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Numeros Complexos De a a Z.Pdf About the Authors 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 Mathemat- ics Competitions (1998–2003), coach of the USA International Mathematical Olympiad Team (IMO) for 10 years (1993–2002), Director of the Mathemat- ical 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 com- petition. 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 prepar- ing 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 chair- man 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 fromAto...Z Birkhauser¨ Boston • Basel • Berlin Dorin Andrica Titu Andreescu “Babes¸-Bolyai” University University of Texas at Dallas Faculty of Mathematics School of Natural Sciences and Mathematics 3400 Cluj-Napoca Richardson, TX 75083 Romania U.S.A. Cover design by Mary Burgess. Mathematics Subject Classification (2000): 00A05, 00A07, 30-99, 30A99, 97U40 Library of Congress Cataloging-in-Publication Data Andreescu, Titu, 1956- Complex 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 eISBN 0-8176-4449-0 Printed on acid-free paper. ISBN-13 978-0-8176-4326-3 c 2006 Birkhauser¨ Boston Complex Numbers from A to...Zis 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 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Birkhauser¨ Boston, c/o Springer Science+Business Media Inc., 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly anal- ysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed in the United States of America. (TXQ/MP) 987654321 www.birkhauser.com The shortest path between two truths in the real domain passes through the complex domain. Jacques Hadamard About the Authors 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 Mathemat- ics Competitions (1998–2003), coach of the USA International Mathematical Olympiad Team (IMO) for 10 years (1993–2002), Director of the Mathemat- ical 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 com- petition. 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 prepar- ing 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 chair- man 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 fromAto...Z Birkhauser¨ Boston • Basel • Berlin Dorin Andrica Titu Andreescu “Babes¸-Bolyai” University University of Texas at Dallas Faculty of Mathematics School of Natural Sciences and Mathematics 3400 Cluj-Napoca Richardson, TX 75083 Romania U.S.A. Cover design by Mary Burgess. Mathematics Subject Classification (2000): 00A05, 00A07, 30-99, 30A99, 97U40 Library of Congress Cataloging-in-Publication Data Andreescu, Titu, 1956- Complex 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 eISBN 0-8176-4449-0 Printed on acid-free paper. ISBN-13 978-0-8176-4326-3 c 2006 Birkhauser¨ Boston Complex Numbers from A to...Zis 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 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Birkhauser¨ Boston, c/o Springer Science+Business Media Inc., 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly anal- ysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed in the United States of America. (TXQ/MP) 987654321 www.birkhauser.com Contents Preface ix Notation xiii 1 Complex Numbers in Algebraic Form 1 1.1 Algebraic Representation of Complex Numbers ............ 1 1.1.1 Definition of complex numbers . ............... 1 1.1.2 Properties concerning addition . ............... 2 1.1.3 Properties concerning multiplication .............. 3 1.1.4 Complex numbers in algebraic form .............. 5 1.1.5 Powers of the number i ..................... 7 1.1.6 Conjugate of a complex number . ............... 8 1.1.7 Modulus of a complex number . ............... 9 1.1.8 Solving quadratic equations ................... 15 1.1.9 Problems ............................ 18 1.2 Geometric Interpretation of the Algebraic Operations . ....... 21 1.2.1 Geometric interpretation of a complex number . ....... 21 1.2.2 Geometric interpretation of the modulus ............ 23 1.2.3 Geometric interpretation of the algebraic operations ...... 24 1.2.4 Problems ............................ 27 vi Contents 2 Complex Numbers in Trigonometric Form 29 2.1 Polar Representation of Complex Numbers .............. 29 2.1.1 Polar coordinates in the plane . ............... 29 2.1.2 Polar representation of a complex number ........... 31 2.1.3 Operations with complex numbers in polar representation . 36 2.1.4 Geometric interpretation of multiplication ........... 39 2.1.5 Problems ............................ 39 2.2 The nth Roots of Unity . ....................... 41 2.2.1 Defining the nth roots of a complex number . ....... 41 2.2.2 The nth roots of unity ...................... 43 2.2.3 Binomial equations ....................... 51 2.2.4 Problems ............................ 52 3 Complex Numbers and Geometry 53 3.1 Some Simple Geometric Notions and Properties ............ 53 3.1.1 The distance between two points . ............... 53 3.1.2 Segments, rays and lines .................... 54 3.1.3 Dividing a segment into a given ratio .............. 57 3.1.4 Measure of an angle ....................... 58 3.1.5 Angle between two lines .................... 61 3.1.6 Rotation of a point ......................
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