Complex Numbers and Geometry

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Author: Liang-shin Hahn

Publisher: Cambridge University Press

ISBN: 9780883855102

Category: Mathematics

Page: 192

View: 4345

The purpose of this book is to demonstrate that complex numbers and geometry can be blended together beautifully. This results in easy proofs and natural generalizations of many theorems in plane geometry, such as the Napoleon theorem, the Ptolemy-Euler theorem, the Simson theorem, and the Morley theorem. The book is self-contained - no background in complex numbers is assumed - and can be covered at a leisurely pace in a one-semester course. Many of the chapters can be read independently. Over 100 exercises are included. The book would be suitable as a text for a geometry course, or for a problem solving seminar, or as enrichment for the student who wants to know more.

Who Gave You the Epsilon?

And Other Tales of Mathematical History

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Author: Marlow Anderson,Victor Katz,Robin Wilson

Publisher: MAA

ISBN: 9780883855690

Category: Mathematics

Page: 431

View: 7613

Follows on from Sherlock Holmes in Babylon to take the history of mathematics through the nineteenth and twentieth centuries.

Non-Euclidean Geometry

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Author: H. S. M. Coxeter

Publisher: Cambridge University Press

ISBN: 9780883855225

Category: Mathematics

Page: 336

View: 3311

A reissue of Professor Coxeter's classic text on non-Euclidean geometry. It surveys real projective geometry, and elliptic geometry. After this the Euclidean and hyperbolic geometries are built up axiomatically as special cases. This is essential reading for anybody with an interest in geometry.

An Episodic History of Mathematics

Mathematical Culture Through Problem Solving

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Author: Steven G. Krantz

Publisher: MAA

ISBN: 0883857669

Category: Mathematics

Page: 381

View: 1535

An Episodic History of Mathematics will acquaint students and readers with mathematical language, thought, and mathematical life by means of historically important mathematical vignettes. It will also serve to help prospective teachers become more familiar with important ideas of in the history of mathematicsboth classical and modern.Contained within are wonderful and engaging stories and anecdotes about Pythagoras and Galois and Cantor and Poincar, which let readers indulge themselves in whimsy, gossip, and learning. The mathematicians treated here were complex individuals who led colorful and fascinating lives, and did fascinating mathematics. They remain interesting to us as people and as scientists.This history of mathematics is also an opportunity to have some fun because the focus in this text is also on the practicalgetting involved with the mathematics and solving problems. This book is unabashedly mathematical. In the course of reading this book, the neophyte will become involved with mathematics by working on the same problems that, for instance, Zeno and Pythagoras and Descartes and Fermat and Riemann worked on.This is a book to be read, therefore, with pencil and paper in hand, and a calculator or computer close by. All will want to experiment; to try things; and become a part of the mathematical process.

Portal Through Mathematics

Journey to Advanced Mathematics

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Author: Oleg Ivanov

Publisher: The Mathematical Association of America

ISBN: 0883856514

Category: Mathematics

Page: N.A

View: 4402

A Portal Through Mathematics is a collection of puzzles and problems mostly on topics relating to secondary mathematics. The problems and topics are fresh and interesting and frequently surprising. One example: the puzzle that asks how much length must be added to a belt around the Earth's equator to raise it one foot has probably achieved old chestnut status. Ivanov, after explaining the surprising answer to this question, goes a step further and asks, if you grabbed that too long belt at some point and raised it as high as possible, how high would that be? The answer to that is more surprising than the classic puzzle's answer. The book is organized into 29 themes, each a topic from algebra, geometry or calculus and each launched from an opening puzzle or problem. There are excursions into number theory, solid geometry, physics and combinatorics. Always there is an emphasis on surprise and delight. And every theme begins at a level approachable with minimal background requirements. With well over 250 puzzles and problems, there is something here sure to appeal to everyone. A Portal Through Mathematics will be useful for prospective secondary teachers of mathematics and may be used (as a supplementary resource) in university courses in algebra, geometry, calculus, and discrete mathematics. It can also be used for professional development for teachers looking for inspiration. However, the intended audience is much broader. Every fan of mathematics will find enjoyment in it.

Euclidean Geometry in Mathematical Olympiads

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Author: Evan Chen

Publisher: The Mathematical Association of America

ISBN: 0883858398

Category: Mathematics

Page: 311

View: 7568

This is a challenging problem-solving book in Euclidean geometry, assuming nothing of the reader other than a good deal of courage. Topics covered included cyclic quadrilaterals, power of a point, homothety, triangle centers; along the way the reader will meet such classical gems as the nine-point circle, the Simson line, the symmedian and the mixtilinear incircle, as well as the theorems of Euler, Ceva, Menelaus, and Pascal. Another part is dedicated to the use of complex numbers and barycentric coordinates, granting the reader both a traditional and computational viewpoint of the material. The final part consists of some more advanced topics, such as inversion in the plane, the cross ratio and projective transformations, and the theory of the complete quadrilateral. The exposition is friendly and relaxed, and accompanied by over 300 beautifully drawn figures. The emphasis of this book is placed squarely on the problems. Each chapter contains carefully chosen worked examples, which explain not only the solutions to the problems but also describe in close detail how one would invent the solution to begin with. The text contains as selection of 300 practice problems of varying difficulty from contests around the world, with extensive hints and selected solutions. This book is especially suitable for students preparing for national or international mathematical olympiads, or for teachers looking for a text for an honor class.

Algebra and Tiling

Homomorphisms in the Service of Geometry

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Author: Sherman Stein,Sandor Szabó,Sandor Szabo,Mathematical Association of America

Publisher: Cambridge University Press

ISBN: 9780883850282

Category: Mathematics

Page: 207

View: 1221

Algebra and Tiling is accessible to undergraduate mathematics majors, as most of the tools necessary to read the book are found in standard upper division algebra courses, but teachers, researchers, and professional mathematicians will find the book equally appealing. Beginners will find the exercises and the appendices especially useful. The unsolved problems will challenge both beginners and experts. The book could serve as the basis of an undergraduate or graduate seminar or a source of applications to enrich an algebra or geometry course.

Lie Groups

A Problem Oriented Introduction Via Matrix Groups

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Author: Harriet Pollatsek

Publisher: MAA

ISBN: 9780883857595

Category: Mathematics

Page: 177

View: 863

The work of the Norwegian mathematician Sophus Lie extends ideas of symmetry and leads to many applications in mathematics and physics. Ordinarily, the study of the "objects" in Lie's theory (Lie groups and Lie algebras) requires extensive mathematical prerequisites beyond the reach of the typical undergraduate. By restricting to the special case of matrix Lie groups and relying on ideas from multivariable calculus and linear algebra, this lovely and important material becomes accessible even to college sophomores. Working with Lie's ideas fosters an appreciation of the unity of mathematics and the sometimes surprising ways in which mathematics provides a language to describe and understand the physical world.Lie Groups is an active learning text that can be used by students with a range of backgrounds and interests. The material is developed through 200 carefully chosen problems. This is the only book in the undergraduate curriculum to bring this material to students so early in their mathematical careers.

Anschauliche Funktionentheorie

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Author: Tristan Needham

Publisher: Oldenbourg Wissenschaftsverlag

ISBN: 9783486709025

Category: Mathematics

Page: 685

View: 2898

Needhams neuartiger Zugang zur Funktionentheorie wurde von der Fachpresse begeistert aufgenommen. Mit über 500 zum großen Teil perspektivischen Grafiken vermittelt er im wahrsten Sinne des Wortes eine Anschauung von der sonst oft als trocken empfundenen Funktionentheorie. 'Anschauliche Funktionentheorie ist eine wahre Freude und ein Buch so recht nach meinem Herzen. Indem er ausschließlich seine neuartige geometrische Perspektive verwendet, enthüllt Tristan Needham viele überraschende und bisher weitgehend unbeachtete Facetten der Schönheit der Funktionentheorie.' (Sir Roger Penrose)

Mathematical Connections

A Companion for Teachers and Others

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Author: Al Cuoco

Publisher: Cambridge University Press

ISBN: 9780883857397

Category: Mathematics

Page: 239

View: 8463

This book contains key topics that form the foundations for high-school mathematics.

Euler

The Master of Us All

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Author: William Dunham

Publisher: MAA

ISBN: 9780883853283

Category: Mathematics

Page: 185

View: 6590

Leonhard Euler was one of the most prolific mathematicians that have ever lived. This book examines the huge scope of mathematical areas explored and developed by Euler, which includes number theory, combinatorics, geometry, complex variables and many more. The information known to Euler over 300 years ago is discussed, and many of his advances are reconstructed. Readers will be left in no doubt about the brilliance and pervasive influence of Euler's work.

Math Through the Ages

A Gentle History for Teachers and Others

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Author: William P. Berlinghoff,Fernando Q. Gouvêa

Publisher: MAA

ISBN: 9780883857366

Category: Mathematics

Page: 273

View: 4441

An informal and accessible overview of the history of mathematics.

Invitation to Complex Analysis

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Author: Ralph P. Boas

Publisher: MAA

ISBN: 0883857642

Category: Mathematics

Page: 327

View: 3873

An ideal choice for a first course in complex analysis, this book can be used either as a classroom text or for independent study. Written in an informal style by a master expositor, the book distills more than half a century of experience with the subject into a lucid, engaging, yet rigorous account. The book reveals both the power of complex analysis as a tool for applications and the intrinsic beauty of the subject as a fundamental part of pure mathematics. Written at the level of courses commonly taught in American universities to seniors and beginning graduate students, the book is suitable for readers acquainted with advanced calculus or introductory real analysis. The treatment goes beyond the standard material of power series, Cauchy's theorem, residues, conformal mapping, and harmonic functions by including accessible discussions of many intriguing topics that are uncommon in a book at this level. Readers will encounter notions ranging from Landau's notation to overconvergent series to the Phragmén-Lindelöf theorem. The flexibility afforded by the supplementary topics and applications makes the book adaptable either to a short, one-term course or to a comprehensive, full-year course.The writing is user-friendly in many ways. Each topic is discussed in a typical, commonly encountered situation rather than in the most general, abstract setting. There are no numbered equations. Numerous exercises interspersed in the text encourage readers to test their understanding of new concepts and techniques as they are presented. Detailed solutions of the exercises, included at the back of the book, both serve as models for students and facilitate independent study. Supplementary exercises at the ends of sections, not solved in the book, provide an additional teaching tool.This second edition of Invitation to Complex Analysis has been painstakingly revised by the author's son, himself an award-winning mathematical expositor.

History of Mathematics

Highways and Byways

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Author: Amy Dahan,Jeanne Peiffer

Publisher: MAA

ISBN: 0883855623

Category: Mathematics

Page: 330

View: 7323

This book is arranged to show the development of the different branches of mathematics over time and contains many illustrations to support the text. In all, a short, innovative and easy-to-read history of mathematics.

Beautiful Mathematics

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Author: Martin Erickson

Publisher: MAA

ISBN: 0883855763

Category: Mathematics

Page: 177

View: 5110

This book is about beautiful mathematical concepts and creations. Mathematical ideas have an aesthetic appeal that can be appreciated by those who have the time and dedication to investigate. Mathematical topics are presented in the categories of words, images, formulas, theorems, proofs, solutions, and unsolved problems. Readers will investigate exciting mathematical topics ranging from complex numbers to arithmetic progressions, from Alcuin's sequence to the zeta function, and from hypercubes to infinity squared.Do you know that a lemniscate curve is the circular inversion of a hyperbola? That Sierpinski's triangle has fractal dimension 1.585….? That a regular septagon can be constructed with straightedge, compass, and an angle trisector? Do you know how to prove Lagrange's theorem that every positive integer is the sum of four squares? Can you find the first three digits of the millionth Fibonacci number? Discover the keys to these and many other mathematical problems. In each case, the mathematics is compelling, elegant, simple, and beautiful.Who should read this book? There is something new for any mathematically-minded person. High school and college students will find motivation for their mathematical studies. Professional mathematicians will find fresh examples of mathematical beauty to pass along to others. Within each chapter, the topics require progressively more prerequisite knowledge. An appendix gives background definitions and theorems, while another gives challenging exercises (with solutions).

The Changing Shape of Geometry

Celebrating a Century of Geometry and Geometry Teaching

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Author: Chris Pritchard,Mathematical Association,Mathematical Association of America

Publisher: Cambridge University Press

ISBN: 9780521531627

Category: Mathematics

Page: 541

View: 6411

Celebrating a century of geometry and geometry teaching, this book will give the reader an enjoyable insight into all things geometrical. There are a wealth of popular articles including sections on Pythagoras, the golden ratio and recreational geometry. Historical items, drawn principally from the Mathematical Gazette, are authored by mathematicians such as G. H. Hardy, Rouse Ball, Thomas Heath and Bertrand Russell as well as some more recent expositors. Thirty 'Desert Island Theorems' from distinguished mathematicians and educationalists give light to some surprising and beautiful results. Contributors include H. S. M. Coxeter, Michael Atiyah, Tom Apostol, Solomon Golomb, Keith Devlin, Nobel Laureate Leon Lederman, Carlo Séquin, Simon Singh, Christopher Zeeman and Pulitzer Prizewinner Douglas Hofstadter. The book also features the wonderful Eyeball Theorems of Peruvian geometer and web designer, Antonio Gutierrez. For anyone with an interest in mathematics and mathematics education this book will be an enjoyable and rewarding read.

Introduction to Complex Analysis

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Author: Hilary A. Priestley

Publisher: Oxford University Press

ISBN: 0198525613

Category: Mathematics

Page: 328

View: 4678

Complex analysis is a classic and central area of mathematics, which is studies and exploited in a range of important fields, from number theory to engineering. Introduction to Complex Analysis was first published in 1985, and for this much-awaited second edition the text has been considerably expanded, while retaining the style of the original. More detailed presentation is given of elementary topics, to reflect the knowledge base of current students. Exercise sets have been substantially revised and enlarged, with carefully graded exercises at the end of each chapter.

Differential Geometry and Its Applications

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Author: John Oprea

Publisher: MAA

ISBN: 9780883857489

Category: Mathematics

Page: 469

View: 7603

Differential geometry has a long, wonderful history it has found relevance in areas ranging from machinery design of the classification of four-manifolds to the creation of theories of nature's fundamental forces to the study of DNA. This book studies the differential geometry of surfaces with the goal of helping students make the transition from the compartmentalized courses in a standard university curriculum to a type of mathematics that is a unified whole, it mixes geometry, calculus, linear algebra, differential equations, complex variables, the calculus of variations, and notions from the sciences. Differential geometry is not just for mathematics majors, it is also for students in engineering and the sciences. Into the mix of these ideas comes the opportunity to visualize concepts through the use of computer algebra systems such as Maple. The book emphasizes that this visualization goes hand-in-hand with the understanding of the mathematics behind the computer construction. Students will not only “see” geodesics on surfaces, but they will also see the effect that an abstract result such as the Clairaut relation can have on geodesics. Furthermore, the book shows how the equations of motion of particles constrained to surfaces are actually types of geodesics. Students will also see how particles move under constraints. The book is rich in results and exercises that form a continuous spectrum, from those that depend on calculation to proofs that are quite abstract.

First Steps for Math Olympians

Using the American Mathematics Competitions

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Author: J. Douglas Faires

Publisher: MAA

ISBN: 9780883858240

Category: Mathematics

Page: 307

View: 4337

Provide students with the tools to solve problems that are found on mathematical problem-solving exams.

New Mexico Mathematics Contest Problem Book

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Author: Liong-shin Hahn

Publisher: UNM Press

ISBN: 9780826335340

Category: Mathematics

Page: 202

View: 2226

The New Mexico Mathematics Contest for high-school students has been held annually since 1966. Each November, thousands of middle- and high-school students from all over New Mexico converge to battle with elementary but tricky math problems. The 200 highest-scoring students meet for the second round the following February at the University of New Mexico in Albuquerque where they listen to a prominent mathematician give a keynote lecture, have lunch, and then get down to round two, an even more challenging set of mathematical mind-twisters. Liong-shin Hahn was charged with the task of creating a new set of problems each year for the New Mexico Mathematics Contest, 1990-1999. In this volume, Hahn has collected the 138 best problems to appear in these contests over the last decades. They range from the simple to the highly challenging--none are trivial. The solutions contain many clever analyses and often display uncommon ingenuity. His questions are always interesting and relevant to teenage contestants. Young people training for competitions will not only learn a great deal of useful mathematics from this book but, and this is much more important, they will take a step toward learning to love mathematics.