Product Description

It is impossible to imagine modern mathematics without complex numbers. Complex Numbers from A to . . . Z introduces the reader to this fascinating subject that, from the time of L. Euler, has become one of the most utilized ideas in mathematics.

The exposition concentrates on key concepts and then elementary results concerning these numbers. The reader learns how complex numbers can be used to solve algebraic equations and to understand the geometric interpretation of complex numbers and the operations involving them.

The theoretical parts of the book are augmented with rich exercises and problems at various levels of difficulty. A special feature of the book is the last chapter, a selection of outstanding Olympiad and other important mathematical contest problems solved by employing the methods already presented.

The book reflects the unique experience of the authors. It distills a vast mathematical literature, most of which is unknown to the western public, and captures the essence of an abundant problem culture. The target audience includes undergraduates, high school students and their teachers, mathematical contestants (such as those training for Olympiads or the W. L. Putnam Mathematical Competition) and their coaches, as well as anyone interested in essential mathematics.

Product Details

• Amazon Sales Rank: #115429 in Books
• Published on: 2005-10-03
• Number of items: 1
• Binding: Paperback
• 321 pages

Review

From the reviews:

"The book is a real treasure trove of nontrivial elementary key concepts and applications of complex numbers developed in a systematic manner with a focus on problem-solving techniques. Much of the book goes to geometric applications, of course, but there are also sections on polynomial equations, trigonometry, combinatorics.... Problems constitute an integral part of the book alongside theorems, lemmas and examples. The problems are embedded in the text throughout the book, partly as illustrations to the discussed concepts, partly as the testing grounds for the techniques just studied, but mostly I believe to emphasize the centrality of problem solving in the authors' world view.... The book is really about solving problems and developing tools that exploit properties of complex numbers.... The reader will find a good deal of elegant and simple sample problems and even a greater quantity of technically taxing ones. The book supplies many great tools to help solve those problems. As the techniques go, the book is truly From 'A to Z'." —MAA

“It is for the readers who seek to harness new techniques and to polish their mastery of the old ones. It is for somebody who made it their business to be solving problems on a regular basis. These readers will appreciate the scope of the methodological detail the authors of the book bring to their attention, they will appreciate the power of the methods and theintricacy of the problems.”(MAA REVIEWS)

"This book is devoted to key concepts and elementary results concerning complex numbers. … It contains numerous exercises with hints and solutions. … The book will serve as a useful source for exercises for an introductory course on complex analysis." (F. Haslinger, Monatshefte für Mathematik, Vol. 149 (3), 2006)

"Both of the authors are well-known for their capacity of an integral point of view about mathematics: from the level of the school, through the university level, to the scientific results. The theory appears strictly connected with the problems, the hardest world contest included. Both of them have a very rich experience in preparing Olympic teams in Romania and in the United States.

"… A significant list of references and two indexes complete the book. I strongly recommend the book for pupils, students and teachers." —Dan Brânzei, Analele Stiintifice

"The main purpose of this book is to stimulate young people to become interested in mathematics … . This book is a very well written introduction to the theory of complex numbers and it contains a fine collection of excellent exercises … . the targeted audience is not standard and it ‘includes high school students and their teachers, undergraduates, mathematics contestants such as those training for Olympiads or the William Lowell Putnam Mathematical Competition, their coaches, and any person interested in essential mathematics." (Vicentiu D. Radulescu, Zentralblatt MATH, Vol. 1127 (4), 2008)

Customer Reviews

Amazing book on complex numbers
This is a complete work about complex numbers. Perfect for Mathematical Olympiads. A lot of difficult problems.

lots of unusual and challenging problems on complex numbers
Comprehensive and yet concise enough to cover lots of material. Lots of wonderful questions to challenge any math problem lovers.
Highly recommended.

A very useful book on complex numbers
Mathematics is amazing not only in its power and beauty, but also in the way that it has applications in so many areas. The aim of this book is to stimulate young people to become interested in mathematics, to enthuse, inspire, and challenge them, their parents and their teachers with the wonder, excitement, power, and relevance of mathematics.
This book is a very well written introduction to the fascinating theory of complex numbers and it
contains a fine collection of excellent exercises ranging in difficulty from the fairly easy, if calculational, to the more challenging. As stated
by the authors, the targeted audience is not standard and it "includes high school students and their teachers,
undergraduates, mathematics contestants such as those training for Olympiads or the William Lowell Putnam Mathematical Competition, their coaches, and any person interested in essential mathematics."
The book is mainly devoted to complex numbers and to their wide applications in various fields, such as geometry, trigonometry or algebraic operations. An important feature of this marvelous book is that
it presents a wide range of problems of all degrees of difficulties, but also
that it includes easy proofs and natural generalizations of many theorems in elementary geometry.
The authors show how to approach the solution of such problems, emphasizing the use of methods rather than the mere use of formulas. Of course, the more sophisticated the problems become, the more specific this approach has to be chosen.

The book is self-contained; no background in complex numbers is assumed and complete
solutions to routine problems and to olympiad-caliber problems are presented in the last chapter of the book.
The aim of the core part of each chapter is to develop key mathematical ideas and to place them in the context of novel, interesting, and unexpected applications to real-world problems.
The first chapter deals with complex numbers in algebraic form and leads up to the geometric interpretations of the modulus and of the algebraic operations. The second chapter deals with various applications to trigonometry,
starting with elementary facts on the polar representation of complex numbers
and going up to more sophisticated properties related to $n$th roots of unity and their applications in solving
binomial equations. Chapter 3 is devoted to the applications of complex numbers in solving problems in Plane and Analytic Geometry. This chapter includes a lot of interesting properties related to collinearity, orthogonality, concyclicity, similar triangles, as well as very useful analytic formulas for the geometry of a triangle and of a circle in the complex plane. Chapter 4 contains much more powerful results such as: the nine-point circle of Euler, some important distances in a triangle, barycentric coordinates, orthopolar triangles, Lagrange's theorem, geometric transformations in the complex plane. This chapter also includes a marvelous theorem known in the mathematical
folklore under the name of "Morley's Miracle" and which simply states that "the three points of intersection
of the adjacent trisectors of any triangle form an equilateral triangle". As stated in the book, this theorem
was mistakenly attributed to Napoleon Bonaparte. The proof of this theorem follows directly from Theorem 3 on page 155, a deep result which was obtained by the celebrated French mathematician Alain Connes (Fields Medal in
1982 and Clay Research Award in 2000),
in connection with his revolutionary results in Noncommutative Geometry. Chapter 5 illustrates the force of the
method of complex numbers in solving several Olympiad-caliber problems where this technique works very efficiently.

This very successful book is the fruit of the prodigious activity of two well-known creators of mathematics problems in various mathematical journals. The big experience of the authors in preparing students for various mathematical competitions allowed them to present a big collection of beautiful problems. This book continues the tradition making national and international mathematical competition problems available to a wider audience and is bound to appeal to anyone interested in mathematical problem solving.
I very strongly recommend this book to all students curious about elementary mathematics, especially those who are bored at school and ready for a challenge. Teachers would find this book to be a welcome resource, as will contest organizers.
This book is meant both to be read and to be used.
All in all, an excellent book for its intended audience!