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227
nou onor6treoov e{etdoerp6Xprt6po to 6rdrpopoe6q opr0trrti,v t;1oupe rtrrvrpuotrtivoproptilv:ro rtrdopoto,rouqopvltlKofq optogo0q e[dtrr(q 'Otrorouto( or oprOgo(po(( ovopd(ov:tar pryto(ap0po(. pr16€v. rorro or opr0po(e(vototre(otoe 6trouqp€ooon6 rtq ouvqtou e(6oug Auroti rou uno6pootrlprdtrlteq,ond tov unotroytogd roOqpeptv€q 0ropCveg, xoto0€oeovulqro vo no((etrove(qpd\or r(ox. rov prp)\rop(rrrv ere(voro e(6qoptOptitvTIouol TI€vo e(pooteCrotpot pd0outrre dv0ptrrnote(te to oyvoodv, e(te od\tiq to ouvdvtqoov ypd-
oto yupvdoro.Qot6oo,uuto( ot v6rprvmo pdOrlpoqg dtryeBpoq opr0po(e(vqton(oteurotKot pnoporivvo €vrurlooldoouvrot vo 6oo rot rouqFrl er6txo0g' rdoorouqpoOqporrroriq pnopo6pe vo 6o0peytor( on6 e(rotiroers mq olyepprrCq Iqpr(6pevor vo ppouv poOrlpotrxo( or xotdqepvov 6ev Drlrtvtrrv, rurvoplo(rrrv mo;g'i 6n 0o enpmevo undp;puv. nou €vtutOov trql0oerqtrrrvnpoptrrlpdttrrv 6ntlq: x + 5 = 0. Me tqv ontr€q e[rotiroetq mfluoov opqtrxo(oprOpo( q ondvqorl e(vorontrdq:-5. Tr ouppoG rtovopvqtmtivoprOFdrv, 6F,lSFero: x' - 2 = 0; Me dllo tr6yrqnoroqopr0p696tov noMontroperov eoutdtoq tootitot pe 2; An6 qv enop! tov llu0oyope(trtv, yvtirpr(ovdn q terpoyovrni p(o tou 2 6ev pnopotioevo
229
KEOAAAIO 9
oupBohme(pe rdnororidopo ror, rord ouv€nerq6evrjrov prltdq oprg_ p6q'Enopdvrrrg, ro o0votrodrov r'v pqttirvoprOpdrv 6evenoprorio€ yrovo eppqvefoer dleq trq (lto0pevrq}foerqrov otryeBprrrirv ymjp_ efrotiroeurv. dpoye dtrIor, dppqror opr0po(nou pnoporloov xov vo itioouvquto6rou e(_ 6ouqlq e{ro6oerq; Aq orerproUpe ndtrrrqv euOe(o rulv oprOptilv. H Errdvo 386e(Xvet ro rgrjpor4q ovdpeoomo 0 ror ro 2. ,EXouU€ ndperqv unorei vouooev6qopo0ytrrv(ou tprytirvou, 4 ndoerdoerqnleupdrou onofourier prlroq1, rr 6xouperonoOer(oer ourr] rqv unore(vouoo ndvurornv eu0e(o rrrrvopr0pdrv. To oprorep6drpo rou euOOypogpou rpr]goroqBp(oreror nd_ vuloro 0. Ie norooqpe(oqq euOe(og owrmorXe( ro dMo drpo;Andro nu_ Ooydpero Oedrpqpo (6pouprdrr to yeogerprrdprjroqourorjtou eu0rjypolr_ poutprjgoroqe(vortE Av orprjooupe vo ndoerndvo onrveu0elorurvoproivo euO0ypoptrro rptipo,rl opXrjtou ono(oupp(orerotmo rilog rou 1.uirv grjrouqy'2, ro orlpe(omo ono(o0o rprjoerqv euOe(o tttrv opr0pdrv 6ev owrmorXef oe p4r6opr0p6.Aurdro orlpe(o0etroupe vo ouoXer(ooup€ U€ro tlT, Onwg orprptirqouoXer(oope dvodllo oqpe(o1.re tov opt0pd2. Oo ovo_ pdooupe uur6vro v€oopr0p6dppqroap0p6.Tre6ouqoprOpdq e(vorqu_ rdgnououoXa[(etor peourdro orlpe(ondvuroqv euOe(o rov opr0pdrv; An6 tqv opXordqrq to y'7roul(6rov pr,€vovon6 ourorlq rouq v€_ yndpyouvdtrtronopo6eryporo ouqdpprlroug opt0porlg. r€rourvdpprlrurv optOpdrv; Arporj dXoupe 46q ono6e(er6u to fe(vor tippqroq,pnoporipe euKolqvq oTro0eqou1ie 6u 0r up€q {t \ft G no06gnoror rerpoyovr_ riq p((rq 6trovr,v rpuorrtivoprOgtirv ilou or 6ro16ev e(vorrerpdytrrvo oprOpdrq e(voren(or1q dpprlroropr0po(.Enop€vog, ot l[i, ,[i, VIZ e(vor pqroioprOgoiotrlddlegor p(eqrurvopr0ptirv nou 6eve(vorretpdyovo
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234
EIKONA 38. To o4pe(orqq euOe(oq rurv opr0pu.0v nou ov{xeroro rCtroq rou pulr ,l[ 6ev e(vor €vo o4;..re(o nou ouoXet((etorpe rdnoro pqrj opr0p6 tfrq onp€b; lo6r], nou vo e(vortr6yoq6rio orCporovoprOptiv)
TOMH TOY NTENTEKINT
i dnetpontr{6tt undpXet Aur6orlgo(vet e(vordppqroroptOpo(. opr0pdrv dpdtrtrouq notrtro0q vo oXqpor(ooupe Mnopotlpe 0ogdppqrovoptOpdrv. too0rot 11 rtpq oflo(ou rou o oprOpdg fto nopd6€tYUo, optOgo0g. pqroug vo to ono6e(oppnroq'Mnopo0pe e(vot vo + en(o4g np€ner Fe(2 {t rot 6efiyo+ opr0g6g pqr6g 6rt o (2 fz) e(vot [ougeout6 unoO€tovrog + o6qye(oe ovt(tpoorl'Av o (2 f4 e(votprp6q' wog6rrouui 11un60eor1 '[i = ol}' opt0po(,o rot B, t€totottita:.:2 * 60o rpuotro( dre undpXouv ro 2 Koton6 ttg 60o ntrtupfqrqg t[(otloqg rot orpotpofpeontrdrg Trirpo, 'Opr.ug, -e(vot ro 6e((or€troqourqqrrlg e((ouroqg ol9 2. {i no(pvoupe otpo0t6oo ro o/p 6oo rot ro 2 e(votp4dvoqpqt6gopr0p6g, npotpovdq otrIdr16qyvop(ouge dtt to tEe(vot pr1t6g, Aut6 orlpo(ver ro(opt0;ro(. 6rr o (2 + f4 npdnervo e(ono6errv0et 6eve(vot.Autri q ovt(
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231
I(EOAAAIO 9
l-ronopd6erypqetuor11rapoyurvtnip(o tou 9 dppqroq{ pqtdqopr0_ pdq;Arpo0ro 9 €ivorro tetpdyurvorou 3, ro ono(oefuorquotrdqoprOpd,g 11tapoyovxr] p(o tou 9 efuorpqr6qopr0;r6g. kegte(te qv ruprnl pr(o tou 9. E(vorpqtdqoudq o opr0pdq;:6poupe6u r1ruBrnlp(o tou 9 6ev efuorguomdqopr0p6g,yrot( ro 2 mov x6poroorirorge g, evtirro 3 mov nlpo roorlrorpe 22. EnopCvr4, 11ruBrni p(o tou 9 np€nervo e(vor6vog opr0g6qovdpeoooro 2 Kolro 3. Kordouvdnerqdnoroqopr0p6qn ov e[_ vor,6€ve(varguorr6qopr0trrdq. Ago06eve(vorguomdqoprOpdq, r6q pe pdorlro nopondvoOerirpqgo, 0o npdnervo e(vordppqrogopr0p6q.flopopofug,q rupm{ pr(orou 8 e(vorprpdqoprOprdq, mer6rjro 8 roorirorpeqv ruprni p(o tou 2 (nou efvorquorr6qopr0p6q),oMd q crpoyrrrvrnip(o tou 8 efuordppqroqopr0p69ogori6eve(vorrdnoroqguorx6qopr0p6q. Ie xd0enep[rnorrl nou dloupeCvovdppr;ro opr0pd,pnoporigevo lqgroupyliooupe€vo yeurpetprrdpdyeOoq ilou vo owrororye(oe ourd tov oprOgdKol vq ro tonoOeuiooupe ndvurmqv eu0e(orurv opr0pdrv, pe to dvo dxpo tou oto trrq6€v. Ere(nou oroupnderro dllo dxpo Clou_ pe €vo oqpe(oorqv euOe(o FoS,ro ono(o6evovnotorye(oe pqrd opr0_ Enog€vurq, eu0e(o turv oprOpdrv np€nervo e(voryepdrq ond rC96. 4 roro(dpprFouoqpe(o. AworiqrougvCouq oprOgo0q ro0qovotrrd(o vpedpp4tougapilpofuq,oAqur6 trd efuorontrdqgto ovopoo(o.lrq vo romvo{oouperohlrepotouq dpprpouqoprOporig npdnervo e{rrdooupe 6rjoloponqprourd rurvprp6v oprOprirv. f-fptinov,to dtr or pqto(oprOpo( e(vatar)d 6nctoylttvot. OPIIMOLOr opr0po(ev6qouvdtrou e(vorontrd6rotetoypCvor ov ro;giouv60o npo0no06oerq: (1) fro onorouo6{nore 6UooprOgorlq, A Kol B, rou ouvdlou ro;trfer 6vo pdvoon6 to nopordrur:A > Brj B > A { A = B (2) Av ol A, B KolI e(voropr0po(tou ouv6}ou,Klov A > B Kor B>f,r6reA>f.'
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go(voworouton66errt€S ylq rous Ot nopondvtrt60o npoiino0Coerq ener6{fiouge ouvr1O(oet t6oo notrf vo uSXprlotponoroUpe opr0porig, 6rov vo no0pe6tr onoror6{no oqptrodpome ;reowo0q.Oo(vetornor6rdmrxo r 60ooxdporor oprO;.ro( npfnervo e(vor(oorrl 6n o 6voqnp6nervo e(vor geyotrdrepoq ond rov dMov. 0o16oo,ouro0 rou e6ouqq 6rdro(r16ev {mv {erdOopo6tr umlpXeoe outo0qrouq nopd(evougdpprlrouqopr& podq.'Orov rorrd(ouperouq pqro0qopr0po0q, q 6rdro[rj tougl6ytrrtou ouonlporoqoprOpdlv TrouXprlo+lonororipe. Ar1tro6ri, roud(ovrdqrouqnpooertrxd,Btr€noupe 6"uo 27.991e(vorpeyolfrepoqond tov Kol6rr o 0,199987 3.990 e(vorprrpdrepog ond rov 0,1999998. Enop€vrrrg, o ouppohop6q rou opr0po0pdq6(verq 6rdro{qtou oprOpo0. loX0er 6pulq Ono (/-D.(/-t = ({-O; Av 61, noroqe(vorpeyol0tepog; An6our6 ro nopd6erypo owrtroppovdpqore 6u ro roX0ovo0orqgooprOpuirv 6evqnorotrrimer ndwo q 6rdro[r1rov rippqnrrvopr0prirvror pnope(vo ovopouy Oofpee(vordpoyeontrd6roctoyp€vor, 6nurqeivoror pqro(oprOpo(; To6e0upoeptitrlpo qer(rror trreqv rleror6rqro.Tqv rlamdqto qv Av 6vo orjvotroe(vorxtramdyro pro oprOpqtmj op(oo1re nporlyoup6vtrrq. npd$1,r6te rd0e gopri nou Xprlorgonororipre ouui tqv npdft mo ororye(o rououv6lou,to onodtreoponou no(pvougeovrirerrr our6 mo otivotro.fro nopd6erypqro orivotromrv quoxtirv oprOgtirv efuqrKtrerord rdoo yro rqv npdoe€on6oo rqr ylq rov noMonlooroogd.Aqlo6{, rd0e gopd nou npoo0froupe66o guotro6qopr0po09to onor€leopoe(vorn
KEOAAAIO 9
roroldpoupeyror( q rleror6rqroe(vot1.rto oqpovrtr( t6r6rqto.rqpo! v€t ort, 6tov Xprlorponororipe rrg rdooeprg npd(ergrqg opt0prltrrr]g, yvtrrp(ouge - €vogdtrtrog rt 0o e(vqrto onor€treopo pqt6qopr0p6g. Tr ougpo(ver dtov ouvOud(oupe dppqrougror pqrodgopr0potiq; E{o_ rotrou0e( vo roxferq rtrerordrnro; Mnopo6pevo 6odpeopdourq 6' or (6roror dpp4totopt0po(6evs(vorrtreroto(, yror(dXoupeOfz).([4 = z. Kordouvdnero, €Xouge pto nep(nttrrorl 6nouro yrv6pevo 66o dppqrov optOptive(vorpqr6goprOgdq. To yevrr6repo eptirr4po e(vorov Xdvoupr qv rleror6rrlrodrov dXoupe po(( dpp4rougror prpodgopr0po6q. Av opX(ooupe vo Xprlorl.lonororipe ug tdooeprg npd(rrgtrlq optOprptr{q oe dppqrougoprOpodq, 0o opX(ooupe dpoyevo no(pvoupe nopd(eveq, trrr1 oprOpqrtrdg ovr6rqreq og onovrrloerg; H yevtrrlovrlouX(o nou 6vurr0ov ot poOqporrro( oXerrrdpe rouqdp_ prlrouqopt0podgnpoXtrrpoUoe nipo ond to eporr]pororrlq rtrerotdr4_ roq Korrrlg 6rdro(qq.H oUyypovq poOqpourrlovdtruoqpoo((etotorov onerpoorrr6 troyropd, Koro or€rpoorrr6g troyropdg dxerro oep€hdrou orrl 0eop(orov op(trrv. AIld 11opXrrr]0etrrp(o rov op(trrvXplorgoror_ ouoe6vvoreq on6 q yeoperp(o, dnoq,yro nopd6erypo, 6u proorotrou_ 0(oo4pe(urv nlrlord(er o'6vo opror6orlpe(o. Mnopotroope 6purgvo 0eurp4ooupe 61 6lo ourd ro oprordorlpe(ooupBdlt(ov 0eptroUq opt0_ porlg;Mrjnurq0o rordppeeo}6rtrqpoqo ndpyogon6 rponouldXopro nou ouyKporoUo€ ro ovtirepopoOqpotrrd, ener6qq Oerrrp(o turvdpp4_ rov oprOptirv 6e oqpr(6rovyepdorrlv€vvororov prlrtiv optOpdrv;
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I TOMH TOY NTENTEI(INT
on6 qv dlyeppq ytot( 6troro owtre(pevo rqq yeopetp(oq to broXurpropd 'Otrttlg, noolyepptrd. qg yerrrperp(oq vo ovonopomooo6v 6cvpnopo0oov npo vo ovltp€ront(et pdro 6roXurprog6 e{orotrouOotiot out6,q yetrrgetp(o nou oqpf ptrriporo. onode(erg tou Eurle6qXprlotponotofoe H yecuperp(o (g{q, eppo6( 6yrot) (ovrovorqv owiAqrpq6rt ro yeogerptrdgeydOrl rjrovouyrp(orpogeto(6toug.Oor6oq Kdnotoon6 ourd ro pteyd0rl {rov ptero[6 roug; g{rq vo ro ouyrpr0o6v prirrl. Mnopotioov ourd ooripperpo pto0eurp(o yroro oo0gpetpop{rq. yeurpftpeq 0r'E}trr1veg Xpetd(owov (nep(nou n.X.) 400-350 Kv(6rog o O E66o[oq 4tov po0qrrlqtou l-ltrdgoOqo'€vov on6 roug onou6otdrepoug'Etrtrqveg TovoKore(etr(X0qre 'Htov prolro0q. en(oqgyrotpdqror ootpov6poq,Kclttou ono6(6etot11 q ono(oe(r1ye( rtttq rpotvogevtrdg o
KEOAAAIO 9
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236
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I TOMH TOY XTENTETINT
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Frorov€TnorqprorfOioq ovdtroyrlF€ rrq rrov6trltdqtou. Og er O€Xrrlre vo 6r6d{erpoO4portrdo'ivo r6rouxdoXotreio 0qtr€trrv. 1869, o Krivroppp{re pro rev4 06o4oro prrpd flovenrouipro tou 0ioq nou e(orolouOorio€ vo pQVe(vorrod(ror4g e(o(peqqertou oro llovenror{prorou Bepotrivou. Eyroroord0qre, Iorn6v,oro Xdtreror dyrveovontrrlptrrrrjq roO4yrprlg (6ro 1872. To €roq,yvrrrpior4xe ror €yrveqiAoqU, €vovdllo veopd po0r1pour6. To 6vopdrou {rov prXopvtNr€vrerrvt- o dvrou UoSi6ooe ro o0yXpovooprop6rov dppqrtrrvopr0ptirv, 14 6rdor1pr1 epyoo(orou rrl Xpovrdnou yvriproetov KdAuro(or 60o po0qporrro(, nou 0o ouvrrodrpepov rdoo noMd oto oro t6trqrou 6€roroudvorououirvo,€yrvovotevo(rpitror ouppepi(ovtov ror unoor{pr(ovo ivoq uq r6€eg tou dtr}ou.
E i v oorp r 0 p f i o r p ootr o A y € p p r r < oo p ir 0 p r o i ; "0rolyepprro( oprOpo( neprtrogBdvouv 6trouq rouqprlrorigopr0godq, ro0tigrr ere(voug ioug dppqrouqopr0porjq nou onotetro0v trdoerq nolutrrvriutrrv. E(voroprOp{orgoro orivo}oturv otryrBprrdrv oprOptirv; Aur6rpo! vetornohj 60orolo np6Blqgo,otrtrdo Kdvropro €luoeedrotro.Aut6 noudxove{rov vo rono0et{oeroe oXiorl6rdro(r1q touq otryeBprrodq pelertirwoq opt0poUq, ro notru6vupo tov ono(urv onotelo0vhioerq. fro rdOenolutitvupo,o Kdvtopdproe€vovoriporo opr0p6,rov H, rovono(oov6pooerirpoq rou noluovdpou.Oupq0e(te 6u €vonolut0vu_ podXetq popgli ooxu + o,xu-'+ orxu'* ... * ou-rX + ou= 0. Op(oupero dQoq tou oq e{rjq:nptirrov,unotroy[(oupe ro v-1, ro ono(oe(vorontrrirq o peyotr0repoq rr0€rqqrou x pe(ov1. Kordnrv,npooO€touge oe ourd qv ondlurr; trgrj 6trurvrurvouweleottirv.Aiyovroqondtruq lp{ evvo_ q Oeurqtrprj rou ouvtetreonj, oUpe ove(dprqtoond ro ov ou{ 11trpq efvorOerrr{{ opvrlnrrioto no}urirvulio. Aurd rdverro 0rfrogrou no_ lutrtv0pou vu roodtorp€ H = (v-t) + lool + lo,l +...+ lo"l, dnouor ypopp€q p€oootrqono(eqpp(orovrotor ouvteleot€q onorrofvvo Xpn_ 275
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orponorodpe trq ondtrureq rrp€qrouq.fro nopd6erygo, to ontrdnolurir= = = = + vupox 1 0 6Xer 6$oq2, yror(oo 1, or I ror v 1. Aurdpoq6(ver H = (1-1) + 1 + i = 2. Mnoporipevo 6oripeop6ourq 6u, yro rdOelprj rou H, dXoupe 6vo pdvonenepoop€vo oprOgdnr0ovrirv notrurrrv0pr.uv. O fl(voroq10nopouord(er ro ovt(ororXo nolutirvugo yto uS tperqnp6teq rrp6gtou H. lto H = 1, €Xoull€ 6vop6vonotrurirvugo, ro x = 0. 'Otovro H e(vot2, dXoupe t€ooeponotrurirvullo ror 6rov e(vor3 no(pvoupe dvtexo6roqopetrrd notrutirvupo. Kord ouv6n€ro, o opr0g6qtov notruov0polv nou ovrrororXoriv oe rd0e rirpogou(dverypriyopo.Oordoo,oe xd0eouyrerprpdvoH ovTlororXe( 6voqg6vonenepoop€voq opr0p6gnotru0v0pov. TtirpoXprlorponoro0pe to 6rdor1trro Oedrpqpo tou Koptrfrdouq,to Oepetrui6egOedlpqpor4g AtryeBpoq. A6erdtr rdOenotruulvuprr{ e{iotrrorl 'Evo 6Xerg(o routrdXrorov l0or1( p(o. n6proporou 0errrprjgor6g tou e(vor6n €vonotrudrvugo €Xer(6roorprBtirg opr0p6}0oeovpe ro po0pd rou nolurrrv0gou. Enop6vrrrq, rd0e notrutirvupo iXerouyrexprpdvo oprOp6 trUoeurv' yro nopd6erypo, ro notrutirvullo x' + 2x - 5 = 0 iXerrperqtrrjorrq,evtirto notrutirvugo x' - 7x + 1 = 0 dXer6rio trrjorrq.fro rdOenoIutirvugorou fl(voro 10,pnoporjpe vo roroypdr[oug€ uS ]0oerqoprivovroqrouqoprOpofqnou enovotropBdvovror. Tdre,yro rd0eo0votronotruovfprrrv rou ovrrorotXoriv oe rdOerrpri tou H, €Xou1le ivo oro0ep6 opr0p6govo6rr6vtrdoeov,6r1lo6qotryeBprrdrv opr0gdrv.
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guotrt0voprOptirv v rot ovdproo€ to N. Enrnldov, ndpe,oq no0ye,to ori_ votrodltovtov 0etrr6vnpoypotrrtivopr0ptilvx rqt ovduoolito R.Aor_ n6v,4 ep6t4o1e(vorontrtiqI e(r1q: pnope(ro N vo ouv6uoote( ueto R pet6torotpdno,droreoe rdOeotorxe(o tori evdqouvdtrou vo ovrrotorxe( 6vo ror g6vo €vo otorXe(o rou dllou ouvdtrou;Me puonp6q proud qnovtderrove(q:<'OXr, our6 e(voroOtlvotov, yror(to N onotele(roron6 (extrrptotdtyritrroroevdlto R E(vordvo ouvex€qorivolon.Autrj 6ptrlgrg €votooq6ev qnooetrvuer t(noto. Kr doo rr ov nrore0od'to N roi R 6evenrtp€nouv 6vovtdtoto ouv6uoopr6, e{orotrou0dr vo prrlvpnopti vo pprrrr4v orr(o.Koroutr]q ort(oe(vornou onooXotrei pnope(vo e(vor ;.re notrriqntrf.5 O Kdvrop, e66, iXer oyvor]oerrouq opvrlrrKoUg npoygorrrodgoptO_ poUq,oltrd out6 6ev onotde( npdptrrlpo.Av pnope(vo pper Uro oilsr_ r6vroq yro dlouq roug 0euroriq npoypourotiq opr0po6gen( rov rpuorr6v opr0pdrv,rdr€, €rr€Kr€(vovtog tqv (6ro reXvrr{, pnope(vq oupfi€pr_ ldper ror 6louq rouq opvquroUgnpoypotrroUqoprOporlq.O Nrdvrerrvr €ypoQeorov Kdvrop dtr 6ev rj(epe vo onovrqoeroro epdrrrlgdtou. Iul ouv€Xero,orrg Z Aereppp(ou1873,orrdr prdvorlpip€S perd ro np6to ypdppo tou, o Kdvrop iypor[e ndtrrorov Nriwerrvr. Teleuto(oefio ro Xp6vo,vonpoXrrrp{ou_r trtyoneproo6tepo rqv etroo(o fiou oou ovErpepo. orlyeponroteUu_r 6tr tetre(trroo ;re out6 to O€po.Av outonotc0pror, 6e 0o pptrrenrerr€otepo rcprtrjon6 eo€vo.flo(pvtrllotndv to Odppoq vo unoBdlrrr otrlv rp(oqoou ourd nou dXrrr ypd{rer,nopd trg otdAeteg nou pnopeivo riXer to nptirtonpooX€6ro.u Ie 6rdorqpo tr(yo neproo6repoon6 pro ep6opd6o, o Kdwop o16_ woge ndvtrrotrlv ovordluQq rrlq (orjg tou _ 6u or npoygolro( oprO_ po( 6ev e(voropr0prlorpor,yrotf e(vorunepBolrrdnotrIo(.oo rorrd{oupe 66o ond trq ono6e(etq tou. H npdrq e(vor11ond6er{r1Troulropou_ o(ooe orov Prxopvt Nt€vte*vt ro Aeripppro rou rgz3 ror 6qpoo(euoe
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Iqv nopondvruorotrouO(q, or 6e(rteq6eiXvouv touqrpuorrofgoprOpodgen(tov ono(ctrv onerrov((ovror or rpoyporrro(opr0po(,6qtro6{ro Ordrpopo o. An6 rqv und0eoqrou Kdvrop,11nopondvur orotrouO(o nepriXer 6trouq touqOetrrodq npoygourodqopr0porlg. Agndpouge6vorp{po rqg euOe(oq rov npoypourdrvoprOprivror og p. ovopdooupe to drpo rou o Kor Iqv npoygorrrdrqro,6ev€Xer oqpooio 7. joseph Warren Dauben, Georg Cantor: His Mathematicsand Philosophyof thetnfinite(t1p(votov, PrincetonUniversityPress,1979),o. 50.
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306
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308
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310
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KEOAAAIO I2
roXrier 6fo ntrrl0txofqoptOgo0q, Pr(oond 1re(6rr, yro onotouo6r]nore rrqrp€tgoX€oetq. rqq rotrrig6tdto(qq 6ev 66tveondvtrloqoe 6Io ro To Oedrprlpo opt0rov uil€pn€nepoopdvtrtv nou unrlpXovorq Oeurp(o npoBtrrlgoro rou Kdvrop.Ot rpuotro(opr0po(rorolr]youvorov np6ro uil€pns1-uirv rorotr4youv opr0trro( opr0p6,rov o. Ot unepnenepoo;tfvor nepool.tivo rot o oe ovr(rpoor1, rdtt r€rotoo6r1ye( dpoyeoro o; To vo uno06ooupe norprdvouv 6e oprOpo( 6tt ot unepnenepoo;r€vot Kdvtopnopo6dyr4re opr0go('Aurd ro 6totdntloeoro t6 oe rdnoto 6pto, 6nurgot
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notorond ouro6groug ovOptirnoug €yrvovtropnpo(snt;.reyoldrvowoq or4povegrot proOq;rortro(, evcO dllor rlrov 6rovorlrrrdro0uorepqg6vor. Aur6ro eupdrpdopo6rovorlrrrcilv rrovor{tcovuno6rltrdrver 6rr 11rrov6rrlro yro voepoUq unotroyropoOq 6evonotetre( prroq yevrXopoKrrlprorrr6 r4g eurputog, otrtrdr(vor6voe(er6rreuprdvo rotr€vro. Irtq p€peqpog,ondvrovuirOouprr 6€oqon6 rov yprjyopovoepdXerprop6peydtrulv opr0pdrv. Aurde(votnr0qv6vvo 6e(Xver €vovr6oponro noldntrororor Bopreorrlpivo, r] prnopr(vo orpeilerororrlv roXdroul ovdmu{q rov unotroyrotirv, pqXovtirv Tr.,Jv 6nlo6n nou rdvouvroug unoloyropodq rorvorol(o.[ror( vo e{oorodpoorenoltron}ootd(ovrog e[orf{rproug opr0goUq pe ro puotr6proq,drov or uno}oyror€g nou €Xoupe oro on(rrpoggnoporivedrotrovo notrtrontrootd(ouv pe6eopr0trro6g ron6vtE{ e(rootCInq(o;O€}urvo nrorEful6rr dXoupe onoporpuv0e( on6 rqv ntrlrrrrrj rupovvfoour6v rov Xpovop6purv unotroytogrirv, yro vo orpo
344
1.StevenB.Smith,The6reat MentalCalculators (N6oy6prr1,ColumbiaUniver_ sitypress, 1e83).
' Y E I EA P I O M O M N H M O N E E
tou NtdpotrvrA' €Xoupeoro BtBtr(o vegprero(0rov rpuXtrdoo0evtirv rou Oov vrt Yye(og torpoU,6teu0uvt]ev6gK€vrpoutUuXrrqg Tpdrpepr, E{atptttxo(dv1punot.' Aor,oro Outordvotv,
0rtrl v ou vn 0 rop€vt- r v A p r0 p r o p v d p o vepsroq g ov0pr,0nov 11npotrton6 ouvqOtop€voug optOpo0g ge peydtroug Or unotroyropo( ond 6vo oovotrot6ro(op€vougovoptirnoug xaporrrlpi(ovrorouvr]or..-rg opt0poot neptoo6repot rot ouv0qr6v'Kor'opXdg, tepov6e(torqturv 11etXov,e(otpertrdov€nruYrot €Xouv, pvripoveg {rov not6rd-0oUpoto ot dv0ponorouro( preydtrrrroov Ie notrIdgneptnrdoetg, pfvq pvr1pr1. rot, pt€ooorq povo(tdtoug, €go0ovvo oyono0vroug onogovopdvot vo perpo0vrut vo rdvouv ovoYKootrt€vot IuXvd,rpo(vovrot opr0po0g. ro}€vro Io toro(repo floltr€gqopiq, 6tov ovorotroTrr€Iot unotroyropo0g. (6tot vo no11dtrlornpoonoOotiv ot opr0gogvrlp6vtrtv, ttrrvt6totputitv opt0gogvrlnotrtro( ro rutr€vrolouq oro rorv6.Enopdvtog, oouordoouv poveg€6tvov rot 6(vouvnopoordottg6toore6d(ovroqorpooqpto Kotrrlv AgePtrrl. otqv Eupdrnq o' €voneptoptoptdvo nou rdvouvepn(nrouv To e(6qrov unotroytoptirv e(votdvorup(opXo0dorivotroroqyoptdrv.flptOrov,o notrtrontrootoopdq pero(6roug Fo rov nopoordoetivrouq.Mnopodvrot notrtrontroord(ouv €vovopr0p6oe plo 60vopq'Aurdgot 11urlrdrvouv opr0pto6g 66ogeydtroug ge tp(o 11rEooepoUnq(o, t(re oprOgodg orpopodvouvqOurg enr6e(erg 116tQqrpte(re rfBo, 11 mov rerpdytlvo oro nou utlttitvovrot Uovotlliqtouq 6qgoEntntr€ov, 6uvdperg. oe geyotrrltepeq nou utfttitvovror ouqoprOgofg opt0ptitv,nop' 6troTIouro rurv pt(tirvgeydtrrrrv rprtrrig e(vorror 4 e(oyrrryri nep(epyoe(vot6u q t(oyuryripr(tirve(votpro 6to6troo(otrty6reponotrUro rorv6flourouqnoporotrouoef. ouvr]otrrg ntrorqon, 6oo ouver6qtonore( People(Nio Y6prr1,Harper& Row Publi2. Darold A. Treffert, Extraordinary shers,1989).
325
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gnope(en(or1q Or opr0gopvrjFovsq nou 6(vouvnopootdoerg vcr€Tn6errv6ouv tqv rrovdtqtdtoug vo pp(orouvpeydlouqnpdtougoprOpofq n vq qnlonotodvpeydtroug o6v0erouq aprOprofg, nop6l 11rrovdrrlro ourrl 6evenrdetrvtieror rdoo ouXvd6oo o notrtrontrooroopdg ror 11 e[oyury(pr(tirv.Kdnororopr0popv(goveg enr6errv0ouv sn(or1g rrlv rKopeydlougoprOporig geydtreg v6rrlrdrous vo onopvrlpoveJouv outrtro4 y€qoptOptirv, 11vo pp(orouvU€Uroporrdrov ntrr10rr6 oprOgdev6qouvdtrouovrrrer;ldvov. Or optOpopv{poveq ondvronopouotd(ouv nopobstyporo np6o0eyro H 6ro(peor1 oqg11orpofpeons. rnv ontrono(qoq, oIXprlorgonore(ror peld ourrl ro0ourrl6evnopouord(eror. Or noltrontrooroour€q oXdoerg rpo(vetor ro{U tov oprOptirv ntoqyorlteriouv rouq oprOgopvripov€q Kor, e(vornro e6rolovo rrq Ouprofvror. enog€voq, lldoo onoubo(eq e(vorour€got rrov6r4reg; Av rrg 6oripe€flrqovsrord, rdvouvro puotrdFoSvo ooor(er. lro nopd6erypo, o T(evrevrdto 'Elprov, Mnd(rov,nou yevvrl0qK€ ro 1702 aro orqv Ayy}(o,ovorpdperot 6rt rpov retre(rr-rg ovotrrpdpqrog rqr nr0ov6vbtovorlrtrdroOuorep4p€vog.Oordoo,U€To 1.ruotrd rou, tirfrooeoro rerpdyrrrvo dvovopr0p6 perprdvroevv€o$r1
YEIE APIOMOMNHMONEE
ooord 60o opr0pofqge 6erorp(ot|4rp(oo ro06voq notrtrontroord(ovroq opr0g6gdvooe e(€vo bcroe{orfqrpto rorno(pvovrcgog onordtreopo tuXo(oon6 unoOt 60o optOpo(enel6yrloov xootort6 6eurep6trento.
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K E O A A A I OT 3
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l-lp t il r oK r o l o 0 v e€roto p r0P o i nrrvofvOerov e(votq ontronoQo4 Toreleuro(orotr€vtonou0o petreqooupe A0vowogoptog€vo rov npdtttrtvopt0ptirv. rot o npoo6toptogdq oprOptirv no(pvouv€vo npoBtrriporqrdnotote(otperrrdypqyoporoprOpopvqpoveq rou. fto nopdmougnpdnouqnopdyow€q peyotro opr0p6Kotrov ovotr0ouv mouq 30 gnope(vq "ovo^ue€i,ri vo nopoyovronorl0e( 6E,yFqo opr0trr6q optOpoi nptirot Kot 5 ovopd(owot 2, 3 or opr0go( 2'3'5. ftor( oprOgo0g oPlIMoL flptitrogap0p1Ee(vorrivogguorr6qopt0p6qnou props( p6voon6rov eourdrou Kotrov opt0g61' r6trsro vo 6totpe0e( ritreto nou 6eve(votnptittot,pnopodvvo 6torpe0o0v Atrtror opt0pto(, opr0go0gon6 ro 1 Kotrov eour6roug.l-ronopdon6 neptoo6rspouq on6 ro l, ro 4 Kotro 2. O optOro 4 pnope(vo6totpe0e(tdtrero 6ery;ro, ovoon6 roug1,6,2 Kat3. Auro(or optOpo( 9696 gnope(vo 6totpe0e( pd(ovroro0v0erotoprOpof. oPlIMoI: IAv1nog apfip6q e(vot€vog rpuorrdgopt0p6gnou n6poon6 pnope(vo 6torpe0t(tdtretorot pt d}trouqoptOgotlg To1 Kotrov eour6rou. 6rt 6troror rpuva 6or1;re E(voterirotro An6rouqnopondvtlopropo0q oe 6dortrdoetg: nou E(votptyoldrepoton6 ro 1 XuJp(ovrot orxo(oprOgo( vo Aevgnopodpe opt0po0g. opt0gotigKotrouso0vOeroug tougnpdrroug opt0pofg,otrtrddtrotot oe ptrpdrepoug €vovnpdro opt0pr6 ovotr0ooupe np6rov oq prooutrtroyr1 gnoporivvo ovonclpooro0o0v optOpo( o0v0eror peto{drouq.Autdo64yefo' 6vovon6 rouq nollonlootoopivurv oprOgdrv o ono(ogtrdetdrr undpXttdvogpdvo turv poOrlporrrdrv, tr(Ooug 0ep€trroug €Xet rdOeopr0p6g €vogoprOgdq'6r1lo6q, rp6nogperov ono(oovotrUetot np6ttlv nopoydvrov. €voxorpdvo€vooOvotro opt0p6g Kd0erpuorr6g THI APIOMHTIKHI: OEOPHMA OEMEAIOAEI on6 to 1 gnope(vo €Kqpoors(tog ytv6pevo trreyotrUtepoq g'€vovror povo6trdrp6no. np6rov opr0gdrv ;
329
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Kdnororrbtoque(g oprOpopvrlpovrs, yrcl vo notrtrqntroordoouv 66o peydtroug optOprodg, ovolfouv npdrrotouq oprOpotiq ror perdopX((ouv vo rdvouvnpd{erg[-l€rous npdrrouqoprOpoOq, 6orE vo o6r1yr10o6v orrlvondvrqorlroug.Oordoo,or po04prouro( ovoyvop((ouv rr €voBoO0reponp6ptrqpo oE 6,1 orpopdrougnptirroug oprOpodq, n€poon6 rrlv ontrr1 YndpXouv dnerporrpuotxo( Xprjorlrouq orougunotroyroporig. oprOgo( nou e(vornpdrror,otrtrdor npd.rror opr0po(oportirvouv ro0tirgor rpuorro(opr0po(y(vovror6Io rot peyoltirepot. To npdptrrlporou vcr ypnyopo geydlor noror opt0po( i(vor npdlror Kornororo6v0eBp(orouv rot rtvE(ro ev6rorp€pov rurvpo0rlporrrdrv e6tirror ot6veg.Tr ouprpo(ver pr rrlv rrovdrnrorou pr€oou ovOptirrou vo ovoyvtrrp((er rougnpdrroug opr0trrofg; O opydprog, o
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'YEIf APIOMOMNHMONEE
Aut{ q teXvrrrle(vornop6goro1.r' ororXq rordlq[r1oro o0v0erooprOpd. yro rrlv e{oyoyqpr(tirv. ere(vq nou Xpqorponore(ror peto nrOovdrorolqfro nopd6erypo, o xotdtroyos rou K6trpnoupv 2.983onoretre(tor rlxd tpqq(orov nopoy6vrovrou orjv0erouoprOpo0 Aqlo6rl,ro 83 gnope(vo oxll.rouoreip6vo onde(roor(e0yqoprOprirv. (efyq. E(rtd(ovroqourd to (efyq, e(6up4rpro onde(roor6rorpoperrrd vorOuvotdvo ovorotr0{roup€ rous nopdyowegrou 2.983.To np6ro (eUyoq rorotrqrcrrtivQqq(ove(vorro 01 Kor83. Arporiro 1 6ev e(vor napdyovroq(tr6yorou 6rt e(vornopdyovroqrdOeopr0pori),nrOovo( prnopo0v vo e(vorro 101Kolro 83.To yrv6yev6rouq6poq nopdyovreg ofioroe(vot€vogopr0p6qU€yoldr€poq on6 ro 8.000.Av enrouvdrlroullr 6r]noredtrlo {qrp(ooro 01 Kor83,ro yrv6pevoe(voror6po peyotr0repnoporlpe poropr0go(.Enog€vog, vo 6roypdrlouge our6 ro nptiro (eriyog.To 6e0tepo(efyoqe(vorro 03 Kor61. Mnopodpe vo 6o0peop6ro Vrlooq 6rr to 3 6eve(votnopdyovroqrou 2.983(ov npoo0iooupe ge ro 3), rr dpo q endpevq rpio0o ovorotr0rloupe 6116e 6rorpodvror nrOovdrqro e(vorro l03 x 61.Aroypdrpoupe 6poqrr our6vro ouv6uoop6,yro rouq (6rouqldyouq lre rov np6to ouv6uoop6.Telrrd,drov rptdvoupe oro 6y6oo(edyoq,ro 19 Kor57,pp(oroupe rougnopdyovrdq poq.Oopr0p6g196evgnope(voy(ver119,yror(r6teto yrv6pevo nou dputg6tt 0o npor0Qer 0o e(voruneppohrdpeydlo.Iuvet6qronotdtvtoq 20 x 150tooUrorp€ 3.000,rorotroBo(voupe opdooq6rr ro 19 x 157 roorirorge ro 6tr6 FoS2.983.fro pog,prpo(og,n Xpt]onevdqr€rorou e(voropyrl ror dXoprl,otrtrdoto X€proevdq16ron(voxoontronoiqoqq quo0qoprOgopvrlgovo, r6ro(tepoev6gor6;rounou €Xeteooreprredoet oye66v e(voreUrologror rpo(vetor r4v 614 6ro6troo(o,o unoloyro;.r69 ourdporoq. o Kdtrgnoupv ror dtrtror O 6uf{rprogn(voroqnou orrogvnp6vEuoe po{0ero yro rlv ontronoiqoq. O enrnldov 6ev enopre(ror Xperd(eror rjrov 6voqrOrorpu{g oprOgopvrlpoT(opr(fldprepMn(vtep(1806-1878) BprlretuXo(odvovoly6pt0poyro uS voqon6 ro M6pergvrqg Ayytr(oq. ontronor{oetq, nou {rov q6r1yvoor6gorouq poOrlpottrofq.M6vo ot oto npoptrrjtrroro onlono(qoqq, 331 neprrro(oprOgo(iXouv ev6rorp€pov
KEOAAAIO I 3
orpouonoroo6rlnore dprroqopr0pdqbrorpe(ror U€ro 2 rot pnope(vo onlonorrlOe( rot vo ovoX0e(o, €vo prrp6repoo0v0eroopt0p6.,Evoq neprrr6gorivOeroq opr0pr6g pnope(vo yporpe(u;gyrv6prevo rov ropo_ roro: (o n F)(o - F) ro ono(o,drov notrtrqntrooroote(, pog 6(ver:q, _ Enop€voq, rd0r neprrrdgoUvOerog B'. opr0p6qpnope(vo yporpe( og 6torpopd 6rjo oprOpdrv uqrop6vov oro rerpdytrrvo, rov o ror 0. O Mn(_ vrsp Xprlorpono(r1oe ourd ro yeyov6gyrctvclpp(orerrougnopdyovreg. Av o x e(voro oprOpdq nou qrdxvougr, r6re x - o, _ B, r] o, _ x = p.. Enopdvrrrg, ovo(nrdgeonltirg€vovopropd(.flo_ poqprlore6rr (o + t)'= o' + 2a+ 1. Enop€voq, yro vo ou(r]oougs Ko_ .l rd ro rerpdyovorou cl,ro pr6vonou Xperd(eror vo rdvouuee(vorvo npoo06ooupr oro 0'To 20 + l. Xprlorponordrvros rrq 66o reXvrr€q rurvrorotrr;(eov rov oprOptirv ror rov brorpoprirv rov r€rpoytirvov, or opr0popvqpoveq pnopotivvo onlo_ nororjvypfiyopoopt0;rodg nou rprdvouv€urgro dvo erorogprtlpro Kor, peprr€g qop€q,6o9ro 6€roeroroppripro.
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400
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