COMPLEX NUMBERS INDEX

Transcription

1 COMPLEX NUMBERS INDEX. The hstory of the complex numers;. The mgnry unt I ;. The Algerc form;. The Guss plne; 5. The trgonometrc form;. The exponentl form; 7. The pplctons of the complex numers. School yer Clss G Techer : Rosell Ntln

2 . The hstory of complex numers Complex numers, of completely dfferent nture from the numers tht we re ccustomed to know, llow gettng to the soluton of n pprently mpossle prolem, lke the prolem of the extrcton of the squre root of negtve numer. The most ncent text, where the negtve numer s root s cted, s y Eron of Alessndr n the frst century A.D. But the prolem out the extrcton of squre root of negtve numer ppered more frequently n the study of the equtons of the second nd thrd degree y Nccolò Trtgl nd Grolmo Crdno, two fmous lgersts of the Renssnce. These two lgersts found formul to determne the solutons of the thrd degree px q 0 equton, whch hs the followng form. x In ths formul, the clculton of the squre root of negtve numer ws often essentl to fnd rel solutons. Ths fct ws extrordnry t tht tme ecuse noody eleved n the rel exstence of these numers, whch were only regrded s some trcks used to rrve t the soluton of the equtons. In the defnton of the complex numers, we ntroduce prtculr unt, clled mgnry unt, ndcted s, whch hs the followng property; Thnks to the ntroducton of the mgnry unt, t s possle to clculte the squre root of ny negtve numer, wrtng t s the product of ts opposte multpled y. An mgnry numer s the product of rel numer multpled y the mgnry unt; the term mgnry ws used, for the frst tme, y Crteso. In the XVIII century, gret mthemtcns worked on the complex numers theory, n prtculr De Movre(79), who found the fmous formul for the clculton of the powers: n ( sen) n n nd Euler (78), who ntroduced the exponentl notton for the complex numers: e ρ( ) ρ But t ws only thnks to the ntroducton of the geometrc nterpretton of the complex numers tht mthemtcns ccepted these numers. Snce the second hlf of the seventeenth century, the mthemtcns John Wlls hd nterpreted mgnry numers on perpendculr strght lne; however, noody hd hd

3 the de to consderte the rel prt nd the coeffcent of the mgnry prt of the complex numer, s the coordntes of the ponts of plne. It ws only n the 797 tht the Dnsh mthemtcn Cspr Wessel hd ths ntuton. However, hs work remned unknown lke tht one of the French mthemtcns Jen Roert Argnd n the 80. It ws Guss, out thrty yers lter, who gve full dgnty to complex numers, wth ther geometrc nterpretton on plne, clled exctly Gussn plne. Fredrch Guss. The mgnry unt To form complex numers you hve to refer to the mgnry unt, wth the symol tht t s chrctered y the followng property :. In order to preserve the forml property of the opertons wth ths new numer you hve to consder lso ( ). Ths mens thn we cn dentfy the squre root of - wth ± : ± Now t s possle to clculte the squre root of negtve numer wrtng ths s product of ts opposte y -. EXAMPLE ( ) ± Numer s ± re mgnry numers. An mgnry numer s the product etween rel numer nd the mgnry unt. nd Wth the mgnry numers t s possle to resolve grde equton s x eng nd of opposte sgn.

4 EXAMPLE x 5 0 It hs got s soluton x ± 5 tht s x ± 5. To operte wth mgnry numers, you hve to use the mgnry unt () such s letter of lgerc computton rememerng tht. EXAMPLES ) ) ( ) ( ) ) 0 / 5 ) () ( ) ( ) ( ) ( ) The propertes of mgnry numers re: The powers of re cyclc of perod ; n fct we hve ; ; nd so on. The sum nd dfference of two or more mgnry numers s lso n mgnry numer. The product nd quotent of two mgnry numers s rel numer. The result of the sum or of the dfference etween rel numer nd n mgnry sn t rel numer or n mgnry one.. The Algerc form The defnton of complex numer s ech equton such s:, R. Where s the rel prt of the complex numer, s the coeffcent of the mgnry prt, C s the set of complex numers. f 0 nd 0 the numer you otn s mgnry; f 0 the numer s rel; two complex numers re equvlent f oth the mgnry prt nd the rel prt re equvlent: ; two complex numers re complex conjugte f they hve got opposte mgnry prts: conjugte -; two complex numers re opposte f nd re opposte: s opposed to --. Opertons wth complex numers To operte wth complex numers we use the sme rules to operte wth polynomls, ug -.

5 Sum ( ) (c d) ( c) ( d) Exmple : (8 ) ( - ) (8 ) ( - ) Multplcton ( )(c d) c c d d (c - d) (c d) Exmple: ( )( 9) 7 5 ( - 5) ( 7) - 5 Dvson c d * c d c d ( )( c d) c d c d ( c d) c d Exmple: * ( )( ) (8 8 ) 8 50 Resoluton of the equtons n C Wth complex numers we cn solve nd degree equtons lso wth negtve dscrmnnt. If the solutons of nd degree equtons re not rel, they're conjugte complex numers. The solutons of nd degree equtons my e: - dfferent nd rel f Δ>0 - equvlents nd rel f Δ0 - complex conjugte f Δ<0 Fundmentl theorem of lger Ech N degrees lgerc equton dmts, n C, N solutons f ech of those solutons s counted wth ts multplcty.. The Gussn plne Guss s plne s Crtesn plne where we cn represent the complex numers s prs of rel numers (,) where s the rel prt nd s the mgnry prt : so the P pont of coordntes (,) represents the complex, tht cn e represented y the OP vector.

6 On the Gussn plne we cn lso represent the ddton of complex numers In the pcture elow we cn see the ddton of complex numers Z 5 nd WG Z W 7 Z ZW W 5. The trgonometrc form On the Crtesn plne we see tht the complex numer hs two coordntes: A s represented on the xs of the scses nd B s represented on the xs of ordntes. The segment tht strts from the centre nd goes to the complex numer s vector tht we cll ρ.so we cn sy tht (ρ, ). We cn conclude tht / ρ nd / ρ we hve: ρ ρ ρ( ).

7 The notton ρ( ) s clled trgonometrc form of the complex numer. ρ s the module of nd t s shown lso y the symol, s the rgument or the nomly. How we cn pss from lgerc form to trgonometrc form ρ ρ ρ tn Multplcton The product of the two complex numer hs s module the product of the modules nd s n rgument the ddton of the rguments. Formul: * ( ) ( ) [ ] ρ ρ * Exmple: 5 5 Dvson The quotent of two complex numers s the complex numer tht hs s module the quotent of modules nd s n rgument the dfference of the rguments of the gven numers. ( ) ( ) [ ] p p Formul: Exmple: Power Formul: ( ) [ ] ( ) ρ ρ n n n n 9 Exmple:

8 . The exponentl form Frst of ll we defne the exponentl functon wth mgnry exponent e Then pplyng the lw of the powers we cn wrte tht e e ( ). If 0 we otn s prtculr cse the exponentl functon wth rel exponent: e ( 0 0) All the forml lws of the powers pply lso to ths type of exponentl functon. Euler found some mportnt reltons etween the exponentl nd trgonometrc functons whch were clled Euler s formuls. If we consder 0 we otn the frst two formuls: e nd e If we dd nd sutrct them we otn the two other reltons: e e e e e These formuls crete connecton etween the gonometrc form of complex numers nd the defned exponentl functon. In fct, pplyng Euler s frst formul, we cn wrte the complex numer ρ( ) s ρe Ths s the exponentl form of the complex numer wth module ρ nd rgument. Opertons such s the multplcton, power nd dvson of complex numers n exponentl form re esly deduced from the defnton. ( If ρ e e ρ e, the product wll e e ) ρ ρ, the quotent of e ρ 0 wll e ρ ( e ) n n n nd the power wll e ρ e. ρ EXAMPLES Algerc form Trgonometrc form ( sen ) Exponentl form e Algerc form Trgonometrc form ( sen ) Exponentl form e

9 7. The pplctons of the complex numers Complex numers fnd pplctons n vrous felds such s Mthemtcs, Physcs nd Engneerng. MATHEMATICS complex numers re used n Complex Anlyss, n Geometry or n other secondry pplctons, s: Numer theory: ths theory uses the Complex Anlyss to fce prolems on the complex numers Improper ntegrls: some of these cn e solved wth the theorem of the resdues of Complex Anlyss Dfferentl equtons: they re solved fndng the complex roots of polynoml ssocted to the equton Frctls: frctl s geometrc oject tht repets tself n ts structure eqully on dfferent scles, so t doesn't chnge spect. Frctls strt wth complex numer. Ech numer produced gves vlue for ech pxel on the screen. Common frctls re sed on the Jul Set nd the Mndelrot Set. Jul Set Mndelrot Set PHYSICS they re used n some sectors s: Flud dynmcs: complex numers re used for descrng the potentl flow n two dmensons Quntum mechncs: n ths feld, complex numers re essentl ecuse the theory s developed n Hlert spce of nfnte dmenson derved y C, complex numers set Reltvty: complex numers re used ecuse some formuls of the metrc spce ecome smpler f the temporl vrle s supposed s n mgnry vrle. Generl Reltvty

10 ENGINEERING complex numers re used n the nlyss of the sgnls nd n ll the felds where the perodc sgnls re studed. In Electrc Engneerng nd Electroncs they re used for pontng out voltge nd current. Wth the complex numers you cn sum up the nlyss of resstnce, cpcty nd nductnce n only one entty, clled mpednce. They cn, esdes, express some reltons tht tke nto consderton the frequences nd the ehvour of the components, ccordng to frequency.