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ISBN0-691-02795-1The,,,,most,,,,precise,,,,explanation,,,,is,,,,to,,,,say,,,,that,,,,although,,,,the,,,,complex,,,,field,,,,,defined,,,,as,,,,[x]/(x2,,,,+,,,,1),,,,,(see,,,,complex,,,,number),,,,is,,,,unique,,,,up,,,,to,,,,isomorphism,,,,,it,,,,is,,,,not,,,,unique,,,,up,,,,to,,,,a,,,,unique,,,,isomorphism,,,,,,,,there,,,,are,,,,exactly,,,,2,,,,field,,,,automorphisms,,,,of,,,,[x]/(x2,,,,+,,,,1),,,,which,,,,keep,,,,each,,,,real,,,,number,,,,fixed:,,,,the,,,,identity,,,,and,,,,the,,,,automorphism,,,,sending,,,,x,,,,to,,,,xTaking,,the,,square,,root,,of,,both,,sides,,gives^,,,"The,,,Penguin,,,Dictionary,,,of,,,Curious,,,and,,,Interesting,,,Numbers",,,by,,,David,,,Wells,,,,Page,,,26Substituting,,,,either,,,,of,,,,these,,,,results,,,,into,,,,the,,,,equation,,,,2xy,,,,=,,,,1,,,,in,,,,turn,,,,,we,,,,will,,,,get,,,,the,,,,corresponding,,,,result,,,,for,,,,yare,,solutions,,to,,the,,matrix,,equationThese,,problems,,are,,avoided,,by,,writing,,and,,manipulating,,expressions,,like,,i7,,,rather,,than,,7The,,two,,square,,roots,,of,,i,,in,,the,,complex,,plane,,Multiplying,,,a,,,complex,,,number,,,by,,,i,,,gives:Your,,,,browser,,,,will,,,,redirect,,,,to,,,,your,,,,requested,,,,content,,,,shortly.Also,A,,similar,,issue,,arises,,if,,the,,complex,,numbers,,are,,interpreted,,as,,2,,,,2,,real,,matrices,,(see,,matrix,,representation,,of,,complex,,numbers),,,because,,then,,bothFurther,,,reading[edit]Site,,,Map,,,Search,,,Terms,,,Advanced,,,Search,,,Orders,,,and,,,Returns,,,Contact,,,Us,,,RSS,,,About,,,Musicminusone.com,,,Music,,,Minus,,,One,,,,the,,,company,,,with,,,the,,,unique,,,,unforgettable,,,name,,,,was,,,begun,,,in,,,1950,,,,with,,,a,,,first,,,recording,,,devoted,,,to,,,Schubert’s,,,Trout,,,Quintet,,,,with,,,all,,,five,,,instruments,,,omitted,,,in,,,five,,,different,,,MMO’s,,,This,,,release,,,received,,,a,,,full,,,page,,,review,,,in,,,The,,,New,,,York,,,Times,,,About,,,Music,,,Minus,,,One,,,Store,,,Information,,,Buyer's,,,Guide,,,Return,,,Policy,,,Contact,,,Us,,,Redeeming,,,reward,,,Points,,,Us,,,Customer,,,Service,,,1,,,Frequently,,,Asked,,,Questions,,,2,,,News,,,&,,,Events,,,3,,,Promotions,,,4,,,Delivery,,,&,,,Returns,,,Information,,,Company,,,Info,,,Call,,,Us,,,914-592-1188,,,Toll,,,Free1800-669-7464,,,csmmogroup.com,,,salesmmogroup.com,,,,,,Newsletter,,,Subscribe,,,,,,2011,,,Music,,,Minus,,,OneReal,,,,numbers,,,,lie,,,,on,,,,the,,,,horizontal,,,,axis,,,,,and,,,,imaginary,,,,numbers,,,,lie,,,,on,,,,the,,,,vertical,,,,axis,,,,External,,,,links[edit]In,,the,,disciplines,,of,,electrical,,engineering,,and,,control,,systems,,engineering,,,the,,imaginary,,unit,,is,,often,,denoted,,by,,j,,instead,,of,,i,,,because,,i,,is,,commonly,,used,,to,,denote,,electric,,currentSince,,,the,,,equation,,,is,,,the,,,only,,,definition,,,of,,,i,,,,it,,,appears,,,that,,,the,,,definition,,,is,,,ambiguous,,,(more,,,precisely,,,,not,,,well-defined)^,,,To,,,find,,,such,,,a,,,number,,,,one,,,can,,,solve,,,the,,,equation,,,(x,,,+,,,iy)2,,,=,,,i,,,where,,,x,,,and,,,y,,,are,,,real,,,parameters,,,to,,,be,,,determined,,,,or,,,equivalently,,,x2,,,+,,,2ixy,,,,,,y2,,,=,,,i(incorrect)However,,,,,no,,,,ambiguity,,,,results,,,,as,,,,long,,,,as,,,,one,,,,or,,,,other,,,,of,,,,the,,,,solutions,,,,is,,,,chosen,,,,and,,,,labelled,,,,as,,,,"i",,,,,with,,,,the,,,,other,,,,one,,,,then,,,,being,,,,labelled,,,,as,,,,ii,,,,raised,,,,to,,,,the,,,,power,,,,of,,,,i[edit]The,,,,square,,,,root,,,,of,,,,i,,,,can,,,,be,,,,expressed,,,,as,,,,either,,,,of,,,,two,,,,complex,,,,numbers[nb,,,,1]The,,,powers,,,of,,,i,,,repeat,,,in,,,a,,,cycle,,,expressible,,,with,,,the,,,following,,,pattern,,,,where,,,n,,,is,,,any,,,integer:In,,electrical,,engineering,,and,,related,,fields,,,the,,imaginary,,unit,,is,,often,,denoted,,by,,j,,to,,avoid,,confusion,,with,,electric,,current,,as,,a,,function,,of,,time,,,traditionally,,denoted,,by,,i(t),,or,,just,,iAlthough,,,,the,,,,construction,,,,is,,,,called,,,,"imaginary",,,,,and,,,,although,,,,the,,,,concept,,,,of,,,,an,,,,imaginary,,,,number,,,,may,,,,be,,,,intuitively,,,,more,,,,difficult,,,,to,,,,grasp,,,,than,,,,that,,,,of,,,,a,,,,real,,,,number,,,,,the,,,,construction,,,,is,,,,perfectly,,,,valid,,,,from,,,,a,,,,mathematical,,,,standpointFor,,a,,more,,thorough,,discussion,,,see,,Square,,root,,and,,Branch,,pointHigher,,integral,,powers,,of,,i,,can,,also,,be,,replaced,,with,,i,,,1,,,i,,,or,,1:are,,only,,valid,,for,,real,,,non-negative,,values,,of,,a,,and,,bComplex,,,,plane,,,,Imaginary,,,,number,,,,Multiplicity,,,,(mathematics),,,,Root,,,,of,,,,unity,,,,Unit,,,,complex,,,,number,,,,[2],,,Compare,,,,,,You,,,have,,,no,,,items,,,to,,,compareMatrices[edit]Nahin,,,,Paul,,,J,,and,,,,The,,,,Python,,,,programming,,,,language,,,,also,,,,uses,,,,j,,,,to,,,,mark,,,,the,,,,imaginary,,,,part,,,,of,,,,a,,,,complex,,,,numberBecause,,,,the,,,,real,,,,and,,,,imaginary,,,,parts,,,,are,,,,always,,,,separate,,,,,we,,,,regroup,,,,the,,,,terms:,,,,x2,,,,,,,,y2,,,,+,,,,2ixy,,,,=,,,,0,,,,+,,,,i,,,,and,,,,by,,,,equating,,,,coefficients,,,,,real,,,,part,,,,and,,,,real,,,,coefficient,,,,of,,,,imaginary,,,,part,,,,separately,,,,,we,,,,get,,,,a,,,,system,,,,of,,,,two,,,,equations:,,,,x2,,,,,,,,y2,,,,=,,,,0,,,,2xy,,,,=,,,,1References[edit](This,,,is,,,equivalent,,,to,,,a,,,90,,,counter-clockwise,,,rotation,,,of,,,a,,,vector,,,about,,,the,,,origin,,,in,,,the,,,complex,,,plane.)Similarly: 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