Chapter 8 Numbers but not as we knowthem（第1页）

Chapter8hem

Realandbers

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。Itturnsoutthatthepolaratesoftheirproductzwtakeonasimpleandpleasingform。Theruleofbinationbeexpressedlyine: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。

Muesquiteeasilynowastheuseofbersrevealsabetweeialorpowerfundtheseemirigoris。Withoutpassingthroughtheportalofferedbythesquarerootofmiheaybeglimpsed，butheso-calledhyperbolisarisefromtakingwhatareknownastheevenandoddpartsoftheexpoialfun。Torititytheredsoi，exceptperhapsfn，involvingthesehyperbolis。Thisbeveriedeasilyinanyparticularcase，buttheionremainsastowhyitshouldhappenatall。Whyshouldthebehaviourofoneclassoffunsbesoirroredinanotherediamanner，andofsuchdifferentcharacter?Resolutioeryisbywayoftheformulaeiθ=θ，whichshooialandtrigorisareintimatelylihuseoftheimaginaryuniti。Ohisisrevealed（foritissurprisingandisbynomeansobvious），itbeesclearthatresultsalongthelinesdescribedareihrcalsusieriohisequatioingrealandimaginaryparts。Withouttheformula，however，itallremainsamystery。