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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 teachesattheUniversityofTexasatDallas.TituispastchairmanoftheUSA MathematicalOlympiad,servedasdirectoroftheMAAAmericanMathemat- icsCompetitions(1998–2003),coachoftheUSAInternationalMathematical OlympiadTeam(IMO)for10years(1993–2002),DirectoroftheMathemat- 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,thegoverningbodyoftheworld’smostprestigiousmathematicscom- 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’scontributionstonumeroustextbooksandproblembooksarerecognized worldwide. DorinAndricareceivedhisPhDin1992from“Babes¸-Bolyai”Universityin Cluj-Napoca,Romania,withathesisoncriticalpointsandapplicationstothe geometry of differentiable submanifolds. Professor Andrica has been chair- manoftheDepartmentofGeometryat“Babes¸-Bolyai”since1995.Dorinhas written and contributed to numerous mathematics textbooks, problem books, articles and scientific papers at various levels. Dorin is an invited lecturer at universityconferencesaroundtheworld—Austria,Bulgaria,CzechRepublic, 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 beenaregularfacultymemberattheCanada–USAMathcampssince2001. Titu Andreescu Dorin Andrica Complex Numbers from A to. . . Z Birkha¨user Boston • Basel • Berlin DorinAndrica TituAndreescu “Babes¸-Bolyai”University UniversityofTexasatDallas FacultyofMathematics SchoolofNaturalSciencesandMathematics 3400Cluj-Napoca Richardson,TX75083 Romania U.S.A. CoverdesignbyMaryBurgess. MathematicsSubjectClassification(2000):00A05,00A07,30-99,30A99,97U40 LibraryofCongressCataloging-in-PublicationData Andreescu,Titu,1956- ComplexnumbersfromAto–Z/TituAndreescu,DorinAndrica. p.cm. “PartlybasedonaRomanianversion...preservingthetitle...andabout35%ofthetext”–Pref. Includesbibliographicalreferencesandindex. ISBN0-8176-4326-5(acid-freepaper) 1.Numbers,Complex.I.Andrica,D.(Dorin)II.Andrica,D.(Dorin)Numerecomplexe QA255.A5582004 512.7’88–dc22 2004051907 ISBN-100-8176-4326-5 eISBN0-8176-4449-0 Printedonacid-freepaper. ISBN-13978-0-8176-4326-3 (cid:2)c2006Birkha¨userBoston ComplexNumbersfromAto...ZisagreatlyexpandedandsubstantiallyenhancedversionoftheRomanian edition,NumerecomplexedelaAla...Z,S.C.EdituraMilleniumS.R.L.,AlbaIulia,Romania,2001 Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewritten permissionofthepublisher(Birkha¨userBoston,c/oSpringerScience+BusinessMediaInc.,233Spring Street,NewYork,NY10013,USA),exceptforbriefexcerptsinconnectionwithreviewsorscholarlyanal- ysis.Useinconnectionwithanyformofinformationstorageandretrieval,electronicadaptation,computer software,orbysimilarordissimilarmethodologynowknownorhereafterdevelopedisforbidden. Theuseinthispublicationoftradenames,trademarks,servicemarksandsimilarterms,eveniftheyare notidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyaresubjectto proprietaryrights. PrintedintheUnitedStatesofAmerica. (TXQ/MP) 987654321 www.birkhauser.com Theshortestpathbetweentwotruthsinthereal domainpassesthroughthecomplexdomain. JacquesHadamard 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 teachesattheUniversityofTexasatDallas.TituispastchairmanoftheUSA MathematicalOlympiad,servedasdirectoroftheMAAAmericanMathemat- icsCompetitions(1998–2003),coachoftheUSAInternationalMathematical OlympiadTeam(IMO)for10years(1993–2002),DirectoroftheMathemat- 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,thegoverningbodyoftheworld’smostprestigiousmathematicscom- 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’scontributionstonumeroustextbooksandproblembooksarerecognized worldwide. DorinAndricareceivedhisPhDin1992from“Babes¸-Bolyai”Universityin Cluj-Napoca,Romania,withathesisoncriticalpointsandapplicationstothe geometry of differentiable submanifolds. Professor Andrica has been chair- manoftheDepartmentofGeometryat“Babes¸-Bolyai”since1995.Dorinhas written and contributed to numerous mathematics textbooks, problem books, articles and scientific papers at various levels. Dorin is an invited lecturer at universityconferencesaroundtheworld—Austria,Bulgaria,CzechRepublic, 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 beenaregularfacultymemberattheCanada–USAMathcampssince2001. Titu Andreescu Dorin Andrica Complex Numbers from A to. . . Z Birkha¨user Boston • Basel • Berlin DorinAndrica TituAndreescu “Babes¸-Bolyai”University UniversityofTexasatDallas FacultyofMathematics SchoolofNaturalSciencesandMathematics 3400Cluj-Napoca Richardson,TX75083 Romania U.S.A. CoverdesignbyMaryBurgess. MathematicsSubjectClassification(2000):00A05,00A07,30-99,30A99,97U40 LibraryofCongressCataloging-in-PublicationData Andreescu,Titu,1956- ComplexnumbersfromAto–Z/TituAndreescu,DorinAndrica. p.cm. “PartlybasedonaRomanianversion...preservingthetitle...andabout35%ofthetext”–Pref. Includesbibliographicalreferencesandindex. ISBN0-8176-4326-5(acid-freepaper) 1.Numbers,Complex.I.Andrica,D.(Dorin)II.Andrica,D.(Dorin)Numerecomplexe QA255.A5582004 512.7’88–dc22 2004051907 ISBN-100-8176-4326-5 eISBN0-8176-4449-0 Printedonacid-freepaper. ISBN-13978-0-8176-4326-3 (cid:2)c2006Birkha¨userBoston ComplexNumbersfromAto...ZisagreatlyexpandedandsubstantiallyenhancedversionoftheRomanian edition,NumerecomplexedelaAla...Z,S.C.EdituraMilleniumS.R.L.,AlbaIulia,Romania,2001 Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewritten permissionofthepublisher(Birkha¨userBoston,c/oSpringerScience+BusinessMediaInc.,233Spring Street,NewYork,NY10013,USA),exceptforbriefexcerptsinconnectionwithreviewsorscholarlyanal- ysis.Useinconnectionwithanyformofinformationstorageandretrieval,electronicadaptation,computer software,orbysimilarordissimilarmethodologynowknownorhereafterdevelopedisforbidden. Theuseinthispublicationoftradenames,trademarks,servicemarksandsimilarterms,eveniftheyare notidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyaresubjectto proprietaryrights. PrintedintheUnitedStatesofAmerica. (TXQ/MP) 987654321 www.birkhauser.com Contents Preface ix Notation xiii 1 ComplexNumbersinAlgebraicForm 1 1.1 AlgebraicRepresentationofComplexNumbers . . . . . . . . . . . . 1 1.1.1 Definitionofcomplexnumbers . . . . . . . . . . . . . . . . . 1 1.1.2 Propertiesconcerningaddition . . . . . . . . . . . . . . . . . 2 1.1.3 Propertiesconcerningmultiplication . . . . . . . . . . . . . . 3 1.1.4 Complexnumbersinalgebraicform . . . . . . . . . . . . . . 5 1.1.5 Powersofthenumberi . . . . . . . . . . . . . . . . . . . . . 7 1.1.6 Conjugateofacomplexnumber . . . . . . . . . . . . . . . . 8 1.1.7 Modulusofacomplexnumber . . . . . . . . . . . . . . . . . 9 1.1.8 Solvingquadraticequations. . . . . . . . . . . . . . . . . . . 15 1.1.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.2 GeometricInterpretationoftheAlgebraicOperations . . . . . . . . . 21 1.2.1 Geometricinterpretationofacomplexnumber . . . . . . . . . 21 1.2.2 Geometricinterpretationofthemodulus . . . . . . . . . . . . 23 1.2.3 Geometricinterpretationofthealgebraicoperations . . . . . . 24 1.2.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

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Complex numbers from A to--Z PDF

336 Pages·2006·1.858 MB·English
by  Titu Andreescu, Dorin Andrica
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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 teachesattheUniversityofTexasatDallas.TituispastchairmanoftheUSA MathematicalOlympiad,servedasdirectoroftheMAAAmericanMathemat- icsCompetitions(1998–2003),coachoftheUSAInternationalMathematical OlympiadTeam(IMO)for10years(1993–2002),DirectoroftheMathemat- 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,thegoverningbodyoftheworld’smostprestigiousmathematicscom- 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’scontributionstonumeroustextbooksandproblembooksarerecognized worldwide. DorinAndricareceivedhisPhDin1992from“Babes¸-Bolyai”Universityin Cluj-Napoca,Romania,withathesisoncriticalpointsandapplicationstothe geometry of differentiable submanifolds. Professor Andrica has been chair- manoftheDepartmentofGeometryat“Babes¸-Bolyai”since1995.Dorinhas written and contributed to numerous mathematics textbooks, problem books, articles and scientific papers at various levels. Dorin is an invited lecturer at universityconferencesaroundtheworld—Austria,Bulgaria,CzechRepublic, 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 beenaregularfacultymemberattheCanada–USAMathcampssince2001. Titu Andreescu Dorin Andrica Complex Numbers from A to. . . Z Birkha¨user Boston • Basel • Berlin DorinAndrica TituAndreescu “Babes¸-Bolyai”University UniversityofTexasatDallas FacultyofMathematics SchoolofNaturalSciencesandMathematics 3400Cluj-Napoca Richardson,TX75083 Romania U.S.A. CoverdesignbyMaryBurgess. MathematicsSubjectClassification(2000):00A05,00A07,30-99,30A99,97U40 LibraryofCongressCataloging-in-PublicationData Andreescu,Titu,1956- ComplexnumbersfromAto–Z/TituAndreescu,DorinAndrica. p.cm. “PartlybasedonaRomanianversion...preservingthetitle...andabout35%ofthetext”–Pref. Includesbibliographicalreferencesandindex. ISBN0-8176-4326-5(acid-freepaper) 1.Numbers,Complex.I.Andrica,D.(Dorin)II.Andrica,D.(Dorin)Numerecomplexe QA255.A5582004 512.7’88–dc22 2004051907 ISBN-100-8176-4326-5 eISBN0-8176-4449-0 Printedonacid-freepaper. ISBN-13978-0-8176-4326-3 (cid:2)c2006Birkha¨userBoston ComplexNumbersfromAto...ZisagreatlyexpandedandsubstantiallyenhancedversionoftheRomanian edition,NumerecomplexedelaAla...Z,S.C.EdituraMilleniumS.R.L.,AlbaIulia,Romania,2001 Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewritten permissionofthepublisher(Birkha¨userBoston,c/oSpringerScience+BusinessMediaInc.,233Spring Street,NewYork,NY10013,USA),exceptforbriefexcerptsinconnectionwithreviewsorscholarlyanal- ysis.Useinconnectionwithanyformofinformationstorageandretrieval,electronicadaptation,computer software,orbysimilarordissimilarmethodologynowknownorhereafterdevelopedisforbidden. Theuseinthispublicationoftradenames,trademarks,servicemarksandsimilarterms,eveniftheyare notidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyaresubjectto proprietaryrights. PrintedintheUnitedStatesofAmerica. (TXQ/MP) 987654321 www.birkhauser.com Theshortestpathbetweentwotruthsinthereal domainpassesthroughthecomplexdomain. JacquesHadamard 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 teachesattheUniversityofTexasatDallas.TituispastchairmanoftheUSA MathematicalOlympiad,servedasdirectoroftheMAAAmericanMathemat- icsCompetitions(1998–2003),coachoftheUSAInternationalMathematical OlympiadTeam(IMO)for10years(1993–2002),DirectoroftheMathemat- 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,thegoverningbodyoftheworld’smostprestigiousmathematicscom- 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’scontributionstonumeroustextbooksandproblembooksarerecognized worldwide. DorinAndricareceivedhisPhDin1992from“Babes¸-Bolyai”Universityin Cluj-Napoca,Romania,withathesisoncriticalpointsandapplicationstothe geometry of differentiable submanifolds. Professor Andrica has been chair- manoftheDepartmentofGeometryat“Babes¸-Bolyai”since1995.Dorinhas written and contributed to numerous mathematics textbooks, problem books, articles and scientific papers at various levels. Dorin is an invited lecturer at universityconferencesaroundtheworld—Austria,Bulgaria,CzechRepublic, 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 beenaregularfacultymemberattheCanada–USAMathcampssince2001. Titu Andreescu Dorin Andrica Complex Numbers from A to. . . Z Birkha¨user Boston • Basel • Berlin DorinAndrica TituAndreescu “Babes¸-Bolyai”University UniversityofTexasatDallas FacultyofMathematics SchoolofNaturalSciencesandMathematics 3400Cluj-Napoca Richardson,TX75083 Romania U.S.A. CoverdesignbyMaryBurgess. MathematicsSubjectClassification(2000):00A05,00A07,30-99,30A99,97U40 LibraryofCongressCataloging-in-PublicationData Andreescu,Titu,1956- ComplexnumbersfromAto–Z/TituAndreescu,DorinAndrica. p.cm. “PartlybasedonaRomanianversion...preservingthetitle...andabout35%ofthetext”–Pref. Includesbibliographicalreferencesandindex. ISBN0-8176-4326-5(acid-freepaper) 1.Numbers,Complex.I.Andrica,D.(Dorin)II.Andrica,D.(Dorin)Numerecomplexe QA255.A5582004 512.7’88–dc22 2004051907 ISBN-100-8176-4326-5 eISBN0-8176-4449-0 Printedonacid-freepaper. ISBN-13978-0-8176-4326-3 (cid:2)c2006Birkha¨userBoston ComplexNumbersfromAto...ZisagreatlyexpandedandsubstantiallyenhancedversionoftheRomanian edition,NumerecomplexedelaAla...Z,S.C.EdituraMilleniumS.R.L.,AlbaIulia,Romania,2001 Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewritten permissionofthepublisher(Birkha¨userBoston,c/oSpringerScience+BusinessMediaInc.,233Spring Street,NewYork,NY10013,USA),exceptforbriefexcerptsinconnectionwithreviewsorscholarlyanal- ysis.Useinconnectionwithanyformofinformationstorageandretrieval,electronicadaptation,computer software,orbysimilarordissimilarmethodologynowknownorhereafterdevelopedisforbidden. Theuseinthispublicationoftradenames,trademarks,servicemarksandsimilarterms,eveniftheyare notidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyaresubjectto proprietaryrights. PrintedintheUnitedStatesofAmerica. (TXQ/MP) 987654321 www.birkhauser.com Contents Preface ix Notation xiii 1 ComplexNumbersinAlgebraicForm 1 1.1 AlgebraicRepresentationofComplexNumbers . . . . . . . . . . . . 1 1.1.1 Definitionofcomplexnumbers . . . . . . . . . . . . . . . . . 1 1.1.2 Propertiesconcerningaddition . . . . . . . . . . . . . . . . . 2 1.1.3 Propertiesconcerningmultiplication . . . . . . . . . . . . . . 3 1.1.4 Complexnumbersinalgebraicform . . . . . . . . . . . . . . 5 1.1.5 Powersofthenumberi . . . . . . . . . . . . . . . . . . . . . 7 1.1.6 Conjugateofacomplexnumber . . . . . . . . . . . . . . . . 8 1.1.7 Modulusofacomplexnumber . . . . . . . . . . . . . . . . . 9 1.1.8 Solvingquadraticequations. . . . . . . . . . . . . . . . . . . 15 1.1.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.2 GeometricInterpretationoftheAlgebraicOperations . . . . . . . . . 21 1.2.1 Geometricinterpretationofacomplexnumber . . . . . . . . . 21 1.2.2 Geometricinterpretationofthemodulus . . . . . . . . . . . . 23 1.2.3 Geometricinterpretationofthealgebraicoperations . . . . . . 24 1.2.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

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