""1.6.3 Binary Quadratic Forms""; ""1.6.4 Cyclotomy""; ""1.7 Algebraic Number Theory""; ""1.7.1 Reciprocity Laws""; ""1.7.2 Fermat's Last Theorem""; ""1.7.3 Dedekind's Ideals""; ""1.7.4 Summary""; ""1.8 Analytic Number Theory""; ""1.8.1 The Distribution of Primes Among the Integers: Introduction""; ""1.8.2 The Prime Number Theorem""; ""1.8.3 The Riemann Zeta Function""
""1.8.4 Primes in Arithmetic Progression""""1.8.5 More on the Distribution of Primes""; ""1.9 Fermat's Last Theorem""; ""1.9.1 Work Prior to That of Wiles""; ""1.9.2 Andrew Wiles""; ""References""; ""Chapter 2 Fermat: The Founder of Modern Number Theory ""; ""2.1 Introduction""; ""2.2 Fermat's Intellectual Debts""; ""2.3 Fermat's Little Theorem and Factorization""; ""2.3.1 A Look Ahead""; ""2.4 Sums of Squares""; ""2.4.1 A Look Ahead""; ""2.5 Fermat's Last Theorem""; ""2.5.1 A Look Ahead""; ""2.6 The Bachet and Pell Equations""; ""2.6.1 Bachet's Equation""; ""2.6.2 A Look Ahead""
""2.6.3 Pell's Equation""""2.6.4 A Look Ahead""; ""2.7 Conclusion""; ""References""; ""Chapter 3 Fermat's Last Theorem: From Fermat to Wiles""; ""3.1 Introduction""; ""3.2 The First Two Centuries""; ""3.3 Sophie Germain""; ""3.4 Lamé""; ""3.4.1 Pythagorean Triples""; ""3.4.2 Lamé's Proof""; ""3.5 Kummer""; ""3.6 Early Decades of the Twentieth Century""; ""3.7 Several Results Related to FLT, 1973�1993""; ""3.8 Some Major Ideas Leading to Wiles' Proof of FLT""; ""3.8.1 Elliptic Curves""; ""3.8.2 Number Theory and Geometry""; ""3.8.3 The Shimura-Taniyama Conjecture""; ""3.9 Andrew Wiles""
""3.10 Tributes to Wiles""""3.11 Is There Life After FLT?""; ""References""; ""Part B Calculus/Analysis""; ""Chapter 4 History of the Infinitely Small and the Infinitely Large in Calculus, with Remarks for the Teacher""; ""4.1 Introduction""; ""4.2 Seventeenth-Century Predecessors of Newton and Leibniz""; ""4.2.1 Introduction""; ""4.2.2 Cavalieri""; ""4.2.3 Fermat""; ""4.3 Newton and Leibniz: The Inventors of Calculus""; ""4.3.1 Introduction""; ""4.3.2 Didactic Observation""; ""4.3.3 Newton""; ""4.3.4 Leibniz""; ""4.3.5 Didactic Observation""; ""4.4 The Eighteenth Century: Euler""
""4.4.1 Introduction""
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