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Chapter 8 Numbers but not as we knowthem（第2页）

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。