.

0 (hodnocen0 x )

BK

Concise edition

Cham : Springer, [2020]

xv, 399 stran : ilustrace (některé barevné) ; 25 cm

ISBN 978-3-030-55192-6 (vázáno)

Undergraduate texts in mathematics, ISSN 0172-6056

Obsahuje bibliografii na stranách 349-376 a rejstřík

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..

001640191

Preface vii // 1 The Theorem of Pythagoras 1 // 1.1 Arithmetic and Geometry 2 // 1.2 Pythagorean Tripies 4 // 1.3 Rational Points on the Circle 6 // 1.4 Right-Angled Triangles 10 // 1.5 Irrational Numbers 12 // 2 Greek Geometry 17 // 2.1 The Deductive Method 18 // 2.2 The Regular Polyhedra 20 // 2.3 Ruler and Compass Constructions 23 // 2.4 Conic Sections 26 // 2.5 Higher-Degree Curves 29 // 3 Greek Number Theory 35 // 3.1 The Role of Number Theory 36 // 3.2 Polygonal, Prime, and Perfect Numbers 36 // 3.3 The Euclidean Algorithm 39 // 3.4 Pell’s Equation 43 // 3.5 The Chord and Tangent Methods 47 // 4 Infinity in Greek Mathematics 51 // 4.1 Fear of Infinity 52 // 4.2 Eudoxus’s Theory of Proportions 54 // 4.3 The Method of Exhaustion 56 // 4.4 The Area of a Parabolic Segment 60 // 5 Polynomial Equations 63 // 5.1 Algebra 64 // 5.2 Linear Equations and Elimination 65 // 5.3 Quadratic Equations 68 // 5.4 Quadratic Irrationals 71 // 5.5 The Solution of the Cubic 73 // 5.6 Angle Division 75 // 5.7 Higher-Degree Equations 77 // 5.8 The Binomial Theorem 79 // 5.9 Fermat’s Little Theorem 82 // 6 Algebraic Geometry 85 // 6.1 Steps Toward Algebraic Geometry 86 // 6.2 Fermat and Descartes 87 // 6.3 Algebraic Curves 89 // 6.4 Newton’s Classification of Cubics 91 // 6.5 Construction of Equations, Bézouťs Theorem 94 // 6.6 The Arithmetization of Geometry 96 // 7 Projective Geometry 99 // 7.1 Perspective 100 // 7.2 Anamorphosis 103 // 7.3 Desargues’s Projective Geometry 105 // 7.4 The Projective View of Curves 108 // 7.5 The Projective Plane 112 // 7.6 The Projective Line 115 // 7.7 Homogeneous Coordinates 118 // 8 Calculus 123 // 8.1 What Is Calculus? 124 // 8.2 Early Results on Areas and Volumes 125 // 8.3 Maxima, Minima, and Tangents 128