Mathematics and Its History: A Concise Edition

Description

This textbook provides a unified and concise exploration of undergraduate mathematics by approaching the subject through its history. Readers will discover the rich tapestry of ideas behind familiar topics from the undergraduate curriculum, such as calculus, algebra, topology, and more. Featuring historical episodes ranging from the Ancient Greeks to Fermat and Descartes, this volume offers a glimpse into the broader context in which these ideas developed, revealing unexpected connections that make this ideal for a senior capstone course. The presentation of previous versions has been refined by omitting the less mainstream topics and inserting new connecting material, allowing instructors to cover the book in a one-semester course. This condensed edition prioritizes succinctness and cohesiveness, and there is a greater emphasis on visual clarity, featuring full color images and high quality 3D models. As in previous editions, a wide array of mathematical topics are covered, from geometry to computation; however, biographical sketches have been omitted. Mathematics and Its History: A Concise Edition is an essential resource for courses or reading programs on the history of mathematics. Knowledge of basic calculus, algebra, geometry, topology, and set theory is assumed. From reviews of previous editions: “Mathematics and Its History is a joy to read. The writing is clear, concise and inviting. The style is very different from a traditional text. I found myself picking it up to read at the expense of my usual late evening thriller or detective novel…. The author has done a wonderful job of tying together the dominant themes of undergraduate mathematics.” Richard J. Wilders, MAA, on the Third Edition "The book...is presented in a lively style without unnecessary detail. It is very stimulating and will be appreciated not only by students. Much attention is paid to problems and to the development of mathematics before the end of the nineteenth century.... This book brings to the non-specialist interested in mathematics many interesting results. It can be recommended for seminars and will be enjoyed by the broad mathematical community." European Mathematical Society, on the Second Edition Preface Contents 1 The Theorem of Pythagoras 1.1 Arithmetic and Geometry 1.2 Pythagorean Triples 1.3 Rational Points on the Circle 1.4 Right-Angled Triangles 1.5 Irrational Numbers 2 Greek Geometry 2.1 The Deductive Method 2.2 The Regular Polyhedra 2.3 Ruler and Compass Constructions 2.4 Conic Sections 2.5 Higher-Degree Curves 3 Greek Number Theory 3.1 The Role of Number Theory 3.2 Polygonal, Prime, and Perfect Numbers 3.3 The Euclidean Algorithm 3.4 Pell's Equation 3.5 The Chord and Tangent Methods 4 Infinity in Greek Mathematics 4.1 Fear of Infinity 4.2 Eudoxus's Theory of Proportions 4.3 The Method of Exhaustion 4.4 The Area of a Parabolic Segment 5 Polynomial Equations 5.1 Algebra 5.2 Linear Equations and Elimination 5.3 Quadratic Equations 5.4 Quadratic Irrationals 5.5 The Solution of the Cubic 5.6 Angle Division 5.7 Higher-Degree Equations 5.8 The Binomial Theorem 5.9 Fermat's Little Theorem 6 Algebraic Geometry 6.1 Steps Toward Algebraic Geometry 6.2 Fermat and Descartes 6.3 Algebraic Curves 6.4 Newton's Classification of Cubics 6.5 Construction of Equations, Bézout's Theorem 6.6 The Arithmetization of Geometry 7 Projective Geometry 7.1 Perspective 7.2 Anamorphosis 7.3 Desargues's Projective Geometry 7.4 The Projective View of Curves 7.5 The Projective Plane 7.6 The Projective Line 7.7 Homogeneous Coordinates 8 Calculus 8.1 What Is Calculus? 8.2 Early Results on Areas and Volumes 8.3 Maxima, Minima, and Tangents 8.4 The Arithmetica Infinitorum of Wallis 8.5 Newton's Calculus of Series 8.6 The Calculus of Leibniz 9 Infinite Series 9.1 Early Results 9.2 From Pythagoras to Pi 9.3 Power Series 9.4 Fractional Power Series 9.5 Summation of Series 9.6 The Zeta Function 10 Elliptic Curves and Functions 10.1 Fermat's Last Theorem 10.2 Rational Points on Cubics of Genus 0 10.3 Rational Points on Cubics of Genus 1 10.4 Elliptic and Circular Functions 10.5 Elliptic Integrals 10.6 Doubling the Arc of the Lemniscate 10.7 General Addition Theorems 10.8 Elliptic Functions 11 Complex Numbers and Curves 11.1 Impossible Numbers 11.2 Cubic Equations 11.3 Angle Division 11.4 The Fundamental Theorem of Algebra 11.5 Roots and Intersections 11.6 The Complex Projective Line 11.7 Branch Points 11.8 Topology of Complex Projective Curves 12 Complex Numbers and Functions 12.1 Complex Functions 12.2 Conformal Mapping 12.3 Cauchy's Theorem 12.4 Double Periodicity of Elliptic Functions 12.5 Elliptic Curves 12.6 Uniformization 13 Non-Euclidean Geometries 13.1 Transcendental Curves 13.2 Curvature of Plane Curves 13.3 Curvature of Surfaces 13.4 Geodesics 13.5 The Parallel Axiom 13.6 Spherical and Hyperbolic Geometry 13.7 Geometry of Bolyai and Lobachevsky 13.8 Beltrami's Conformal Models 13.9 The Complex Interpretations 14 Group Theory 14.1 The Group Concept 14.2 Subgroups and Quotients 14.3 Permutations and Theory of Equations 14.4 Permutation Groups 14.5 Polyhedral Groups 14.6 Groups and Geometries 14.7 Combinatorial Group Theory 14.8 Finite Simple Groups 15 Topology 15.1 Geometry and Topology 15.2 Polyhedron Formulas of Descartes and Euler 15.3 The Classification of Surfaces 15.4 Surfaces and Planes 15.5 The Fundamental Group 16 Commutative Algebra 16.1 Linear Algebra 16.2 Vector Spaces 16.3 Fields 16.4 Algebraic Numbers and Algebraic Integers 16.5 Rings 16.6 Fields as Vector Spaces 16.7 Fields of Algebraic Numbers 16.8 Ideals 16.9 Ideal Prime Factorization 17 Sets, Logic, and Computation 17.1 Sets 17.2 Ordinals 17.3 Measure 17.4 Axiom of Choice and Large Cardinals 17.5 The Diagonal Argument 17.6 Computability 17.7 Logic and Gödel's Theorem 17.8 Provability and Truth Image Credits Bibliography Index