Complex Numbers from A to ...Z

336 Pages·2005·2.81 MB·English

by  Titu Andreescu

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