## Overview

- Learn how complex numbers may be used to solve algebraic equations as well as their geometric interpretation
- Theoretical aspects are augmented with rich exercises and problems at various levels of difficulty
- A special feature is a selection of outstanding Olympiad problems solved by employing the methods presented
- May serve as an engaging supplemental text for an introductory undergrad course on complex numbers or number theory
- Includes supplementary material: sn.pub/extras

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## About this book

It is impossible to imagine modern mathematics without complex numbers. The second edition of *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. Many new problems and solutions have been added in this second edition. 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. Itdistills 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 undergraduate high school students and their teacher's 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.

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## Table of contents (6 chapters)

## Reviews

“I would warmly recommend this book to anyone interested in competitive mathematics or exploring an algebraic approach to Euclidean geometry, and teachers will find it a treasure trove of beautiful questions for enthusiastic students. Both authors richly deserve their reputations as problem selectors of great taste and discernment.” (Dominic Rowland, The Mathematical Gazette, Vol. 100 (549), November, 2016)

“The target audience of the book is high school and undergraduate students. There is a large list of exercises with solutions, some taken from Mathematical Olympiad competitions. Thus the book is also a valuable resource for teachers and those interested in mathematical competitions. The first half of the book presents the complex numbers and their geometric properties in depth. The second half is a collection of exercises with solutions.” (Stefan Ulrych, Mathematical Reviews, October, 2014)

"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)

"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)

"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)

"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*

## Authors and Affiliations

## Bibliographic Information

Book Title: Complex Numbers from A to ... Z

Authors: Titu Andreescu, Dorin Andrica

DOI: https://doi.org/10.1007/978-0-8176-8415-0

Publisher: Birkhäuser Boston, MA

eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

Copyright Information: Springer Science+Business Media New York 2014

Softcover ISBN: 978-0-8176-8414-3Published: 17 February 2014

eBook ISBN: 978-0-8176-8415-0Published: 17 February 2014

Edition Number: 2

Number of Pages: XVII, 391

Number of Illustrations: 83 b/w illustrations

Topics: Number Theory, Algebraic Geometry, Geometry