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Chapter8hem
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Realandbers
&ruplexnumbersismuchsimplerandgoesmuoothlytharuoftherealhefirststageinprodugtherealsisdevelopmeionals，atwhitlainwhatismeantbyafra.Afra，suchas23isjustapairofintegers，whichwerepresentinthisfamiliarbutpeheideaoffraalpartsisnotdiffiderstand，althoughthediakesrealefforttthewayyourteachersexplaininpassingthatsuchfrasas23，46，69etc.are‘equal’–theyarenotthesamenumberpairsbuttheydorepresentequalslicesofpie.Thisisnothardtoacceptbutitdoesdrawourattentiohatarationalnumberisiyaofequivalentfras，eachrepresentedbyapairofihissoundsintimidatiprefernottothinktoomuchaboutthis，fortheprospeanipulatingiionsofpairshtleaveusfeelihereisonesavianyfrahasauniquereducedrepresentatioheoraoraree，whibegotbyganyonfathefrawithwhichyinallybegaheless，oniliarwiththepropertiesoffradtherulesfthem，nothingsheventhoughihat，asyoudoyoursums，youareimpliipulatingiionsofintegerpairs.
Itistemptingthallthisfrettingaboutparticularequationsandsimplydeclarethatwealreadyknowwhattherealheyaretheofallpossibledecimalexpansions，bothpositiveaheseareveryfamiliar，inpractiowhowtousethem，andsowefeelonsafegrouilweasksomeverybasiaiureofhatyouadd，subtract，multiply，a，forexample，howareyousupposedtomultiplytwoinfinitendecimals?Wedependondecimalsbeihsothatyou‘startfrht-hathereisnosugwithaninfinitedecimalexpansion.Ite，butitisplicatedbothintheoryandinpraumbersystemwhereyletoexplainholydoesisfactory.
Youmayfiioionsraisedaboryoumaygrowimpatientwithalltheiioobemakingtroubleforourselveswhenpreviouslyallwassmoothsailing.Thereisaseriouspoihematisappreciatethat，whehematicalobjetroduced，itimportanttostructthemfromkicalobjects，theway，foriioofaspairsers.Inthisway，wemaycarefullybuilduptherulesthatgovereemandkafoundatiowillebatuslater.Forexample，therapiddevelopmentofcalculus，whichwasbornoutofthestudyofmotioospectacularresults，suchasprediovemes.Houlationofihingsasiftheywerefiimesprovidedamazinginsightsaimespatentingyourmathematicalsystemsonafirmfoundation，wehowtotellthedifferenpractice，mathematisoftenindulgein‘formal’manipulatiooseeifsheoffieisworthyofattebeprorouslybygoingbacktobasidbyihathavebeeablishedearlier.
ThisiswhyJuliusDedekind（1831–1916）tookthetroubleofformallystrugtherealembasedoisoasDedekindcutsofthereallihemati，however，tosuccessfullydealwiththedilemmacausedbytheexisteionalnumberswasEudoxusofidus（fl380BC）whoseTheoryofProportionsallowedArchimedestousetheso-calledMethodofExhaustiorouslyderiveresultsonareasandvolumesofcurvedshapesbeforetheadventofcale1，900yearslater.
Thefihenumberjigsaw–theimaginaryunit
13.Additionofbersbyaddiedlis
&iberspresentsitselfveryheplexplahinkofthebera+biasbeihepoint（a，b）intheateplawobersz=（a，b）andw=（c，d），wesimplyaddtheirfirstariestiveusz+w=（a+c，b+d）.Ifwemakeuseofthesymboli，wehaveforexample（2+i）+（1+3i）=3+4i.
Thisdstowhatiskoradditionintheplaedliors）areaddedtogether，toptotail（seeFigure13）.Webeginatthein，whichhasatesof（0，0），andinthisexamplewelaydownourfirstarrowfromtheretothepoint（2，1）.Toaddtheedby（1，3），wegotothepoint（2，1），anddrawanarroresentsmoving1unitrightialdire（thatisthedireoftherealaxis），and3unitsupiioical（theimaginaryaxis）.Weendupatthepointwithates（3，4）.Inmuchthesameway，weesubtraplexnumbersbysubtragtherealandimaginarypartssothat，forexample，（11+7i）-（2+5i）=9+2i.Thisbepicturedasstartingwiththevector（11，7），andsubtragthevector（2，5），tofinishatthepoint（9，2）.
Multipliisaer.Formallyitiseasytodo:wemultiplytwoberstogetherbymultiplyis，rememberingthati2=-1.AssumiributiveLawuestohold，whichisthealgebraicrulethatallowsustoexpasintheusualway，thenmultipliproceedsasfollows:
（a+bi）（c+di）=a（c+di）+bi（c+di）=
ac+adi+bci+bdi2=（ac-bd）+（ad+bc）i
Byusiherthanspeberswethesameway，fieofageneraldivisionofbersiheirrealandimaginarypartsaswehavedoneabeneralultipli.Hastheteiqueisuhereisnoproduorizetheresultingformula.
14.Thepositionofaberinpolarates
Furtherces
Thereareahostofappliplexheelemeheiweeangularandpolarrepresentatiooplayinasurprisingandadvantageousway.Foriandardexerciseforstudeionofimportahaturallybytakingarbitrarybersofunitmodulus（i.e.r=1），andgpbularandthenpolarates.Equatiwoformsoftheaherigoion.
&hesameinpives:
&ively，thepolarformforultiplibederivedusirigoriulas.Ihatwehavestatedhere，withoutproof，formultiplipolarformisusuallyfirstderivedfrularformbyusingtrigoriulas.
bersandmatrices
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