Titu Andreescu received his BA, MS, and PhD from the West University of Timisoara, Romania. The topic of his doctoral dissertation was “Research on Diophantine Analysis and Applications.” Professor Andreescu currently teaches at the University of Texas at Dallas. Titu is past chairman of the USA Mathematical Olympiad, served as director of the MAA American Mathematics Competitions (1998–2003), coach of the USA International Mathematical Olympiad Team (IMO) for 10 years (1993–2002), Director of the Mathematical Olympiad Summer Program (1995–2002) and leader of the USA IMO Team (1995–2002). In 2002 Titu was elected member of the IMO Advisory Board, the governing body of the world’s most prestigious mathematics competition. Titu received the Edyth May Sliffe Award for Distinguished High School Mathematics Teaching from the MAA in 1994 and a “Certiﬁcate of Appreciation” from the president of the MAA in 1995 for his outstanding service as coach of the Mathematical Olympiad Summer Program in preparing the US team for its perfect performance in Hong Kong at the 1994 IMO. Titu’s contributions to numerous textbooks and problem books are recognized worldwide. Dorin Andrica received his PhD in 1992 from “Babes-Bolyai” University in ¸ Cluj-Napoca, Romania, with a thesis on critical points and applications to the geometry of differentiable submanifolds. Professor Andrica has been chairman of the Department of Geometry at “Babes-Bolyai” since 1995. Dorin has ¸ written and contributed to numerous mathematics textbooks, problem books, articles and scientiﬁc papers at various levels. Dorin is an invited lecturer at university conferences around the world—Austria, Bulgaria, Czech Republic, Egypt, France, Germany, Greece, the Netherlands, Serbia, Turkey, and USA. He is a member of the Romanian Committee for the Mathematics Olympiad and member of editorial boards of several international journals. Dorin has been a regular faculty member at the Canada–USA Mathcamps since 2001.

Titu Andreescu Dorin Andrica

Complex Numbers from A to. . . Z

Birkh¨ user a • Basel • Berlin Boston

Titu Andreescu University of Texas at Dallas School of Natural Sciences and Mathematics Richardson, TX 75083 U.S.A.

Dorin Andrica “Babes-Bolyai” University ¸ Faculty of Mathematics 3400 Cluj-Napoca Romania

Cover design by Mary Burgess. Mathematics Subject Classiﬁcation (2000): 00A05, 00A07, 30-99, 30A99, 97U40

Library of Congress Cataloging-in-Publication Data Andreescu, Titu, 1956Complex numbers from A to–Z / Titu Andreescu, Dorin Andrica. p. cm. “Partly based on a Romanian version . . . preserving the title. . . and about 35% of the text”–Pref. Includes bibliographical references and index. ISBN 0-8176-4326-5 (acid-free paper) 1. Numbers, Complex. I. Andrica, D. (Dorin) II. Andrica, D. (Dorin) Numere complexe QA255.A558 2004 512.7’88–dc22 2004051907

ISBN-10 0-8176-4326-5 ISBN-13 978-0-8176-4326-3

eISBN 0-8176-4449-0

Printed on acid-free paper.

c 2006 Birkh¨ user Boston a Complex Numbers from A to. . . Z is a greatly expanded and substantially enhanced version of the Romanian edition, Numere complexe de la A la. . . Z, S.C. Editura Millenium S.R. L., Alba Iulia, Romania, 2001

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