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Muesquiteeasilynowastheuseofbersrevealsabetweeialorpowerfundtheseemirigoris.Withoutpassingthroughtheportalofferedbythesquarerootofmiheaybeglimpsed，butheso-calledhyperbolisarisefromtakingwhatareknownastheevenandoddpartsoftheexpoialfun.Torititytheredsoi，exceptperhapsfn，involvingthesehyperbolis.Thisbeveriedeasilyinanyparticularcase，buttheionremainsastowhyitshouldhappenatall.Whyshouldthebehaviourofoneclassoffunsbesoirroredinanotherediamanner，andofsuchdifferentcharacter?Resolutioeryisbywayoftheformulaeiθ=θ，whichshooialandtrigorisareintimatelylihuseoftheimaginaryuniti.Ohisisrevealed（foritissurprisingandisbynomeansobvious），itbeesclearthatresultsalongthelinesdescribedareihrcalsusieriohisequatioingrealandimaginaryparts.Withouttheformula，however，itallremainsamystery.
bersandmatrices
&usexaminesomecesoftherevelationthatmultiplibyirepresentsarotatiharightahetreoftheateplane.Ifz=x+iy，wehavethroughexpasaiplisthati（x+iy）=-y+ix，sothatthepoint（x，y）istakento（-y，x）uhisrotation;seeFigure15.Inthislibyiberegardedasonpoihisoperatiohespecialpropertythatforanytwopointszandwandanyrealnumbera，wehavei（z+w）=iz+iw，andi（a>
Moreover，ifwemultiplyarealnumberabyaberx+iy，wegeta（x+iy）=ax+i（ay）.Intermsofpointsintheplexplahat（x，y）ismovedto（ax，ay），ortowriteitanotherway，a（x，y）=（ax，ay）.
15.Multiplibyirotatesaberbyarightangle
Thekihatewopropertiesareknownaslinearandareofparamountimportahroughoutallmathematics.Here，Iwishonlytodrawtoyourattentiohattheeffectofsuoperatioermisathetwopoints（1，0）and（0，1），forletussupposethatL（1，0）=（a，b）andL（0，1）=（c，d）.Thenforanypoint（x，y）wehave（x，y）=x（1，0）+y（0，1），andsousiiesofaliioain:
L（x，y）=L（x（1，0）+y（0，1））=xL（1，0）+yL（0，1）=
=x（a，b）+y（c，d）=（ax，bx）+（cy，dy）=（ax+cy，bx+dy）.
Thisinformationmaybesummarizedbywhatisknownasamatrixequation:
Herewehavedraleofmatrixmultipli，whidicateshoerationiscarriedoutirixisjustaregulararrayofrowsandbers.Matrices，however，representanotherkindoftwo-dimensionalnumericalobjed，whatismore，theypervadenearlyallofhighermathematics，bothpureaheyrepresentawhebra，andmuathematicsstrivestorepresehroughmatrices，sousefulhavetheyprovedtobe.Twomatriceswiththesamenumberofrowsandthesamenumberofherareaddedery:forexample，tofihesedrowandthirdoftwomatrices，wesimplyaddthediriesiriquestion.Itismatrixmultipli，hivesthesubjeeortantcharadhowitisductedhasemergedofitsownathepreviousexample–eatryimatrixisformedbytakiproductofarowofthefirstmatrixwithanoftheseeaningthattheentryisthesumofthedingprodutherowofthefirstmatrixisplatopofthenofthesed.
MatricesfollowalltheusuallawsofalgebraexutativityofmultiplieaningthatfortwomatridBitisruethatAB=BA.However，matrixmultipliisassociative，meaningthatproduylengthmaybewrittenunambiguouslywithouttheneedf.
&ransformationsoftheplaypicallyrotationsabiiohroughtheisandtrasabin，andsocalledshears（），whichmovepointsparalleltoafixedaxisbyanamountproportioahataxisinamannersimilartotheagesofabookslidepastoher.Ahesetransformatioedbymultiplyingalloftherelevahertorevealasihathasthesameasallthosetransformationsaturn.Therowsoftheresultantmatrixaresimplytheimagesofthetwopoints（1，0）and（0，1），aswesawabove，knownasbasisvectors.
ItisnownaturaltolookatthematrixJthatrepresentsananticlockwiserotatileabinasitshouldmimicthebehaviouremultiplybytheimaginaryuniti.Si（1，0）istake（0，1）bytherotationandsimilarlythepoint（1，0）movesto（-1，0），thesetwovetherowsofourmatrixJ.TheresultJwillbeamatrixthathasthegeometriceffectph2×90°=180°abiethisbelowbymatrixmultipli.TofihebhtentryofJ2wetakethedotproductofthesedrowandsen，whichgives（-1）×1+0×0=-1+0=-1.Thepletecalhasthefollowingoute:
&rixIwithrows（10）and（01）istheidentitymatrix，socalledasitactslikethehatwhenmultipliedbyarixAtheresultisA.Thematrix-I，whichrepresentsafullhalfturnrotationabin，doesbehavelike-1inthat（-I）2=I.TheupshotofallthisisthatthematricesaI+bJ，whereaandbarerealhfullymimibersa+biwithrespecttoadditionandmultiplidsogiveamatrixrepresentationofthebereld.Thematrixdiypibera+biis
&ricesthatrepresentthebersdoutewitho，aswasmentiohisdenerallyapplytoallmatrixproduotherwayinwhimmatrimisbehaveisthatnotallofthemverted’.FormostsquarematricesA（amatrixwithequalnumbersofrowsandns），wemayndaurixBsuchthatAB=BA=I，theidentitymatrix.Theexisteheirixhoweverdependsuponasinglenumberassociatedwithasquarematrixknownasitsdeterminahisisasumofsigsformedbytakiryfromeadnofthearray.Forthetypical22matrixarrayasintrodupage118，thedeterminantisthenumber△=adbc.Determinantshavemanyusesandagreeableproperties.Forinstastheareascalefactoroftheatrixtransformation:ashapeofareaawillbetraooneofarea△awhenundergoingatransformationbythatmatrix（andif△isheshapealsoesaree，reversingtheiion）.Whatismore，thedeterminaoftwosquarematricesistheproductofthedeterminantsofthosematrices.AsquarematrixAwillhaveainthecasewhere△=0，inwhichcaseitwillerminantetricallytoadegeransformationwhereareasarecollapsedbythematrixtoguresofzeroareasuchasalioreve.
Forthematrixofaberz=a+bi，we△=a2+b2，whieverzeroexz=0–butofcoursethenumber0neverhadareciprocalbefore，ahethewiderarenaofthebers.Thisdoeshoweverthateverynon-zeroberpossessesamultipliverse.
&aheedgeofthevastworldsebra，representationtheory，andappliulti-dimensionalcaldthisisogofurther.However，thereadershouldbeawarethatmatricesapplytothreedimensionsaon-dimensioypiatrices.Althoughthearraysbeelargerandmoreplicated，thematricesthemselvesyetremaintwo-dimensionalnumericalobjects.
heplexplane
Thefieldplexnumbersispleteintwoimportantways.Aninnitesequenbersinwhichthetermstoeversmallercirclesofradiusthatapproaches0isvergent.Asequenbersapproachesalimitingber.Thisisalsotrueoftherealionals–thesuccessivedecimalapproximationstoanyirrationalasequeioapproachalimitoutsideoftherationals.Moreover，plete（orthealgebraisethatitbeshoolyionp（z）=a+bz+=0hasions，z1，z2…zn，whiallowsp（z）itselftobefullyfactorizedasp（z）=（z-z1）（z-z2）…（z-zn）.
Thisaunheberslargelyobviatetheheemfurtherbeyondtheplexplaisnotpossibletostruaugmeemthatsdalsoretainsallthenormallawsofalgebra.Moreover，thereareoeretainmuchalgebraicstructureatall，thesebeiernioonions.Althoughtheiruseisnotnearlysowidespreadasthatofthebers，thequaterowork，forexample，inthree-dimensionalputergraphics.Theos，whibethoughtofaspairsofquaternions，lalytheutativepropertybutalsotheassociativepropertyofmultipli.
Aquaternionisaheformz=a+bi+cj+dk，wherethefirstparta+biisanordinaryberaernionunitsjandkalsosatisfyj2=k2=-1.Iodomultipliwithquateroknowhowtheunitsmultiplywithohisisdetermiherulesij=k，jk=i，ki=jbutthereversedproductscarrytheoppositesign，sothat，forexample，ji=-k（iheseproductsmaybederivedfromthesiioion:ijk=-1）.Thequaternionsthenformanenhancedalgebraicsystemthatsatisfiesallthelawsofalgebraexutativityofmultipli，duetothesigiohereversedproducts.Thecyofthesystemohroughrepresentationby2×2matrices，butthistimelexrathertharies.Thenumber1isoifiedwithI，theidentitymatrixbuttheunitsi，j，aheirmatrixterparts:
&ypicalquaternionzhasasitsmatrix:
Thisrepresentatioernionsbymatriotunique，however，aherepresentationofthebersbymatricesalsohasequivaleives.Moreover，itispossibletorepreseernionswithoutemployingbersbutonlyattheexpenseofusirixarrays:thequaternionsberepreseain4×4matriceswithonlyrealries.
Newkindsofheextensionsofoldsystemshaveeabhtheoperformcalstheouteofwhiotbeaodatedbytheemasitstood.Everycivilizatiohtheumbers，butcalsinvmeofras，thoseinvolvioives，andasPythagorasdiscovered，thoseinvolvihsleadtoirratiohoughaveryaiohatnotallterscouldbedealtwithusingwholeheirratioswasasubtlediscoveryofadeeperkind.Asscebecamemoresophisticated，theemsrequiredhaveureiodealwiththeseadvasdonotgenerallylooktoewemsinawhimsicalfashiorary，theyareielyainglyatfirst，todealroblems.Forexample，althoughfirstihe19thtury，matricesaroseirresistiblyinquantummetheearly20thturywhenphysiteredaquantityoftheformq=AB-BAthatwaszero.Inanyutativesystemofnumbers，qwouldofcoursebe0，sothenumericalobjeeededherewerenotofakibefore:theywerematrices.
&hattheworldofmathematidphysibertypes.Althoughtherearekindsofmehisbook，thehatareohroughoutmathematidsotogeagreatdealsihalfofthe20thtury.
&ions，hourmathematicalballoos.WebeganatgroundlevelandhaveasdedtowhereIhopethereadergazedownupoheridmysteriousworldofnumbers.
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