P5047G@TEMPLEVM.BITNET (Apostolis Apostolopoulos) (02/10/90)
Geia sas Hellasites Gia na spasei ligo i monotonia twn minimatwn peri Kouvatsou, Tourkwn, kai twn diaforwn melanxolikwn poinmatwn, idou ena provlimataki gia to Savvatokiriko. 1=sqrt(1)=sqrt((-1)(-1))=sqrt(-1)*sqrt(-1)=i*i=-1====> 1=-1 Pou einai to lathos????????!!!! Pou spaei n eksiswsi??????????????????? Apostolis Apostolopoulos Temple U.
ANEZIRIS@SUHEP.BITNET (02/10/90)
Oi migadikoi arithmoi ehoun apo dio tetragonikes rizes o kathenas, an y*y=x
tote (-y)*(-y)=x. Me tous pragmatikous dialegoume ti thetiki riza, me tous mi-
gadikous ta pragmata den einai toso apla. Otan grafeis 1=sqrt(-1)*sqrt(-1),
sqrt(-1) einai kai to i kai to -i, ara i ekfrasi den einai kai toso safis. To
mono pou simenei einai oti 1 anikei sto sinolo {i*i,i*(-i),(-i)*i,(-i)*(-i)}.
Afto ontos simvenei. Me trik tetragonikon rizon "apodiknieis" o,ti thes, p.h.
b*b=(-b)*(-b), ara a+b=a-b.
Tora mia alli erotisi: orizoume to i na einai i tetragoniki riza tou -1. An
kaneis mas rotisei "poia apo tis dio" ti apantame;
Telos: to polionimo ax**n + bx**(n-1) + ... + fx + g opou a,b,...,g einai
n+1 pragmatikoi sintelestes, mporei na min ehei kamia pragmatiki riza, p.h.
x**2 + 1 . An oi sintelestes einai migadikoi, ehei panta migadiki riza; Nomizo
oti i apantisi einai katafatiki kai i apodiksi topologiki.
Xarilaos AnezirisST401843@BROWNVM.BITNET (thanasis kehagias) (02/10/90)
opoiosdhpote migadikos ariqmos exei n rizes niostes tachs. auto sumbainei
dioti an to z einai niosth riza tou b, tote z**n=b => z**n -b=0.
alla auth einai mia poluwnumikh exiswsh . to b mporei na einai migadikos
ariqmos. genika opoiadhpote eciswsh z**n+a(n-1)*z**(n-1)+...+a(1)*z+a(0)=0,
exei n migadikes rizes. auto einai to qemeliwdes qewrhma ths algebras
(pou ston Ntziwra, Algebra Pempths Gumnasiou, parousiazotan ws aciwma ?).
h apodeich den einai topologikh. basizetai sthn qewria migadikwn sunarthsewn.
sugkekrimena, to qewrhma tou liouville mas leei oti mia migadikh sunarthsh
ANALUTIKH kai FRAGMENH sto migadiko epipedo einai staqerh (apo auto
eukola apodeiknuetai to Qemeliwdes Qewrhma ...). to qewrhma liouville
meta seira tou basizetai sto QEIKO qewrhma tou Cauchy, pou leei
oti oi times mias analutikhs sunarthshs mesa se ena xwrio, einai
plhrws prosdiorismenes apo tis times sto sunoro tou xwriou!!!
otan milame gia niostes rizes enos migadikou ariqmou, ceroume oti
an h r*e**(i*theta) einai mia apo autes tis rizes, tote
rizes einai kai oi r*e**(i*[theta+2*k*pi]/n), k=0,1,2,... .
autos o tupos dinei mia apeiria rizwn, alla ousiastika pairnoume
mono n diaforetikes rizes, dioti otan k=m kai otan k=m+n pairnoume
thn idia timh. oi rizes autes einai summetrika topoqethmenes se ena
kuklo me aktina r, kai apexoun metacu tous 2*k*pi/n aktinia.
ola auta uparxoun kai pali sto biblio tou Ntziwra. epishs ola auta
uparxoun kai sto biblio tou Schaum Outline Series , Complex Variables,
by M. Spiegel.
thanasisgeogiou@GN.ECN.PURDUE.EDU (Ioannes T Georgiou) (02/10/90)
1=sqrt(1)*sqrt(1)=sqrt(1)**2=(sqrt(-1)*(-1))**2=(i**2)**2=(-1)**2=1 ===> 1=1 1=sqrt(1) odhgei se paradokso.
nbp@SWORD.BELLCORE.COM (Nikolaos-John Pronios) (02/11/90)
APOPSTOLI: Quiz: 1=sqrt(1)=sqrt((-1)(-1))=sqrt(-1)*sqrt(-1)=i*i=-1====> 1=-1 APANTHSH: 1=exp(j2pi+2kpi) k integer (2k even) sqrt(1)=exp(jpi+kpi), k integer, even or odd Gia non-complex pairnoume k even (zygo) esti wste k=2l, 2lpi=> plhrhs peristrofh, pali stous pragmatikous. H ypo0esh (assumption) einai oti douleuoume se grammh kai oxi se epipedo. An einai ka0aro oti douleueis se migadikous, 0a prepri na pareis oles tis lyseis (k=-inf, 0, inf akeraios) oxi mono tous zygous (even). -1=exp(jpi+2kpi) k integer odd or even (-1)(-1)=exp(j2pi+2lpi)=1, l integer, akomh douleueis ston aksona twn pragmatikwn. sqrt(-1)sqrt(-1)=exp(jpi/2+kpi) exp(jpi/2+kpi) k integer (vale k1, k2 an 0eleis) Se auto to shmeio, douleueis sto epipedo kai oxi se grammh (eu0eia) -------------------------- exp(jpi/2+kpi) exp(jpi/2+kpi) = i * i Edw agnoeis ta -i pou pairneis gia k=odd, pairneis mono k=even, pou keintai ston aksona twn fantastikwn. Etsi auto pou ginetai einai oti eksiswneis ena apo ta dyo apotelesmata pou einai dynata an douleueis sto complex plane, me apotelesma pou pairneis an douleueis ston aksona twn pragmatikwn. En pasei periptwsei, 1=sqrt(1)=sqrt((-1)(-1))=sqrt(-1)*sqrt(-1)=i*i=-1====> 1=-1 ^^^ Ypo0etwntas oti eixes ksekinhsei me migadikous, edw einai to la0os. H isothta den isxyei. ^^^^ 0a mporousa na to valw kai edw, mia kai edw ginetai h metavash apo pragmatiko se migadiko. AN synexizes kanonika, exp(jpi/2+kpi) exp(jpi/2+lpi) =exp(jpi+(k+l)pi) kai gia k+l=odd pairneis 1 gia k+l=even pairneis -1 Auta, elpizw na mhn sas kourasa, Nikos P.
patrinos@RODAN.ACS.SYR.EDU (Anthony J. Patrinos) (02/11/90)
Sto gumnasio thumamai eihame mathei oti akribws gia na apofeuhthoun asafeies autou tou eidous, i tetragwniki riza, alla kai opoiadipote ypsosi se dinami ehei noima (i orizetai) mono gia thetikous arithmous. Auta - geia hara Antonis Patrinos
PETROPOULOS@NUACC.BITNET (02/12/90)
Ston orismo 1=sqrt(1) kai sthn mh thrhsh tou stis epomenes isothtes.
P5047G@TEMPLEVM.BITNET (Apostolis A.) (02/12/90)
Geia sas hellasites prwta tha nthela na anaferthw sto filo (den thumamai onoma) pou eipe oti sto gimnasio mathenamai otu prepei na ipsonoume mono thetikous arithmous se diafores dinameis. Den kserw alla prepei o en logw filos na einai panw apo 190 xronwn. Twra oson afora to filo Niko P. Tha elega oti tha prepei na ksanakitakseis tin apantisi pou esteila, deftero, hrafeis:1=exp(j2pi+2kpi) afto mporei na ginei: 1=exp(2pi(j+k))=exp(j2pi)*exp(2kpi)=1*(kapoion arithmo dedomenou tou ti einai to k) epomenos olos o sillogismos sou einai lanthasmenos. Apo tin alli safws doulevoume sto C logw tis genikis tou morfis alla kai tou oti mporoume na kanoume xrisi tis enoias branch kai branch cut gia na orisoume katallila ti migadiki mas sinartisi. R iposinilo C Apostolis A.
geogiou@GN.ECN.PURDUE.EDU (Ioannes T Georgiou) (02/12/90)
Ean kaneis arxisei me -1 + 0*i = sqrt(1 + 0*i) den kataligei se asafeia. 1+0*i = exp(2*k*p*i) ----> sqrt(1+0*i)=exp(k*p*i)=-1 + 0*i
geogiou@GN.ECN.PURDUE.EDU (Ioannes T Georgiou) (02/12/90)
Dior0wsh: 1+0*i = exp(2*p*i) -----> sqrt(1+0*i) = exp(p*i) = -1 + 0*i (k=1)
ST401843@BROWNVM.BITNET (thanasis kehagias) (02/12/90)
epanalambanw oti h ennoia "tetragwnikh riza enos migadikou ariqmou" kai genikotera "niosth riza enos migadikou ariqmou" den einai monoshmanta orismenh. gia na exoume monoshmanto orismo, mporoume na poume "prwteuousa niosth riza", pou qa pei oti dialegoume, p.x., thn timh me to mikrotero qetiko orisma ...
geogiou@GN.ECN.PURDUE.EDU (Ioannes T Georgiou) (02/12/90)
Symfwnw apolyta. Gia ayto gialeksa k=1 wste -1 + 0*i = sqrt(1+0*i)= sqrt(exp(2*p*i))
geogiou@GN.ECN.PURDUE.EDU (Ioannes T Georgiou) (02/12/90)
Giati sqrt(1)=sqrt((-1)*(-1))=sqrt(-1)*srt(-1) ====> 1=-1 kai sqrt(1) = sqrt((-1)/(-1))=sqrt(-1)/sqrt(-1) ===> 1=1 ; Mporei kapoios na dwsei gewmetrikh ermhneia;
nbp@SWORD.BELLCORE.COM (Nikolaos-John Pronios) (02/12/90)
Apostoli, Den 0ymamai akrivws ti eixa grapsei. Pantws an eixa typo la0os, 1=exp(j(2pi+2kpi)) Mporei na mhn eixa thn deuterh paren0esh, mia kai einai profanes oti to j 0a paei gia olo ton ek0eth. Nikos P. P.S. An exeis thn apanthsh MOU steile thn gia na thn dw, kai na 0ymy0w.