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Itistemptingthallthisfrettingaboutparticularequationsandsimplydeclarethatwealreadyknowwhattherealheyaretheofallpossibledecimalexpansions，bothpositiveaheseareveryfamiliar，inpractiowhowtousethem，andsowefeelonsafegrouilweasksomeverybasiaiureofhatyouadd，subtract，multiply，a，forexample，howareyousupposedtomultiplytwoinfinitendecimals?Wedependondecimalsbeihsothatyou‘startfrht-hathereisnosugwithaninfinitedecimalexpansion.Ite，butitisplicatedbothintheoryandinpraumbersystemwhereyletoexplainholydoesisfactory.
Youmayioionsraisedaboryoumaygrowimpatientwithalltheiioobemakingtroubleforourselveswhenpreviouslyallwassmoothsailing.Thereisaseriouspoihematisappreciatethat，whehematicalobjetroduced，itimportanttostructthemfromkicalobjects，theway，foriioofaspairsers.Inthisway，wemaycarefullybuilduptherulesthatgovereemandkafoundatiowillebatuslater.Forexample，therapiddevelopmentofcalculus，whichwasbornoutofthestudyofmotioospectacularresults，suchasprediovemes.Houlationofihingsasiftheywereimesprovidedamazinginsightsaimespatentingyourmathematicalsystemsonarmfoundation，wehowtotellthedifferenpractice，mathematisoftenindulgein‘formal’manipulatiooseeifsheoffieisworthyofattebeprorouslybygoingbacktobasidbyihathavebeeablishedearlier.
ThisiswhyJuliusDedekind（1831-1916）tookthetroubleofformallystrugtherealembasedoisoasDedekindcutsofthereallimathemati，however，tosuccessfullydealwiththedilemmacausedbytheexisteionalnumberswasEudoxusofidus（fl380BC）whoseTheoryofProportionsallowedArchimedestousetheso-calledMethodofExhaustiorouslyderiveresultsonareasandvolumesofcurvedshapesbeforetheadventofcale1，900yearslater.
Thefiheheimaginaryunit
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，weesubtrabersbysubtragtherealandimaginarypartssothat，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
Multiplihasageometriterpretationthatisrevealedifwealterouratesystemfromtheularatestopolarates.Inthissystem，apointzisonspecifiedbyanorderedpairofnumbers，riteas（r，θ）.Thehedistanceofourpointzfrihistextthepole）.Thereforerisaivequantityandallpointswiththesamevalueofrformacircleofradiusrtredatthepole.Weusethesedateθtospethiscirclebytakiheangle，measuredinananti-clockwisediretherealaxistothelihenumberriscalledthemodulus（pluralmoduli）ofz，whiletheangleθiscalledtheargumentofz.
Supposewobers，zandw，whosepolaratesare（r1，θ1）and（r2，θ2）respectively.ItturnsoutthatthepolaratesoftheirproductzwtakeonasimpleandpleasingforTheruleofbinationbeexpressedlyine:themodulusoftheproductzwistheproduoduliofzandw，whiletheargumentofzwisthesumumentsofzandw.Insymbols，zolarates（r1r2，θ1+θ2）.Themultiplioftherealnumbersissubsumeduhismeneralwayoflookingatthings:apositiverealnumberr，forinstance，haspolarates（r，0），aiplybyaheresultistheexpected（rs，totherealnumberrs.
Muchmoreofthecharaultipliplexnumbersisrevealedthroughthisrepresentation.Thepolaratesoftheplexunitiaregivenby（1，9lesarenreesinsuchthenaturalmathematiitoftheradian:thereare2πradiansiaturnofoneradiaomovialongthecirceoftheuredatthein.Oneradianisabout57.3°.）Ifwenolexnumberz=（r，θ）andmultiplybyi=（1，90°），wefindthatzi=（r，θ+90°）.Thatistosay，multiplibyidstorotatiharightaheplexplaherwords，therightamostfualgeometricidea，berepresentedasanumber.
&heeffectofaddingormultiplyingbyaberzosinagiveheplexplauredgeometrically.Imagineanyregionyoufantheplaoeverypointinsideyion，wesimplymoveeatthesamedistanddireihearrow，orvectorasweoftencallit，represehatistosaywetraosomeotherpositionintheplatheshapeandsizeareexatained，asisitsattitude，bywhitheregionhasnoatioion.Multiplyiinyionbyz=（r，θ）hastwoeffects，however，onecausedbyraherbyθ.Themodulusofeatintheregionisincreasedbyafactorr，soallthedimensionsionareincreasedbyafactorofralso（soitsareaismultipliedbyafactorofr2）.Ofcourse，ifr〈1thenthis‘expansioerdescribedasatraasthenewregionwillbesmallerthantheinal.Theregionwill，however，maintainitsshape-foririangleismappedontoasimilartriahesameanglesasbefore.Theeffectofθ，aslaiorotatetheregihaiclockwiseaboutthepole.Theheninmultiplyingallpointsionbyzistoexpandandrionaboutthepole.Thenewregionwillstillhavethesameshapeasbeforebutwillbeofadiffereerminedbyr，andwillbelyingiitudeasdetermiionangleθ.
Furtherces
Thereareahostofappliplexheelemeheiweeangularandpolarrepresentatiooplayinasurprisingandadvantageousway.Foriandardexerciseforstudeionofimportahaturallybytakingarbitrarybersofunitmodulus（i.e.r=1），andgpbularandthenpolarates.Equatiwoformsoftheaherigoion.
&hesameinpives:
&herealandimaginarypartsofthetwoversionsofthisothenpainlesslyyieldsthestandardanglesumformulasory:
&ively，thepolarformforultiplibederivedusirigoriulas.Ihatwehavestatedhere，withoutproof，formultiplipolarformisusuallyrstderivedfrularformbyusingtrigoriulas.
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