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El. knyga: Mathematics and Its History: A Concise Edition

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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 tothe 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

Recenzijos

This is a beautiful book. it would be fun to teach a course from it and I hope to get that chance. This book flows so well that I did not feel anything was lacking from it. I am confident that the many readers of Stillwells work will find it satisfying and worthwhile to update their libraries with this edition. Those currently unfamiliar with his work will find this a delightful place to begin. (Michele Intermont, MAA Reviews, October 18, 2021)

Preface vii
1 The Theorem of Pythagoras
1(16)
1.1 Arithmetic and Geometry
2(2)
1.2 Pythagorean Triples
4(2)
1.3 Rational Points on the Circle
6(4)
1.4 Right-Angled Triangles
10(2)
1.5 Irrational Numbers
12(5)
2 Greek Geometry
17(18)
2.1 The Deductive Method
18(2)
2.2 The Regular Polyhedra
20(3)
2.3 Ruler and Compass Constructions
23(3)
2.4 Conic Sections
26(3)
2.5 Higher-Degree Curves
29(6)
3 Greek Number Theory
35(16)
3.1 The Role of Number Theory
36(1)
3.2 Polygonal, Prime, and Perfect Numbers
36(3)
3.3 The Euclidean Algorithm
39(4)
3.4 Pell's Equation
43(4)
3.5 The Chord and Tangent Methods
47(4)
4 Infinity in Greek Mathematics
51(12)
4.1 Fear of Infinity
52(2)
4.2 Eudoxus's Theory of Proportions
54(2)
4.3 The Method of Exhaustion
56(4)
4.4 The Area of a Parabolic Segment
60(3)
5 Polynomial Equations
63(22)
5.1 Algebra
64(1)
5.2 Linear Equations and Elimination
65(3)
5.3 Quadratic Equations
68(3)
5.4 Quadratic Irrationals
71(2)
5.5 The Solution of the Cubic
73(2)
5.6 Angle Division
75(2)
5.7 Higher-Degree Equations
77(2)
5.8 The Binomial Theorem
79(3)
5.9 Fermat's Little Theorem
82(3)
6 Algebraic Geometry
85(14)
6.1 Steps Toward Algebraic Geometry
86(1)
6.2 Fermat and Descartes
87(2)
6.3 Algebraic Curves
89(2)
6.4 Newton's Classification of Cubics
91(3)
6.5 Construction of Equations, Bezout's Theorem
94(2)
6.6 The Arithmetization of Geometry
96(3)
7 Projective Geometry
99(24)
7.1 Perspective
100(3)
7.2 Anamorphosis
103(2)
7.3 Desargues's Projective Geometry
105(3)
7.4 The Projective View of Curves
108(4)
7.5 The Projective Plane
112(3)
7.6 The Projective Line
115(3)
7.7 Homogeneous Coordinates
118(5)
8 Calculus
123(16)
8.1 What Is Calculus?
124(1)
8.2 Early Results on Areas and Volumes
125(3)
8.3 Maxima, Minima, and Tangents
128(2)
8.4 The Arithmetica Infinitorum of Wallis
130(3)
8.5 Newton's Calculus of Series
133(3)
8.6 The Calculus of Leibniz
136(3)
9 Infinite Series
139(18)
9.1 Early Results
140(3)
9.2 From Pythagoras to Pi
143(3)
9.3 Power Series
146(3)
9.4 Fractional Power Series
149(2)
9.5 Summation of Series
151(2)
9.6 The Zeta Function
153(4)
10 Elliptic Curves and Functions
157(24)
10.1 Fermat's Last Theorem
158(4)
10.2 Rational Points on Cubics of Genus 0
162(3)
10.3 Rational Points on Cubics of Genus 1
165(3)
10.4 Elliptic and Circular Functions
168(2)
10.5 Elliptic Integrals
170(3)
10.6 Doubling the Arc of the Lemniscate
173(2)
10.7 General Addition Theorems
175(2)
10.8 Elliptic Functions
177(4)
11 Complex Numbers and Curves
181(24)
11.1 Impossible Numbers
182(1)
11.2 Cubic Equations
183(2)
11.3 Angle Division
185(4)
11.4 The Fundamental Theorem of Algebra
189(4)
11.5 Roots and Intersections
193(3)
11.6 The Complex Projective Line
196(4)
11.7 Branch Points
200(1)
11.8 Topology of Complex Projective Curves
201(4)
12 Complex Numbers and Functions
205(20)
12.1 Complex Functions
206(4)
12.2 Conformal Mapping
210(2)
12.3 Cauchy's Theorem
212(3)
12.4 Double Periodicity of Elliptic Functions
215(3)
12.5 Elliptic Curves
218(4)
12.6 Uniformization
222(3)
13 Non-Euclidean Geometries
225(32)
13.1 Transcendental Curves
226(3)
13.2 Curvature of Plane Curves
229(3)
13.3 Curvature of Surfaces
232(3)
13.4 Geodesies
235(2)
13.5 The Parallel Axiom
237(3)
13.6 Spherical and Hyperbolic Geometry
240(3)
13.7 Geometry of Bolyai and Lobachevsky
243(5)
13.8 Beltrami's Conformal Models
248(4)
13.9 The Complex Interpretations
252(5)
14 Group Theory
257(26)
14.1 The Group Concept
258(3)
14.2 Subgroups and Quotients
261(2)
14.3 Permutations and Theory of Equations
263(4)
14.4 Permutation Groups
267(2)
14.5 Polyhedral Groups
269(3)
14.6 Groups and Geometries
272(3)
14.7 Combinatorial Group Theory
275(4)
14.8 Finite Simple Groups
279(4)
15 Topology
283(14)
15.1 Geometry and Topology
284(1)
15.2 Polyhedron Formulas of Descartes and Euler
285(2)
15.3 The Classification of Surfaces
287(3)
15.4 Surfaces and Planes
290(4)
15.5 The Fundamental Group
294(3)
16 Commutative Algebra
297(26)
16.1 Linear Algebra
298(1)
16.2 Vector Spaces
299(3)
16.3 Fields
302(3)
16.4 Algebraic Numbers and Algebraic Integers
305(3)
16.5 Rings
308(3)
16.6 Fields as Vector Spaces
311(2)
16.7 Fields of Algebraic Numbers
313(3)
16.8 Ideals
316(2)
16.9 Ideal Prime Factorization
318(5)
17 Sets, Logic, and Computation
323(24)
17.1 Sets
324(2)
17.2 Ordinals
326(3)
17.3 Measure
329(2)
17.4 Axiom of Choice and Large Cardinals
331(3)
17.5 The Diagonal Argument
334(1)
17.6 Computability
335(4)
17.7 Logic and Godel's Theorem
339(4)
17.8 Provability and Truth
343(4)
Image Credits 347(2)
Bibliography 349(28)
Index 377
John Stillwell is Professor Emeritus at the University of San Francisco and formerly Associate Professor at Monash University in Melbourne, Australia. He is an accomplished author, whose Springer books include: The Four Pillars of Geometry, Elements of Algebra, Naive Lie Theory, and many more.
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