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JohSnt illwell numBERSAnD GEOMETRY I I I I I I I 'Spdng« UndergraduatThexts inM athematics Rl:ading inM athematics E.ditml S.Axl er F.W. Gehring K.A.Ribe t Springer New York Berlin Heidelberg Barcelona HongK ong London Milan Paris Singapore Tokyo Th.S.a One III II 11 l1I1111l11 11111l111l111l11 11111111 WD1G-SKK-FLUF • GraduatTee xtisn M athematics ReadinignMs a thematics Ebbinghaenuns/eHs/HirzebruchlKoecher/Mainzer/Neukin:h/Prestel/Remmert: Numbers Fulton/Harris:R epresentatiThoneo ry: A FirsCto urse Remrncrt: Theory ofC omplexF unctions WalterO:rd inaDryi fferentiEqaula tions UndergraduTaetxeti sn M athematics ReadinignMs a thematics AngliMna:t hematicsA: C oncisHeis torya ndP liilosoplty Anglin/Lambek:Th e Heritaogef Th ales Brnsoud: Second YearC alculus Haim/Wanner:A nalybsyiI stH si story Hllmmerlin/HotlinanNnu:m ericMaalt hematics Isaac: TheP leasures ofPro bability Laubenbacher/PengeMlaltehye:m atExpediictailo ns:C hronicblyet sh Expel orers SamueProlj:e ctiGveeo metry StillwNeulmlbers: andG eometry Toth:G limpseosf A lgebraan dG eomett')' John Stillwell Numberasn d Geometry With 95n tustrations Springer JohnS tillwell Monash University Department ofM athematics ClaytonV,i ctoria 3168A ustralia Editorial Board S.Ax ler F.W. Gehring K.BR.i bet Departmento f Department of DepartmenotfM athematics Mathematics Mathematics UniversitoyfC alifornia MichigSant atUen iversity UniversitoyfM ichigan at Berkeley EasLta nsinMgI, 4 8824 Ann ArborM,l 48109 Berekley,C A 94720 USA USA USA Cover illustorfat thieeo xnp onentimaalp byTi istanN eedham.Re printefrdo m Visual Complex Analbyyspe irsm issiono fTiistaNne edhama nd OxfordU niversity 1P99r7e.s s, Mathematics ubject Classific0Ia tion (2000): I 1-0 I, 51- Libraroyf C ongressC ataloging-in-PubliDcaattai on StillweJlolh,n . Numbersa ndg eometry p. cm. - (Undergradutaetxeti sn ma thematicsRe. adigns in mathematics) Includbeibs liograprheifecalr enc(eps. )a ndi n.dell. ISBN0 -387-98289(-h2e ; paaleprk). 1.M athematics. I. TitlIeI.Se. r ies. QA39.2.S755 1997 510-dc21 97-22858 CIP Printeodn acid-free paper. o 199S8p ringer-VerNleawg Y ork, Inc. 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Printinet hde U nited StaotfAme ersi ca. 98765432 ISB0N· 387-98289S-p2r inger-NVeew rYloargk B rlin Heid7 6,I9tble5r g SPIN 10 'lb my mothearnd my latfea ther whot aughmet howt oc ount + Preface What shouledv erya spirimnagt hematickinaonw ?Th e answeforr mosto ft he2 0tche ntuhrya sb eenc:a lculFuosr2. 0 00y earsb efore thatth,e a nswewra s EucliIdtn. o ws eemsa good timet or aisteh e question abgeacianu,set he olda nswersa ren ol ongecro nvincing. Mathematitcosd aiys m uchm oreth an Euclibdu,ti ti sa lsmou ch moret hanc alculaunsd;t hec alculnuosw taught iss,a dlmyu,c h lestsh aint u sedt ob e.L ittblyel ittclael,c ulhuassb eend eprivoefd thea lgebrgae,o metrya,n dl ogiictn eedtso s ustaiint u,n timla ny instituthiaovnehs a dt op uti to nh igh-tleicfhe -supspyoserttm s.A subjescttru gglingt os urvivies h ardlay g oodi ntroducttoit ohne vigoorfr eamla thematics. Butif itw ereo nlay m atteorfp uttithneg gu ts backin to calculus itw oulndo tb en ecessaryt ow ritaen ewb ookI.tw oulbde e nougtho recommenfodr,e xamplSep,i vakC'alsc uloursH ,a rdyPu'rse Mathe matics. Int hec urrensti tuatwieon ne,e dt ore vivneo to nlcya lculus, buta lsaol gebgreao,m etrya,n dth e wholied etah amta thematiicsas rigorocuusm,u latidivcsei pline inw hiche achm athematicsitaann ds ont hes houldeorsfg iants. The bestw ay tot eacrhe amla thematiIc bse,l ieviest, o s tart deepedro wn,w itht hee lementaryi deaosf n umbera nds paceE.v eryonceo ncedtehsa tth esaer efu ndamentablu,tt heyh aveb een vii viiiP reface scandalounselgyl ectpeedr,h apisn t hen aivbee lietfh ata nyone learnicnagl culus ouhtagsr owtnh em.I nfa ct,a rithmetailcg,e bra, andg eometrcya nn everh e. outgrowann,d t hem ostr ewarding patht oh ighemra thematiscuss taitnhse idre velopmeanlto ngside the" advancberda"n chseusc ha sc alculAulss.o b,y m aintaintiinegs betweetnh esdei scipliintie ssp ,o ssibtloep resena tm oreu nified viewo fm athematiycest,a tt hes amet imet oi ncludme.o res pice andv ariety. Thea imo ft hibso okt,h eni,st og ivae b roavdi ewo fa rithmetic, geometrayn,d a lgebartat hel eveolfc alculwuist,h oubte inag c al culubso ok( ora "precalcublouosk")I .t rso otasr ei na rithmeatnidc geometrtyh,et woo pposiptoel eosf m athematiacnsd,t hes ourcoef histocroincc eptucaoln fliTchte.r esolutoifot nh icso nfliacntd,i ts rolien t hed evelopmeonfmt a thematiicsos n,e o ft hem ains tories int heb ookT.h ek eyi sa lgebrwah,i chb ringasr ithmeatnidcg eom etrtyo gethaenrd a llowtsh emt of louriasnhd b rancohu ti nn ew directiTuo nkse.e pt hes torayss implaes p ossibIll ei,n ekv erything tot hea lgebrtahiecm eosf l ineaanrd q uadraetqiuca tions. Ther estricttoil oonw -degreequea tionsi sn otas dreadfuals h igh schooall gebmriag hts uggestE.v enl ineeaqru atioanrsei nteresting wheno nlyi ntegseorl utiaorness oughatn,d t heyn eatlmyo tivaat e wholei ntroducctoourrys ienn umbert heorfryo,m theE uclideaaln gorithtmou nique prfaicmteo rizatQiuoand.r ateiqcu atioanrsee ven morei nterestfrionmg t hei ntegpeori notf v ieww,i thP ythagorean triplaensdP elle'qsu atiloena didnege pi ntaol gebrgae,o metrya,n d analysFirso.m t hep oinotf v iewo fg eometrqyu,a drateiqcu ations represetnhtec onisce ctionsfa-sac inatitnogp iicn i tself-tahned areabso undebdy t hesceu rvedse finet he circluolgaarr,i thamnidc , hyperbofluincc tioInnst .h iwsa yw e arel edt ot hes ubjemcatt ter ofa firscta lculcuosu rsbeu,tw ithl esmsa chinearny dm oret imet o probfeu ndamentqaule stiosnusc ha st hen aturoefn umbers,c urves, anda reaI.ti sw ortmhe ntionitnhga wte alscoo vetrh em aini deaosf Euclid-geomeatrriyt,h metaincdt, h et heoroyf r eanlu mbers-but with2 000y earosf e xtrian sighatdsd ed. Infa ct,t hibso okc oulbde describaesda deepelro oka to rdi nary thingMso.s to fm athematiicsas b ountu mbersc,u rveasn,d functioannsd,t hel inkbse tweetnh esceo ncepctasn b es uggested bya thorousgthu doyf s impleex amplessu,c ha st hec ircalned t he Preface ix squarIeh .o pet os howt hamta thematilcisk,te h ew orldfi,t sW ilalmi Blaked'ess cription: Tu seea W orlid na Graionf S and, Anda Heaveni na WildF lower, HoldI nfintiyi nt he palmo fy ourh and And Eterniitnay n h our. Becausieti sv irtuaimlplyos sbilet ol earmna thematiscb ym erer ead ingt,h ibso oki ncludmeasn ye xerciasnedse vereyn couraegmentt o dot hemT.h erei sa seto fe xercisaetts h e endo fe achse ction,s o newi deacsa nb ei nstanttelsyt ecdl,a rifiaendd,r einforceTdh.e e x ercisaerse o ftevna riatioorng esn eriazaltionso ft her esulitnst h e maint exti;n s omec aseIs t hintkh eya ree venm orei ntereinsgt! Inp aritculra,t hey include simplfiied arrangemeonftm sa nyc lassic proofbsy greamta thematicifraonms ,Eu clidt oH ilbertE.ac h set ofe xerciissea sc companibeyda commentary tom ake itpsu rpose ands ignificancec leaIrh .o pet hiwsi lble u sefupla,r ticulwahrelny severeaxle rcihsaevset ob el inketodge thert op roducea b igr esult. Who ist hibso okfo r?B ecausietp resupposoensl yh ighs chool algebriatc, a ni np rincipel be readb y anyw ellprepaerd student - enteriunngiv erstyi.I tc omplementthseu suaclo ursehse,n cei tc an be offereads a" harodp tiotno�s tudenwthso aren ot sufficiently extendedb y thes tandamradte riala tt halte veOln. theo thehra nd, ith ass ol ittilnce o mmon witht hec alculaunsd l ineaarl gebtrhaa t dominattehse standacrudr ricultuhma itt m ay wellb e a revela tioenv ent os enior undergraduatMeasn.y s tudenntosw graduate inm athematiwcist houhtav ing donea coursien n umbert heory, geometroyr,fo undations-efoxra mplwei,ht out havnig seent he fundamenttahle oreomfa rithmentoinc-,E ucligdeeoamne troyrt, h e definitioofrn e anlu mbersF.o rs uchs tudentthsi,bs o okc oulsde rve asa capstocnoeu rsien t hes enioyre arp,r esentian ugn ifieadp- · proactho m athematiacnsd p rovinmga nyo ft hec lassriecs ultthsa t aren ormalyl onlym entioned. Itc ouladl sboe u sedi nc onventiocnoaulr seCsh.a pteIr,s4 ,6 ,7 , andp artosf8 and9 contani mosto ft he standamradt erifoarl a first numbert heorcyou rse.C hapters 2,3 ,5 ,a nd8 coulsde rvea sa first coures ing eometrByu.tn aturailtwl oyu ldb eb esti fh ett woc ourses