TREATISE OF PLANE GEOMETRY THROUGH GEOMETRIC ALGEBRA

Transcription

1 TRETISE OF PLNE GEOMETRY THROUGH GEOMETRI LGER Ramon Gonzálz alt

2 TRETISE OF PLNE GEOMETRY THROUGH GEOMETRI LGER Ramon Gonzálz alt Th gomtric algbra, initiall iscor b Hrmann Grassmann (89-877) was rformulat b William Kingon liffor ( ) through th snthsis of th Grassmann s tnsion thor an th quatrnions of Sir William Rowan Hamilton (85-865). In this wa th bass of th gomtric algbra wr stablish in th XIX cntur. Notwithstaning, u to th prmatur ath of liffor, th ctor analsis a rmak of th quatrnions b Josiah Willar Gibbs (89-9) an Olir Haisi (85-95) bcam, aftr a long contrors, th gomtric languag of th XX cntur; th sam ctor analsis whos baut attract th attntion of th author in a cours on lctromagntism an l him -bing still unrgrauat- to ra th Hamilton s Elmnts of Quatrnions. Mawll himslf alra appli th quatrnions to th lctromagntic fil. Howr th quations ar not writtn so nicl as with ctor analsis. In 986 Ramon contact Josp Manl Parra i Srra, tachr of thortical phsics at th Unirsitat arclona, who acquaint him with th liffor algbra. In th framwork of th summr courss on gomtric algbra which th ha taught for grauats an tachrs sinc 994, th plan of writing som books on this subjct appar in a r natural mannr, th first sampl bing th Tractat gomtria plana mitjançant l àlgbra gomètrica (996) now out of print. Th goo rcption of th rars has ncourag th author to writ th Tratis of plan gomtr through gomtric algbra (a r nlarg translation of th Tractat) an publish it at th Intrnt sit writing it not onl for mathmatics stunts but also for an prson intrst in gomtr. Th plan gomtr is a basic an as stp to ntr into th liffor-grassmann gomtric algbra, which will bcom th gomtric languag of th XXI cntur. Dr. Ramon Gonzálz alt (964) is high school tachr of mathmatics sinc 987, fllow of th Socitat atalana Matmàtiqus ( an also of th Socitat atalana Gnomònica (

3 TRETISE OF PLNE GEOMETRY THROUGH GEOMETRI LGER Dr. Ramon Gonzálz alt Mathmatics Tachr I.E.S. Pr alrs, ranola l Vallès I

4 To m son Pr, born with th book. Ramon Gonzálz alt ( rgonzal@tllin.s ) This is an lctronic ition b th author at th Intrnt sit ll th rights rsr. n lctronic or papr cop cannot b rprouc without his prmission. Th rars ar authoris to print th fils onl for his prsonal us. Sn our commnts or opinion about th book to ramon.gonzalzc@campus.uab.s. ISN: First atalan ition: Jun 996 First English ition: Jun to Jun II

5 PROLOGUE Th book I am so plas to prsnt rprsnts a tru innoation in th fil of th mathmatical iactics an, spcificall, in th fil of gomtr. as on th long nglct iscoris ma b Grassmann, Hamilton an liffor in th nintnth cntur, it prsnts th gomtr -th lmntar gomtr of th plan, th spac, th spactim- using th bst algbraic tools sign spcificall for this task, thus making th subjct mocraticall aailabl outsi th narrow circl of iniiuals with th high isual imagination capabilitis an th tru mathmatical insight which wr rquir in th abanon classical Euclian traition. Th matrial pos in th book offrs a wi rprtor of gomtrical contnts on which to bas powrful, rasonabl an up-to-at rintrouctions of gomtr to prsnt-a high school stunts. This long-for rintrouctions ma (or bttr shoul) tak aantag of a combin us of smbolic computr programs an th cross isciplinar rlationships with th phsical scincs. Th propos introuction of th gomtric liffor-grassmann algbra in high school (or n bfor) follows rightl from a pagogical principl pos b William Kingon liffor ( ) in his projct of taching gomtr, in th Unirsit ollg of Lonon, as a practical an mpirical scinc as oppos to ambrig Euclian aiomatics:... for gomtr, ou know, is th gat of scinc, an th gat is so low an small that on can onl ntr it as a littl chil. Fllow of th Roal Socit at th ag of 9, liffor also ga a st of Lcturs on Gomtr to a lass of Lais at South Kngsinton an was pl concrn in loping with MacMillan ompan a sris of inpnsi r goo lmntar schoolbook of arithmtic, gomtr, animals, plants, phsics.... Not forign to this proposal ar Fli Klin lcturs to tachrs collct in his book Elmntar mathmatics from an aanc stanpoint an th aic of lfr North Whitha saing that th harst task in mathmatics is th stu of th lmnts of algbra, an t this stag must prc th comparati simplicit of th iffrntial calculus an that th postponmnt of ifficult mis no saf clu for th maz of ucational practic. larl nough, whn th fat of psuo-mocratic ucational rforms, isguis as a back to basic litmotifs, has bn answr b such an acut analsis b R. Noss an P. Dowling unr th titl Mathmatics in th National urriculum: Th Empt St?, th tim ma b ripn for a rappraisal of tru pagogical rforms bas on a ral knowlg, of substanti contnts, rlant for ach iniiual worliw construction. W bli that th introuction of th ital or printial plan, spac an spac-tim gomtris along with its propr algbraic structurs will b a substantial part of a succssful (high) school scintific curricula. Knowing, tlling, larning wh th sign rul, or th compl numbrs, or matrics ar mathmatical structurs corrlat to th human rprsntation of th ral worl ar worth objctis in mass ucation projcts. n this is possibl toa if w larn to stan upon th shoulrs of giants such as Libniz, Hamilton, Grassmann, liffor, Einstin, Minkowski, tc. To this aim this book, offr an opn to suggstions to th whol worl of concrn popl, ma b a most but most aluabl stp towars ths r goo schoolbooks that constitut on of th chrful liffor's aims. Fli Klin, Elmntar mathmatics from an aanc stanpoint. Dor (N. Y., 94)..N. Whitha, Th aims of ucation. MacMillan ompan (99), Mntor ooks (N.Y., 949). P. Dowling, R. Noss, s., Mathmatics rsus th National urriculum: Th Empt St?. Th Falmr Prss (Lonon, 99). III

6 Finall, som wors borrow from Whitha an Russll, that I am sur con som of th pst flings, thoughts an critical concrns that Dr. Ramon Gonzálz has ha in min whil writing th book, an that full justif a work that appars to b quit rmo from toa high school taching, at last in ataluna, our countr. Whr attainabl knowlg coul ha chang th issu, ignoranc has th guilt of ic. Th uncritical application of th principl of ncssar antcnc of som subjcts to othrs has, in th hans of ull popl with a turn for organisation, prouc in ucation th rnss of th Sahara. Whn on consirs in its lngth an in its brath th importanc of this qustion of th ucation of a nation's oung, th brokn lis, th fat hops, th national failurs, which rsult from th friolous inrtia with which it is trat, it is ifficult to rstrain within onslf a saag rag. tast for mathmatics, lik a tast for music, can b gnrat in som popl, but not in othrs.... ut I think that ths coul b much fwr than ba instruction maks thm sm. Pupils who ha not an unusuall strong natural bnt towars mathmatics ar l to hat th subjct b two shortcomings on th part of thir tachrs. Th first is that mathmatics is not hibit as th basis of all our scintific knowlg, both thortical an practical: th pupil is conincingl shown that what w can unrstan of th worl, an what w can o with machins, w can unrstan an o in irtu of mathmatics. Th scon fct is that th ifficultis ar not approach grauall, as th shoul b, an ar not minimis b bing connct with asil apprhn cntral principls, so that th ific of mathmatics is ma to look lik a collction of tach hols rathr than a singl tmpl mboing a unitar plan. It is spciall in rgar to this scon fct that liffor's book (ommon Sns of th Eact Scincs) is aluabl.(russll) 4. n apprciation that liffor himslf ha formulat, in his funamntal papr upon which th prsnt book rlis, rlati to th ushnungslhr of Grassmann, prssing m coniction that its principls will rcis a ast influnc upon th futur of mathmatical scinc. Dpartamnt Física Fonamntal Unirsitat arclona Josp Manl Parra i Srra, Jun 4 W. K. liffor, ommon Sns of th Eact Scincs. lfr. Knopf (946), Dor (N.Y., 955). IV

7 «On man n scon liu, laqull s u qualités oit êtr préféré ans s élémns, la facilité, ou la rigour act. J répons qu ctt qustion suppos una chos fauss; ll suppos qu la rigour act puiss istr sans la facilité & c st l contrair; plus un éuction st rigouraus, plus ll st facil à ntnr: car la riguur consist à ruir tout au princips ls plus simpls. D où il s nsuit ncor qu la riguur proprmnt it ntraîn nécssairmnt la métho la plus naturll & la plus irct. Plus ls principls sront isposés ans l orr connabl, plus la éuction sra rigouras; c n st pas qu absolumnt ll n pût l êtr si on suionit un métho plus composé, com a fait Eucli ans ss élémns: mais alors l mbarras la march froit aisémnt sntir qu ctt riguur précair & forcé n sroit qu improprmnt tll.» 5 [ Sconl, on rqusts which of th two following qualitis must b prfrr within th lmnts, whthr th asinss or th act rigour. I answr that this qustion implis a falshoo; it implis that th act rigour can ist without th asinss an it is th othr wa aroun; th mor rigorous a uction will b, th mor asil it will b unrstoo: bcaus th rigour consists of rucing rthing to th simplst principls. Whnc follows that th proprl call rigour implis ncssaril th most natural an irct mtho. Th mor th principls will b arrang in th connint orr, th mor rigorous th uction will b; it os not man that it cannot b rigorous at all if on follows a mor composit mtho as Eucli ma in his lmnts: but thn th ifficult of th march will mak us to fl that this prcarious an forc rigour will onl b an impropr on. ] Jan l Ron D'lmbrt (77-78) 5 «Elémns s scincs» in Encclopéi, ou ictionair raisonné s scincs, s arts t s métirs (París, 755). V

8 PREFE TO THE FIRST ENGLISH EDITION Th first ition of th Tratis of Plan Gomtr through Gomtric lgbra is a r nlarg translation of th first atalan ition publish in 996. Th goo rcption of th book (now out of print) ncourag m to translat it to th English languag rwriting som chaptrs in orr to mak asir th raing, nlarging th othrs an aing thos ot to th non-euclian gomtr. Th gomtric algbra is th tool which allows to stu an sol gomtric problms through a simplr an mor irct wa than a purl gomtric rasoning, that is, b mans of th algbra of gomtric quantitis insta of snthtic gomtr. In fact, th gomtric algbra is th liffor algbra gnrat b th Grassmann's outr prouct in a ctor spac, although for m, th gomtric algbra is also th art of stating an soling gomtric quations, which corrspon to gomtric problms, b isolating th unknown gomtric quantit using th algbraic ruls of th ctors oprations (such as th associati, istributi an prmutati proprtis). Following Pano 6 : Th gomtric alculus iffrs from th artsian Gomtr in that whras th lattr oprats analticall with coorinats, th formr oprats irctl on th gomtric ntitis. Initiall propos b Libniz 7 (charactristica gomtrica) with th aim of fining an intrinsic languag of th gomtr, th gomtric algbra was iscor an lop b Grassmann 8, Hamilton an liffor uring th XIX cntur. Howr, it i not bcom usual in th XX cntur ought to man circumstancs but th ctor analsis -a rcasting of th Hamilton quatrnions b Gibbs an Haisi- was grauall accpt in phsics. On th othr han, th gomtr follow its own wa asi from th ctor analsis as Gibbs 9 point out: n th growth in this cntur of th so-call snthtic as oppos to analtical gomtr sms u to th fact that b th orinar analsis gomtrs coul not asil prss, cpt in a cumbrsom an unnatural mannr, th sort of rlations in which th wr particularl intrst 6 Giuspp Pano, «Saggio i alcolo gomtrico». Translat in Slct works of Giuspp Pano, 69 (s th bibliograph). 7. I. Grhart, G. W. Libniz. Mathmatical Schriftn V, 4 an Dr rifwchsl on Gottfri Wilhlm Libniz mit Mathmatikr, In 844 a priz (45 gol ucats for 846) was offr b th Fürstlich Jablonowski'schn Gssllschaft in Lipzig to whom was capabl to lop th charactristica gomtrica of Libniz. Grassmann won this priz with th mmoir Gomtric nalsis, publish b this socit in 847 with a forwor b ugust Frinan Möbius. Its contnts ar ssntiall thos of Di ushnungslhr (844). 9 Josiah Willar Gibbs, «On Multipl lgbra», rprouc in Scintific paprs of J.W. Gibbs, II, 98. VI

9 Th work of rision of th histor an th sourcs (s J. M. Parra ) has allow us to snthsis th contributions of th iffrnt authors an compltl rbuil th olution of th gomtric algbra, rmoing th concptual mistaks which l to th ctor analsis. This prfac has not nough tnsion to plain all th histor, but on must rmmbr somthing usuall forgottn: uring th XIX cntur sral points of iw or what shoul bcom th gomtric algbra cam into comptition. Th Gibbs' ctor analsis was on of ths bing not th bttr. In fact, th gomtric algbra is a fil of knowlg whr iffrnt formulations ar possibl as Pano show: In ths arious mthos of gomtric calculus o not at all contraict on anothr. Th ar arious parts of th sam scinc, or rathr arious was of prsnting th sam subjct b sral authors, ach stuing it inpnntl of th othrs. It follows that gomtric calculus, lik an othr mtho, is not a sstm of conntions but a sstm of truth. In th sam wa, th mthos of iniisibls (aaliri), of infinitsimals (Libniz) an of fluions (Nwton) ar th sam scinc, mor or lss prfct, plain unr iffrnt forms. Th gomtric algbra owns som funamntal gomtric facts which cannot b ignor at all an will b rcognis to it, as Grassmann hop: For I rmain compltl confint that th labour which I ha pan on th scinc prsnt hr an which has man a significant part of m lif as wll as th most strnuous application of m powrs will not b lost. It is tru that I am awar that th form which I ha gin th scinc is imprfct an must b imprfct. ut I know an fl oblig to stat (though I run th risk of sming arrogant) that n if this work shoul again rmain unus for anothr sntn ars or n longr, without ntring into th actual lopmnt of scinc, still th tim will com whn it will b brought forth from th ust of obliion, an whn ias now ormant will bring forth fruit. I know that if I also fail to gathr aroun m in a position (which I ha up to now sir in ain) a circl of scholars, whom I coul fructif with ths ias, an whom I coul stimulat to lop an nrich furthr ths ias, nrthlss thr will com a tim whn ths ias, prhaps in a nw form, will aris anw an will ntr into liing communication with contmporar lopmnts. For truth is trnal an iin, an no phas in th lopmnt of truth, howr small ma b th rgion ncompass, can pass on without laing Josp Manl Parra i Srra, «Gomtric algbra rsus numrical artsianism. Th historical trn bhin liffor s algbra», in rack t al.., liffor lgbras an thir pplications in Mathmatical Phsics, 7-6,. r complt rfrnc is Michal J. row, Histor of Vctor nalsis. Th Eolution of th Ia of a Vctorial Sstm. Giuspp Pano, op. cit., 68. VII

10 a trac; truth rmains, n though th garmnt in which poor mortals cloth it ma fall to ust. s an othr aspct of th human lif, th histor of th gomtric algbra was conition b man fortuitous nts. Whil Grassmann uc th tnsion thor from philosophic concpts unintlligibl for authors such as Möbius an Gibbs, Hamilton intifi ctors an bictors -th starting point of th grat tangl of ctor analsis- using a ha notation 4. liffor ha foun th corrct algbraic structur 5 which intgrat th sstms of Hamilton an Grassmann. Howr u to th prmatur ath of liffor in 879, his opinion was not takn into account 6 an a long pistolar war was carri out b th quatrnionists (spciall Tait) against th fnrs of th ctor analsis, crat b Gibbs 7, who i not rcognis to b influnc b Grassmann an Hamilton: t all nts, I saw that th mthos which I was using, whil narl thos of Hamilton, wr almost actl thos of Grassmann. I procur th two E. of th ushnungslhr but I cannot sa that I foun thm as raing. In fact I ha nr ha th prsranc to gt through with ithr of thm, an ha prhaps got mor ias from his miscllanous mmoirs than from thos works. I am not howr conscious that Grassmann's writings rt an particular influnc on m Vctor nalsis, although I was gla nough in th introuctor paragraph to shltr mslf bhin on or two istinguish nams [Grassmann an liffor] in making changs of notation which I flt woul b istastful to quatrnionists. In fact if ou ra that pamphlt carfull ou will s that it all follows with th inorabl logic of algbra from th problm which I ha st mslf long bfor m acquaintanc with Grassmann. I ha no oubt that ou consir, as I o, th mthos of Grassmann to b suprior to thos of Hamilton. It thus sm to m that it might [b] intrsting to ou to know how commncing with som knowlg of Hamilton's mtho an influnc simpl b a sir to obtain th simplst algbra for th prssion of th rlations of Gom. Phs. tc. I was l ssntiall to Grassmann's algbra of ctors, inpnntl of an influnc from him or an on ls. 8 Hrmann Gunthr Grassmann. Prfac to th scon ition of Di ushnungslhr (86). Th first ition was publish on 844, hnc th "sntn ars". Translat in row, op. cit. p Th Lcturs on Quatrnions was publish in 85, an th Elmnts of Quatrnions posthumousl in William Kingon liffor lft us his snthsis in «pplications of Grassmann's Etnsi lgbra». 6 S «On th lassification of Gomtric lgbras», unfinish papr whos abstract was communicat to th Lonon Mathmatical Socit on March th, Th first Vctor nalsis was a priat ition of Draft of a lttr snt b Josiah Willar Gibbs to Victor Schlgl (888). Rprouc b row, op. cit. p. 5. VIII

11 for its bginning th contrors was alra suprfluous 9. Notwithstaning th pistolar war continu for twl ars. Th ctor analsis is a proisional solution (which spnt all th XX cntur!) aopt b rbo ought to its asinss an practical notation but haing man troubls whn bing appli to thr-imnsional gomtr an unabl to b gnralis to th Minkowski s four-imnsional spac. On th othr han, th gomtric algbra is, b its own natur, ali in an imnsion an it offrs th ncssar rsourcs for th stu an rsarch in gomtr as I show in this book. Th rar will s that th thortical planations ha bn complt with problms in ach chaptr, although this splitting is somwhat fictitious bcaus th problms ar monstrations of gomtric facts, bing on of th most intrsting aspcts of th gomtric algbra an a proof of its powr. Th usual numric problms, which our pupils lik, can b asil outlin b th tachr, bcaus th gomtric algbra alwas ils an immiat prssion with coorinats. I'm inbt to profssor Josp Manl Parra for ncouraging m to writ this book, for th ialctic intrchang of ias an for th bibliographic support. In th framwork of th summr courss on gomtric algbra for tachrs that w taught uring th ars in th Escola stiu scunària organis b th ol lgi Oficial Doctors i Llicnciats n Filosofia i Lltrs i n ièncis ataluna, th projct of som books on this subjct appar in a natural mannr. Th first book ot to two imnsions alra lis on our hans an will probabl b follow b othr books on th algbra an gomtr of th thr an four imnsions. Finall I also acknowlg th suggstions rci from som rars. Ramon Gonzálz alt ranola l Vallès, Jun 9 S lfr M. ork «Vctors rsus quatrnions Th lttrs in Natur». Th ctor analsis bass on th ualit of th gomtric algbra of th thr-imnsional spac: th fact that th orintation of lins an plans is trmin b thr numric componnts in both cass. Howr in th four-imnsional tim-spac th sam orintations ar rspctil trmin b four an si numbrs. IX

12 ONTENTS First Part: Th ctor plan an th compl numbrs. Th ctors an thir oprations. (Jun 4 th, upat March 7 th ) Vctor aition,.- Prouct of a ctor an a ral numbr,.- Prouct of two ctors,.- Prouct of thr ctors: associati proprt, 5. Prouct of four ctors, 7.- Inrs an quotint of two ctors, 7.- Hirarch of algbraic oprations, 8.- Gomtric algbra of th ctor plan, 8.- Erciss, 9.. bas of ctors for th plan. (Jun 4 th ) Linar combination of two ctors,.- as an componnts,.- Orthonormal bass,.- pplications of th formula for th proucts,.- Erciss,.. Th compl numbrs. (ugust st, upat Jul st ) Subalgbra of th compl numbrs,.- inomial, polar an trigonomtric form of a compl numbr,.- lgbraic oprations with compl numbrs, 4.- Prmutation of compl numbrs an ctors, 7.- Th compl plan, 8.- ompl analtic functions, 9.- Th funamntal thorm of algbra, 4.- Erciss, Transformations of ctors. (ugust 4 th, upat Jul st ) Rotations, 7.- Rflctions, 8.- Inrsions, 9.- Dilatations,.- Erciss, Scon Part: Th gomtr of th Euclian plan 5. Points an straight lins. (ugust 9 th, upat Sptmbr 9 th ) Translations,.- oorinat sstms,.- arcntric coorinats,.- Distanc btwn two points an ara,.- onition of alignmnt of thr points, 5.- artsian coorinats, 6.- Vctorial an paramtric quations of a lin, 6.- lgbraic quation an istanc from a point to a lin, 7.- Slop an intrcpt quations of a lin, 4.- Polar quation of a lin, 4.- Intrsction of two lins an pncil of lins, 4.- Dual coorinats, 4.- Th Dsargus thorm, 47.- Erciss, ngls an lmntal trigonomtr. (ugust 4 th, upat Jul st ) Sum of th angls of a polgon, 5.- Dfinition of trigonomtric functions an funamntal intitis, 54.- ngl inscrib in a circl an oubl angl intitis, 55.- ition of ctors an sum of trigonomtric functions, 56.- Prouct of ctors an aition intitis, 57.- Rotations an D Moir's intit, 58.- Inrs trigonomtric functions, 59.- Erciss, Similaritis an singl ratio. (ugust 6 th, upat Jul st ) Dirct similarit, 6.- Opposit similarit, 6.- Th thorm of Mnlaus, 6.- Th thorm of a, 64.- Homotht an singl ratio, 65.- Erciss, Proprtis of th triangls. (Sptmbr r, upat Jul st ) ra of a triangl, 68.- Mians an cntroi, 69.- Prpnicular bisctors an circumcntr, 7.- ngl bisctors an incntr, 7.- ltitus an orthocntr, 7.- Eulr's lin, 76.- Th Frmat's thorm, 77.- Erciss, 78. X

13 9. ircls. (Octobr 8 th, upat Jul 6 th ) lgbraic an artsian quations, 8.- Intrsctions of a lin with a circl, 8.- Powr of a point with rspct to a circl, 8.- Polar quation, 8.- Inrsion with rspct to a circl, 8.- Th nin-point circl, 85.- clic an circumscrib quarilatrals, 87.- ngl btwn circls, 89.- Raical ais of two circls, 89.- Erciss, 9.. ross ratios an rlat transformations. (Octobr 8 th, upat Jul st ) ompl cross ratio, 9.- Harmonic charactristic an rangs, 94.- Th homograph (Möbius transformation), 96.- Projcti cross ratio, 99.- Th points at th infinit an homognous coorinats,.- Prspctiit an projctiit,.- Th projctiit as a tool for thorms monstration, 8.- Th homolog,.- Erciss, 5.. onics (Nombr th, upat Jul st ) onic sctions, 7.- Two foci an two irctrics,.- Vctorial quation,.- Th hasls' thorm,.- Tangnt an prpnicular to a conic, 4.- ntral quations for th llips an hprbola, 6.- Diamtrs an pollonius' thorm, 8.- onic passing through fi points,.- onic quations in barcntric coorinats an tangntial conic,.- Polaritis, 4.- Ruction of th conic matri to a iagonal form, 6.- Using a bas of points on th conic, 7.- Erciss, 7. Thir Part: Psuo-Euclian gomtr. Matri rprsntation an hprbolic numbrs. (Nombr n, upat Ma st ) Rotations an th rprsntation of compl numbrs, 9.- Th subalgbra of th hprbolic numbrs, 4.- Hprbolic trigonomtr, 4.- Hprbolic ponntial an logarithm, 4.- Polar form, powrs an roots of hprbolic numbrs, 44.- Hprbolic analtic functions, 47.- nalticit an squar of conrgnc of th powr sris, 5.- bout th isomorphism of liffor algbras, 5.- Erciss, 5.. Th hprbolic or psuo-euclian plan (Januar st, upat Jul st ) Hprbolic ctors, 54.- Innr an outr proucts of hprbolic ctors, 55.- ngls btwn hprbolic ctors, 56.- ongrunc of sgmnts an angls, 58.- Isomtris, 58.- Thorms about angls, 6.- Distanc btwn points, 6.- ra on th hprbolic plan, 6.- Diamtrs of th hprbola an pollonius' thorm, 6.- Th law of sins an cosins, 64.- Hprbolic similarit, 67.- Powr of a point with rspct to a hprbola with constant raius, 68.- Erciss, 69. Fourth Part: Plan projctions of triimnsional spacs 4. Sphrical gomtr in th Euclian spac. (March r, upat ugust 5 th ) Th gomtric algbra of th Euclian spac, 7.- Sphrical trigonomtr, 7.- Th ual sphrical triangl, 75.- Right sphrical triangls an Napir s rul, 76.- ra of a sphrical triangl, 76.- Proprtis of th projctions of th sphrical surfac, 77.- Th XI

14 cntral or gnomonic projction, 77.- Strographic projction, 8.- Orthographic projction, 8.- Sphrical coorinats an clinrical quiistant (Plat arré) projction, 8.- Mrcator's projction, 8.- Ptr's projction, 84.- onic projctions, 84.- Erciss, Hprboloial gomtr in th psuo-euclian spac (Lobachsk's gomtr). (pril th, upat ugust st ) Th gomtric algbra of th psuo-euclian spac, 88.- Th hprboloi of two shts, 9.- Th cntral projction (ltrami isk), 9.- Hprboloial (Lobachskian) trigonomtr, 96.- Strographic projction (Poincaré isk), 98.- zimuthal quialnt projction,.- Wirstrass coorinats an clinrical quiistant projction,.- linrical conformal projction,.- linrical quialnt projction,.- onic projctions,.- On th congrunc of gosic triangls, 5.- ommnt about th nams of th non-euclian gomtr, 5.- Erciss, Solutions of th propos rciss. (pril 8 th an Ma 7 th, upat Jul th ). Th ctors an thir oprations, 7.-. bas of ctors for th plan, 8.-. Th compl numbrs, Transformations of ctors,.- 5. Points an straight lins, ngls an lmntal trigonomtr,.- 7. Similaritis an singl ratio, Proprtis of th triangls, ircls, 6.-. ross ratios an rlat transformations, 4.-. onics, Matri rprsntation an hprbolic numbrs, 5.-. Th hprbolic or psuo-euclian plan, Sphrical gomtr in th Euclian spac, Hprboloial gomtr in th psuo- Euclian spac (Lobachsk's gomtr), 6. ibliograph, 66. In, 7. hronolog, 75. XII

15 TRETISE OF PLNE GEOMETRY THROUGH GEOMETRI LGER FIRST PRT: THE VETOR PLNE ND THE OMPLEX NUMERS Points an ctors ar th main lmnts of th plan gomtr. point is conci (but not fin) as a gomtric lmnt without tnsion, infinitl small, that has position an is locat at a crtain plac on th plan. ctor is fin as an orint sgmnt, that is, a pic of a straight lin haing lngth an irction. ctor has no position an can b translat anwhr. Usuall it is call a fr ctor. If w plac th n of a ctor at a point, thn its ha trmins anothr point, so that a ctor rprsnts th translation from th first point to th scon on. Taking into account th istinction btwn points an ctors, th part of th book ot to th Euclian gomtr has bn ii in two parts. In th first on th ctors an thir algbraic proprtis ar stui, which is nough for man scintific an nginring branchs. In th scon part th points ar introuc an thn th affin gomtr is stui. ll th lmnts of th gomtric algbra (scalars, ctors, bictors, compl numbrs) ar not with lowrcas Latin charactrs an th angls with Grk charactrs. Th capital Latin charactrs will not points on th plan. s ou will s, th gomtric prouct is not commutati, so that fractions can onl b writtn for ral an compl numbrs. Sinc th gomtric prouct is associati, th inrs of a crtain lmnt at th lft an at th right is th sam, that is, thr is a uniqu inrs for ach lmnt of th algbra, which is inicat b th suprscript. lso u to th associati proprt, all th factors in a prouct ar writtn without parnthsis. In orr to mak as th raing I ha not numrat thorms, corollaris nor quations. Whn a finition is introuc, th finit lmnt is mark with italic charactrs, which allows to irct attntion an hlps to fin again th finition.. THE VETORS ND THEIR OPERTIONS ctor is an orint sgmnt, haing lngth an irction but no position, that is, it can b plac anwhr without changing its orintation. Th ctors can rprsnt man phsical magnitus such as a forc, a clrit, an also gomtric magnitus such as a translation. Two algbraic oprations for ctors ar fin, th aition an th prouct, that gnralis th aition an prouct of th ral numbrs. Vctor aition Th aition of two ctors u is fin as th ctor going from th n of th ctor u to th ha of whn th ha of u contacts th n of (uppr triangl in th figur. ). Making th construction for u, that is, placing th n of u at th ha of (lowr triangl in th figur.) w s that th aition ctor is th sam. Thrfor, th ctor aition has th commutati proprt: Figur.

16 RMON GONZLEZ LVET u u an th paralllogram rul follows: th aition of two ctors is th iagonal of th paralllogram form b both ctors. Th associati proprt follows from this finition bcaus (u)w or u(w) is th ctor closing th polgon form b th thr ctors as shown in th figur.. Th nutral lmnt of th ctor aition is th null ctor, which has zro lngth. Hnc th opposit ctor of u is fin as th ctor u with th sam orintation but opposit irction, which a to th initial Figur. ctor gis th null ctor: u ( u) Prouct of a ctor an a ral numbr On fins th prouct of a ctor an a ral numbr (or scalar) k, as a ctor with th sam irction but with a lngth incras k tims (figur.). If th ral numbr is ngati, thn th irction is th opposit. Th gomtric finition implis th commutati proprt: k u u k Figur. Two ctors u, with th sam irction ar proportional bcaus thr is alwas a ral numbr k such that k u, that is, k is th quotint of both ctors: k u u Two ctors with iffrnt irctions ar sai to b linarl inpnnt. Prouct of two ctors Th prouct of two ctors will b call th gomtric prouct in orr to b istinguish from othr ctor proucts currntl us. Nrthlss I hop that ths othr proucts will pla a sconar rol whn th gomtric prouct bcoms th most us, a nar nt which this book will forwar. t that tim, th ajcti «gomtric» will not b ncssar. Th following proprtis ar man to th gomtric prouct of two ctors:

17 TRETISE OF PLNE GEOMETRY THROUGH GEOMETRI LGER ) To b istributi with rgar to th ctor aition: u ( w ) u u w ) Th squar of a ctor must b qual to th squar of its lngth. finition, th lngth (or moulus) of a ctor is a positi numbr an it is not b u : u u ) Th mi associati proprt must ist btwn th prouct of ctors an th prouct of a ctor an a ral numbr. k ( u ) ( k u ) k u k ( l u ) ( k l ) u k l u whr k, l ar ral numbrs an u, ctors. Thrfor, parnthsis ar not n. Ths proprtis allows us to uc th prouct. Lt us suppos that c is th aition of two ctors a, b an calculat its squar appling th istributi proprt: c a b c ( a b ) ( a b ) ( a b ) a a b b a b W ha to prsr th orr of th factors bcaus w o not know whthr th prouct is commutati or not. If a an b ar orthogonal ctors, th Pthagoran thorm applis an thn: a b c a b a b b a a b b a That is, th prouct of two prpnicular ctors is anticommutati. If a an b ar proportional ctors thn: a b b k a, k ral a b a k a k a a b a bcaus of th commutati an mi associati proprtis of th prouct of a ctor an a ral numbr. Thrfor th prouct of two proportional ctors is commutati. If c is th aition of two ctors a, b with th sam irction, w ha: c a b c a b a b a b a b angl(a, b) ut if th ctors ha opposit irctions: c a b

18 4 RMON GONZLEZ LVET c a b a b a b a b angl(a, b) π How is th prouct of two ctors with an irctions? Du to th istributi proprt th prouct is rsol into on prouct b th proportional componnt b an anothr b th orthogonal componnt b : a b a ( b b ) a b a b Th prouct of on ctor b th proportional componnt of th othr is call th innr prouct (also scalar prouct) an not b a point (figur.4). Taking into account that th projction of b onto a is proportional to th cosin of th angl btwn both ctors, on fins: a b a b a b cos α Th innr prouct is alwas a ral numbr. For ampl, th work ma b a forc acting on a bo is th innr prouct of th forc an th walk spac. Sinc th commutati proprt has bn uc for th prouct of ctors with th sam irction, it follows also for th innr prouct: Figur.4 a b b a Th prouct of on ctor b th orthogonal componnt of th othr is call th outr prouct (also trior prouct) an it is not with th smbol : a b a b Th outr prouct rprsnts th ara of th paralllogram form b both ctors (figur.5): a b a b a b sin α Sinc th outr prouct is a prouct of orthogonal ctors, it is anticommutati: Figur.5 a b b a Som ampl of phsical magnitus which ar outr proucts ar th angular momntum, th torqu, tc. Whn two ctors ar prmut, th sign of th orint angl is chang. Thn th cosin rmains qual whil th sin changs th sign. caus of this, th innr prouct is commutati whil th outr

19 TRETISE OF PLNE GEOMETRY THROUGH GEOMETRI LGER 5 prouct is anticommutati. Now, w can rwrit th gomtric prouct as th sum of both proucts: a b a b a b From hr, th innr an outr proucts can b writtn using th gomtric prouct: Figur.6 a b b a a b a b b a a b In conclusion, th gomtric prouct of two proportional ctors is commutati whras that of two orthogonal ctors is anticommutati, just for th pur cass of outr an innr proucts. Th outr, innr an gomtric proucts of two ctors onl pn upon th mouli of th ctors an th angl btwn thm. Whn both ctors ar rotat prsring th angl that th form, th proucts ar also prsr (figur.6). How is th absolut alu of th prouct of two ctors? Sinc th innr an outr prouct ar linarl inpnnt an orthogonal magnitus, th moulus of th gomtric prouct must b calculat through a gnralisation of th Pthagoran thorm: a b a b a b a b a b a b a b a b ( cos α sin α ) a b That is, th moulus of th gomtric prouct is th prouct of th moulus of ach ctor: a b a b Prouct of thr ctors: associati proprt It is man as th fourth proprt that th prouct of thr ctors b associati: 4) u ( w ) ( u ) w u w Hnc w can rmo parnthsis in multipl proucts an with th forgoing proprtis w can uc how th prouct oprats upon ctors. W wish to multipl a ctor a b a prouct of two ctors b, c. W ignor th rsult of th prouct of thr ctors with iffrnt orintations cpt whn two ajacnt factors ar proportional. W ha sn that th prouct of two ctors pns onl on th angl btwn thm. Thrfor th paralllogram form b b an c can b

20 6 RMON GONZLEZ LVET rotat until b has, in th nw orintation, th sam irction as a. If b' an c' ar th ctors b an c with th nw orintation (figur.7) thn: b c b' c' a ( b c ) a ( b' c' ) an b th associati proprt: Figur.7 a ( b c ) ( a b' ) c' Sinc a an b' ha th sam irction, a b' a b is a ral numbr an th tripl prouct is a ctor with th irction of c' whos lngth is incras b this amount: a ( b c ) a b c' It follows that th moulus of th prouct of thr ctors is th prouct of thir mouli: a b c a b c On th othr han, a can b firstl multipli b b, an aftr this w can rotat th paralllogram form b both ctors until b has, in th nw orintation, th sam irction as c (figur.8). Thn: Figur.8 ( a b ) c a'' ( b'' c ) a'' b c lthough th gomtric construction iffrs from th forgoing on, th figurs clarl show that th tripl prouct ils th sam ctor, as pct from th associati proprt. ( a b ) c a'' b c c b a'' c b'' a'' c ( b a ) That is, th tripl prouct has th proprt: a b c c b a which I call th prmutati proprt: r ctor can b prmut with a ctor locat two positions farthr in a prouct, although it os not commut with th nighbouring ctors. Th prmutati proprt implis that an pair of ctors in a prouct sparat b an o numbr of ctors can b prmut. For ampl: a b c a c b c a b c b a

21 TRETISE OF PLNE GEOMETRY THROUGH GEOMETRI LGER 7 Th prmutati proprt is charactristic of th plan an it is also ali for th spac whnr th thr ctors ar coplanar. This proprt is rlat with th fact that th prouct of compl numbrs is commutati. Prouct of four ctors Th prouct of four ctors can b uc from th formr rasoning. In orr to multipl two pair of ctors, rotat th paralllogram form b a an b until b' has th irction of c. Thn th prouct is th paralllogram form b a' an but incras b th moulus of b an c: a b c a' b' c a' b c b c a' Now lt us s th spcial cas whn a c an b. If both ctors a, b ha th sam irction, th squar of thir prouct is a positi ral numbr: a b ( a b ) a b > If both ctors ar prpnicular, w must rotat th paralllogram through π/ until b' has th sam irction as a (figur.9). Thn a' an b ar proportional but haing opposit signs. Thrfor, th squar of a prouct of two orthogonal ctors is alwas ngati: a b ( a b ) a' b' a b a' b a b a b < Figur.9 Likwis, th squar of an outr prouct of an two ctors is also ngati. Inrs an quotint of two ctors Th inrs of a ctor a is that ctor whos multiplication b a gis th unit. Onl th ctors which ar proportional ha a ral prouct. Hnc th inrs ctor has th sam irction an inrs moulus: a a a a a a a Th quotint of ctors is a prouct for an inrs ctor, which pns on th orr of th factors bcaus th prouct is not commutati: a b b a Obiousl th quotint of proportional ctors with th sam irction an sns is qual to th quotint of thir mouli. Whn th ctors ha iffrnt irctions, thir quotint can b rprsnt b a paralllogram, which allows to tn th concpt of

22 8 RMON GONZLEZ LVET ctor proportionalit. W sa that a is proportional to c as b is to whn thir mouli ar proportional an th angl btwn a an c is qual to th angl btwn b an : a c b a c b an α(a, c) α(b, ) Thn th paralllogram form b a an b is similar to that form b c an, bing α(a, c) th angl of rotation from th first to th scon on. Th inrs of a prouct of sral ctors is th prouct of th inrss with th chang orr, as can b asil sn from th associati proprt: ( a b c ) c b a Hirarch of algbraic oprations Lik th algbra of ral numbrs, an in orr to simplif th algbraic notation, I shall us th following hirarch for th ctor oprations plain abo: ) Th parnthsis, whos contnt will b firstl oprat. ) Th powr with an ponnt (squar, inrs, tc.). ) Th outr an innr prouct, which ha th sam hirarch ll but must b oprat bfor th gomtric prouct. 4) Th gomtric prouct. 5) Th aition. s an ampl, som algbraic prssions ar gin with th simplifi prssion at th lft han an its maning using parnthsis at th right han: a b c ( a b ) ( c ) a b c ( ( a ) ( b c ) ) a b c a ( ( b c ) ) Gomtric algbra of th ctorial plan Th st of all th ctors on th plan togthr with th oprations of ctor aition an prouct of ctors b ral numbrs is a two-imnsional spac usuall call th ctor plan V. Th gomtric prouct gnrats nw lmnts (th compl numbrs) not inclu in th ctor plan. So, th gomtric (or liffor) algbra of a ctorial spac is fin as th st of all th lmnts gnrat b proucts of ctors, for which th gomtric prouct is an innr opration. Th gomtric algbra of th Euclian ctor plan is usuall not as l, (R) or simpl as l. Making a paralllism with probabilit, th sampl spac is th st of lmntal rsults of a crtain William Rowan Hamilton fin th quatrnions as quotints of two ctors in th wa that similar paralllograms locat at th sam plan in th spac rprsnt th sam quatrnion (Elmnts of Quatrnions, posthumousl it in 866, hlsa Publishrs 969, ol I, s p. an fig. 4). In th ctorial plan a quatrnion is ruc to a compl numbr. Th quatrnions wr iscor b Hamilton (Octobr 6 th, 84) bfor th gomtric prouct b liffor (878).

23 TRETISE OF PLNE GEOMETRY THROUGH GEOMETRI LGER 9 ranom primnt. From th sampl spac Ω, th union an intrsction gnrat th ool algbra (Ω), which inclus all th possibl nts. In th sam mannr, th aition an gomtric prouct gnrat th gomtric algbra of th ctorial spac. Thn both sampl an ctorial spac pla similar rols as gnrators of th ool an gomtric (liffor) algbras rspctil. Erciss. Pro that th sum of th squars of th iagonals of an paralllogram is qual to th sum of th squars of th four sis. Think about th sis as ctors.. Pro th following intit: ( a b ) ( a b ) a b. Pro that: a b c a ( b c ) ( a b c ).4 Pro that: a b c a c b a b c.5 Pro th prmutati proprt rsoling b an c into th componnts which ar proportional an orthogonal to th ctor a..6 Pro th Hron s formula for th ara of th triangl: s ( s a ) ( s b ) ( s c ) whr a, b an c ar th sis an s th smiprimtr: s a b c

24 RMON GONZLEZ LVET. SE OF VETORS FOR THE PLNE Linar combination of two ctors Er ctor w on th plan is alwas a linar combination of two inpnnt ctors u an : w a u b a, b ral caus of this, th plan has imnsion qual to. In orr to calculat th cofficints of linar combination a an b, w multipl w b u an at both sis an subtract th rsults: to obtain: u w a u b u an w u a u b u u w w u b ( u u ) w a u b an w a u b w w a ( u u ) w a u u b u w Th rsolution of a ctor as a linar combination of two inpnnt ctors is a r frqunt opration an also th founation of th coorinats mtho. as an componnts n st of two inpnnt ctors {, } can b takn as a bas of th ctor plan. Er ctor u can b writtn as linar combination of th bas ctors: u u u Th cofficints of this linar combination u, u ar th componnts of th ctor in this bas. Thn a ctor will b rprsnt as a pair of componnts: u ( u, u ) Th componnts pn on th bas, so that a chang of bas las to a chang of th componnts of th gin ctor. W must onl a componnts to a ctors: (, ) u ( u, u ) Th prssion of th gomtric prouct with componnts is obtain b mans of th istributi proprt:

25 TRETISE OF PLNE GEOMETRY THROUGH GEOMETRI LGER u ( u u ) ( ) u u u u u u u ( u u ) ( u u ) Hnc th prssion of th squar of th ctor moulus writtn in componnts is: u u u u u u Orthonormal bass n bas is ali to scrib ctors using componnts, although th orthonormal bass, for which both an ar unitar an prpnicular (such as th canonical bas shown in th figur.), ar th mor connint an suitabl: Figur. For r orthonormal bas : Th prouct rprsnts a squar of unit ara. Th squar powr of this prouct is qual to : ( ) For an orthonormal bas, th gomtric prouct of two ctors bcoms: u u u ( u u ) Not that th first an scon summans ar ral whil th thir is an ara. Thrfor it follows that th ar rspctil th innr an outr proucts: u u u u ( u u ) lso, th moulus of a ctor is calculat from th slf innr prouct: u u u pplications of th formula for th proucts Th first application is th calculation of th angl btwn two ctors: u u cosα u u u sin α u

26 RMON GONZLEZ LVET Th alus of sin an cosin trmin a uniqu angl α in th rang <α<π. Th angl btwn two ctors is a sns magnitu haing positi sign if it is countrclockwis an ngati sign if it is clockwis. Thus this angl pns on th orr of th ctors in th outr (an gomtric) prouct. For ampl, lt us consir th ctors (figur.) u an : u (, ) u ( 4, ) 5 cos 5 α ( u, ) cosα(, u) sin α ( u, ) sin α(, u) α (u, ) α(, u ) lso w ma tak th angl α( u, ) π Th angl so obtain is alwas that going from th first to th scon ctor, bing uniqu within a prio. Othr application of th outr prouct is th calculus of aras. Using th prssion with componnts, th ara (consir as a positi ral numbr) Figura. of th paralllogram form b u an is: u u u 4 Whn calculating th ara of an triangl w must onl ii th outr prouct of an two sis b. Erciss. Lt (u, u ) an (, ) b th componnts of th ctors u an in th canonical bas. Pro gomtricall that th ara of th paralllogram form b both ctors is th moulus of th outr prouct u u u.. alculat th ara of th triangl whos sis ar th ctors (, 5), (, ) an thir aition (, ).. Pro th prmutati proprt using componnts: a b c c b a..4 alculat th angl btwn th ctors u an 4 in th canonical bas..5 onsir a bas whr has moulus, has moulus an th angl btwn both ctors is π/. alculat th angl btwn u an 4..6 In th canonical bas (, 5). alculat th componnts of this ctor in a nw bas { u, u } if u (, ) an u (5, ).

27 TRETISE OF PLNE GEOMETRY THROUGH GEOMETRI LGER. THE OMPLEX NUMERS Subalgbra of th compl numbrs If {, } is th canonical bas of th ctor plan V, its gomtric algbra is fin as th ctor spac gnrat b th lmnts {,,, } togthr with th gomtric prouct, so that th gomtric algbra l has imnsion four. Th unitar ara is usuall not as. Du to th associati charactr of th gomtric prouct, th gomtric algbra is an associati algbra with intit. Th complt tabl for th gomtric prouct is th following: Not that th subst of lmnts containing onl ral numbrs an aras is clos for th prouct: This is th subalgbra of compl numbrs., th imaginar unit, is usuall not as i. Th ar call compl numbrs bcaus thir prouct is commutati lik for th ral numbrs. inomial, polar an trigonomtric form of a compl numbr Er compl numbr z writtn in th binomial form is: z a b a, b ral whr a an b ar th ral an imaginar componnts rspctil. Th moulus of a compl numbr is calculat in th sam wa as th moulus of an lmnt of th gomtric algbra b mans of th Pthagoran thorm: z a b a b Sinc r compl numbr can b writtn as a prouct of two ctors u an forming a crtain angl α:

28 4 RMON GONZLEZ LVET z u u ( cos α sin α ) Figur. w ma rprsnt a compl numbr as a paralllogram with sis bing th ctors u an. ut thr ar infinit pairs of ctors u' an ' whos prouct is th compl z proi that: u u' ' an α α' ll th paralllograms haing th sam ara an obliquit rprsnt a gin compl (th ar quialnt) inpnntl of th lngth an orintation of on si.(figur.). Th trigonomtric an polar forms of a compl numbr z spcifis its moulus z an argumnt α : z z ( cos α sin α ) z α compl numbr can b writtn using th ponntial function, but firstl w must pro th Eulr s intit: p(α ) cos α sin α α ral Th ponntial of an imaginar numbr is fin in th sam mannr as for a ral numbr: Figur. p ( α ) lim α n n s shown in th figur. (for n 5), th limit is a powr of n rotations with angl α,/n or quialntl a rotation of angl α. Now a compl numbr writtn in ponntial form is: z z p(α ) n lgbraic oprations with compl numbrs Each algbraic opration is mor asil calculat in a form than in anothr accoring to th following schm: aition / subtraction binomial form prouct / quotint binomial or polar form powrs / roots polar form

29 TRETISE OF PLNE GEOMETRY THROUGH GEOMETRI LGER 5 Th binomial form is suitabl for th aition bcaus both ral componnts must b a an also th imaginar ons,.g.: z 4 t 5 z t If th compl numbrs ar writtn in anothr form, th must b conrt to th binomial form,.g. z π / 4 t 4 π / 6 π π π z t π / 4 4π / 6 cos sin 4 cos π sin 6 ( ) In orr to writ th rsult in polar form, th moulus must b calculat: z t ( ) ( ) 4 ( 6) z t an also th argumnt from th cosin an sin obtain as quotint of th ral an imaginar componnts rspctil ii b th moulus: cosα 5 6 sinα α z t.98. Whn multipling compl numbrs in binomial form, w appl th istributi proprt takn into account that th squar of th imaginar unit is,.g.: ( 5 ) ( 4 ) ( ) 6 4 Th ponntial of an aition of argumnts (ral or compl) is qual to th prouct of ponntial functions of ach argumnt. ppling this charactristic proprt to th prouct of compl numbrs in ponntial form, w ha: z t z p(α ) t p( β ) z t p[ (α β ) ] from whr th prouct of compl numbrs in polar form is obtain b multipling both mouli an aing both argumnts.:

30 6 RMON GONZLEZ LVET z α t β z t α β On ma subtract π to th rsulting argumnt in orr to kp it btwn an π. Th prouct of two compl numbrs z an t is th gomtric opration consisting in th rotation of th paralllogram rprsnting th first compl numbr until it touchs th Figur. paralllogram rprsnting th scon compl numbr. Whn th contact in a unitar ctor (figur.), th paralllogram form b th othr two ctors is th prouct of both compl numbrs: z u t w z t u w u w This gomtric construction is alwas possibl bcaus a paralllogram can b lngthn or win maintaining th ara so that on si has unit lngth. Th conjugat of a compl numbr (smbolis with an astrisk) is that numbr whos imaginar part has opposit sign: z a b z* a b Th gomtric maning of th conjugation is a prmutation of th ctors whos prouct is th compl numbr (figur.4). In this cas, th innr prouct is prsr whil th outr prouct changs th sign. Th prouct of a compl numbr an its conjugat is th squar of th moulus: Figur.4 z z* u u u z Th quotint of compl numbrs is fin as th prouct b th inrs. Th inrs of a compl numbr is qual to th conjugat ii b th squar of th moulus: z * a b z z a b Thn, lt us s an ampl of quotint of compl numbrs: ( 5 )( 4 ) 6 ( 4)

31 TRETISE OF PLNE GEOMETRY THROUGH GEOMETRI LGER 7 With th polar form, th quotint is obtain b iiing mouli an subtracting argumnts: z t α β z t α β Th bst wa to calculat th powr of a compl numbr with natural ponnt is through th polar form, although for low ponnts th binomial form an th Nwton formula is oftn us,.g.: ( ) 8 6 From th charactristic proprt of th ponntial function it follows that th moulus of a powr of a compl numbr is th powr of its moulus, an th argumnt of this powr is th argumnt of th compl numbr multipli b th ponnt: n ( z ) z n nα α This is a r usful rul for larg ponnts,.g.: ( ) ( π/4 ) [( ) ] 5 π 5 5 Whn th argumnt cs π, ii b this alu an tak th rmainr, in orr to ha th argumnt within th prio <α <π. Sinc a root is th inrs opration of a powr, its alu is obtain b tracting th root of th moulus an iiing th argumnt b th in n. ut a compl numbr of argumnt α ma b also rprsnt b th argumnts α π k. Thir iision b th in n ils n iffrnt argumnts within a prio, corrsponing to n iffrnt roots: n z n α z k,... n ( α π k )/ n For ampl, th cubic roots of 8 ar {,, } 8 π / 4π/. In th compl plan, th n-th roots of r compl numbr ar locat at th n rtics of a rgular polgon. Prmutation of compl numbrs an ctors Th prmutati proprt of th ctors is intimatl rlat with th commutati proprt of th prouct of compl numbrs. Lt z an t b compl numbrs an a, b, c an ctors fulfilling: z a b t c Thn th following qualitis ar quialnt: z t t z a b c c a b

32 8 RMON GONZLEZ LVET compl numbr z an a ctor c o not commut, but th can b prmut b conjugating th compl numbr: z c a b c c b a c z* Er ral numbr commut with an ctor. Howr r imaginar numbr anticommut with an ctor, bcaus th imaginar unit anticommut with as wll as with : z c c z z imaginar Th compl plan In th compl plan, th compl numbrs ar rprsnt taking th ral componnt as th abscissa an th imaginar componnt as th orinat. Th ctorial plan iffrs from th compl plan in th fact that th ctorial plan is a plan of absolut irctions whras th compl plan is a plan of rlati irctions with rspct to th ral ais, to which w ma assign an irction. s plain in mor tail in th following chaptr, th unitar compl numbrs ar rotation oprators appli to ctors. Th following qualit shows th ambialnc of th artsian coorinats in th Euclian plan: ( ) Du to a carlss us, oftn th compl numbrs ha bn improprl thought as ctors on th plan, furnishing th confusion btwn th compl an ctor plans to our pupils. It will b argu that this has bn r fruitful, but this argumnt cannot satisf gomtrs, who sarch th funamntals of th gomtr. On th othr han, som phsical magnitus of a clarl ctorial kin ha bn takn improprl as compl numbrs, spciall in quantum mchanics. caus of this, I m astonish whn sing how th innr an outr proucts transform in a spcial commutati an anticommutati proucts of compl numbrs. Th rlation btwn ctors an compl numbrs is stat in th following wa: If u is a fi unitar ctor, thn r ctor a is mapp to a uniqu compl z fulfilling: a u z with u lso othr ctor b is mapp to a compl numbr t: b u t Th outr an innr proucts of th ctors a an b can b writtn now using th compl numbrs z an t: a b ( a b b a ) ( u z u t u t u z ) u ( z* t t* z )

33 TRETISE OF PLNE GEOMETRY THROUGH GEOMETRI LGER 9 ( z* t t* z ) ( zr t I z I t R ) a b ( a b b a ) ( u z u t u z u t ) ( z* t t* z ) zr t R z I t I whr z R, t R, z I, t I ar th ral an imaginar componnts of z an t. Ths proucts ha bn call improprl scalar an trior proucts of compls. So, I rpat again that compl quantitis must b istinguish from ctorial quantitis, an rlati irctions (compl numbrs) from absolut irctions (ctors). gui for oing this is th rrsion, unr which th ctors ar rrs whil th compl numbr ar not. ompl analtic functions Th compl numbrs ar a commutati algbra whr w can stu functions as for th ral numbrs. function f() is sai to b analtical if its compl riati ists: lim f () z f ( z ) f(z) analtical at z z z z z This mans that th riati masur in an irction must gi th sam rsult. If f() a b an z, th riatis following th abscissa an orinat irctions must b qual: f' () z a b a b whnc th auch-rimann conitions ar obtain: a b an a b Du to its linarit, now th riati in an irction ar also qual. consqunc of ths conitions is th fact that th sum of both scon riatis (th Laplacian) anishs, that is, both componnts ar harmonic functions: a a b b phsical ampl is th altrnating currnt. Th oltag V an intnsit I in an lctric circuit ar continuousl rotating ctors. Th nrg E issipat b th circuit is th innr prouct of both ctors, E V I. Th impanc Z of th circuit is of cours a compl numbr (it is inariant unr a rrsion). Th intnsit ctor can b calculat as th gomtric prouct of th oltag ctor multipli b th inrs of th impanc I V Z. If w tak as rfrnc a continuousl rotating irction, thn V an I ar rplac b psuo compl numbrs, but proprl th ar ctors.