千米小说网

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

Chapter8hem

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

&usexaminesomecesoftherevelationthatmultiplibyirepresentsarotatiharightahetreoftheateplane。Ifz=x+iy，wehavethroughexpasaiplisthati（x+iy）=-y+ix，sothatthepoint（x，y）istakento（-y，x）uhisrotation;seeFigure15。Inthislibyiberegardedasonpoihisoperatiohespecialpropertythatforanytwopointszandwandanyrealnumbera，wehavei（z+）=a（iw）。Moreover，ifwemultiplyarealnumberabyaberx+iy，wegeta（x+iy）=ax+i（ay）。Intermsofpointsintheplexplahat（x，y）ismovedto（ax，ay），ortowriteitanotherway，a（x，y）=（ax，ay）。