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Nvmeri complexi: operationes et aeqvationes

Svmmarivm

Haec lectio nvmeros complexos constrvit vt paria ordinata nvmerorvm realivm, cvm additione et mvltiplicatione formvlis explicitis definitis, ita vt aeqvalitas i2=−1i^2 = -1 postvlatvm esse desinat et theorema fiat. Formam binomialem, planvm complexvm, potentias vnitatis imaginariae, nvmervm conivgatvm et modvlvm introdvcit, cvm proprietatibvs demonstratis, inter qvas inaeqvalitas triangvlaris, nec non divisionem per nvmervm conivgatvm. His instrvmentis aeqvationes secvndi gradvs discriminantis negativi solvit, demonstrat radices non reales polynomii coefficientivm realivm per paria conivgata apparere, et aeqvationes solvit vt z3=1z^3 = 1 vel eas qvae zz cvm svo conivgato conivngvnt.

Proposita discendi

  1. Nvmeros complexos vt paria realivm constrvere, ex definitione prodvcti dedvcere i2=−1i^2 = -1, et in forma binomiali operari.
  2. Nvmeros complexos in plano repraesentare et svmmam, conivgatvm, modvlvm prodvctvmqve per ii geometrice interpretari.
  3. Proprietates conivgati et modvli demonstrare, inter qvas inaeqvalitatem triangvlarem, et dividere mvltiplicando per conivgatvm.
  4. In C\mathbb{C} aeqvationes secvndi gradvs, aeqvationes polynomiales simplices et aeqvationes lineares in zz eivsqve conivgato solvere.

Nvmeri complexi vt paria nvmerorvm realivm

In lectione de generibvs nvmerorvm vnitas imaginaria ii introdvcta est vt nvmervs novvs cvivs qvadratvm est −1-1, et cvm expressionibvs a+bia + bi eisdem regvlis operati svmvs atqve cvm realibvs. Hic modvs procedendi dvas qvaestiones apertas relinqvit: nvm obiectvm ea proprietate re vera exsistat, et nvm regvlae realivm ei sine contradictione applicari possint. Responsio, qvam Hamilton anno 1837 edidit, in eo consistit vt complexi ex realibvs constrvantvr, nvllo novo postvlato: nvmervs complexvs est par ordinatvm nvmerorvm realivm.

Definitio (nvmervs complexvs). Nvmervs complexvs est par ordinatvm (a,b)(a, b) nvmerorvm realivm; eorvm copia signo C\mathbb{C} notatvr. Dvo complexi aeqvales svnt cvm primae eorvm componentes congrvvnt et secvndae componentes congrvvnt. Svmma et prodvctvm formvlis seqventibvs definivntvr.
(a,b)+(c,d)=(a+c,b+d),(a,b)⋅(c,d)=(ac−bd,ad+bc)(a, b) + (c, d) = (a + c, b + d), \qquad (a, b) \cdot (c, d) = (ac - bd, ad + bc)

Svmma per singvlas componentes operatvr. Prodvctvm avtem non est prodvctvm per singvlas componentes, qvod (ac,bd)(ac, bd) daret, eivsqve forma arbitraria videtvr; conseqventiis svis ivstificatvr. Prodvcto per singvlas componentes, prodvctvm ex (1,0)(1, 0) et (0,1)(0, 1) esset (0,0)(0, 0), qvamvis nevter factor nvllvs esset, nec vlla divisio esse posset; prodvcto definito, contra, par (0,1)(0, 1) qvadratvm habet par (−1,0)(-1, 0), et paria formae (a,0)(a, 0) prorsvs vt reales se gervnt.

Reales intra complexos

Paria qvorvm secvnda componens nvlla est eodem modo addvntvr et mvltiplicantvr atqve reales:

(a,0)+(c,0)=(a+c,0),(a,0)⋅(c,0)=(ac,0)(a, 0) + (c, 0) = (a + c, 0), \qquad (a, 0) \cdot (c, 0) = (ac, 0)

Ergo par (a,0)(a, 0) cvm reali aa identificari potest, nvlla operatione eventvm mvtante, et R\mathbb{R} in C\mathbb{C} continetvr, vt principivm amplificationvm in lectione de generibvs nvmerorvm envntiatvm postvlat. Mvltiplicare par per realem, praeterea, ambas eivs componentes mvltiplicat, qvia (r,0)⋅(a,b)=(ra,rb)(r, 0) \cdot (a, b) = (ra, rb). Vnitas imaginaria appellatvr par i=(0,1)i = (0, 1).

TheoremaVnitas imaginaria et forma binomialis

Qvadratvm nvmeri i=(0,1)i = (0, 1) est realis −1-1, et omnis complexvs (a,b)(a, b) vno tantvm modo scribitvr vt a+bia + bi, cvm aa et bb realibvs. Demonstratio nihil adhibet nisi definitionem prodvcti et identificationem cvivsqve realis aa cvm pari (a,0)(a, 0). Figvra qvodqve par (a,b)(a, b) repraesentat vt pvnctvm plani coordinatarvm aa et bb, qvae notio in capite seqventi explicatvr.

Demonstratio

  1. i2=(0,1)⋅(0,1)=(0⋅0−1⋅1,0⋅1+1⋅0)=(−1,0)i^2 = \resaltar{(0, 1) \cdot (0, 1)} = (0 \cdot 0 - 1 \cdot 1, 0 \cdot 1 + 1 \cdot 0) = \resaltar{(-1, 0)}

    Definitionem prodvcti adhibemvs cvm a=c=0a = c = 0 et b=d=1b = d = 1.

  2. i2=−1\resaltar{i^2} = \resaltar{-1}

    Par (−1,0)(-1, 0) est realis −1-1: vnitas imaginaria est radix qvadrata nvmeri −1-1. Altera est eivs oppositvm, −i=(0,−1)-i = (0, -1), qvia (−i)2=i2(-i)^2 = i^2.

  3. b⋅i=(b,0)⋅(0,1)=(b⋅0−0⋅1,b⋅1+0⋅0)=(0,b)b \cdot i = \resaltar{(b, 0) \cdot (0, 1)} = (b \cdot 0 - 0 \cdot 1, b \cdot 1 + 0 \cdot 0) = \dato{b}{\resaltar{(0, b)}}

    Realis bb per ii mvltiplicatvs dat par cvivs prima componens nvlla est et secvnda componens bb.

  4. (a,b)=(a,0)+(0,b)=a+bi(a, b) = \resaltar{(a, 0) + (0, b)} = \dato{f}{\resaltar{a + bi}}

    Omne par est svmma paris in axe primo et alterivs in secvndo.

  5. a+bi=c+di↔(a=c∧b=d)a + bi = c + di \leftrightarrow \resaltar{(a = c \wedge b = d)}

    Scriptvra vnica est: si a+bi=c+dia + bi = c + di, paria (a,b)(a, b) et (c,d)(c, d) aeqvalia svnt, et ex definitione aeqvalitatis eorvm componentes congrvvnt.

Expressio a+bia + bi est forma binomialis complexi z=(a,b)z = (a, b). Realis aa est pars realis nvmeri zz et realis bb eivs pars imaginaria; animadvertendvm est partem imaginariam esse nvmervm realem bb, non prodvctvm bibi. Complexi partis imaginariae nvllae svnt reales, et ii partis realis nvllae, vt 3i3i, imaginarii pvri appellantvr. Ex postrema theorematis sententia, aeqvalitas inter complexos aeqvivalet dvabvs aeqvalitatibvs inter reales; inde seqvitvr omnem aeqvationem complexam in systema dvarvm aeqvationvm realivm converti posse, qvod est praecipvvm instrvmentvm capitis postremi.

Proprietates operationvm

Forma binomialis vtilis est tantvm si cvm ea operari licet vt cvm expressionibvs algebraicis realivm, prodvcta evolvendo et pro i2i^2 ponendo −1-1. Id postvlat vt operationes in C\mathbb{C} easdem proprietates habeant atqve eae in R\mathbb{R}.

TheoremaProprietates operationvm

Additio et mvltiplicatio complexorvm commvtativae et associativae svnt, mvltiplicatio respectv additionis distribvtiva est, 0=(0,0)0 = (0, 0) et 1=(1,0)1 = (1, 0) svnt elementa nevtra additionis et mvltiplicationis, et omnis complexvs (a,b)(a, b) oppositvm habet (−a,−b)(-a, -b). Proprietates additionis a proprietatibvs realivm per singvlas componentes hereditate accipivntvr. Eae mvltiplicationis confirmantvr vtroqve membro proprietatibvs realivm evolvto. Exsistentia inversi mvltiplicationis infra demonstratvr, ope conivgati. Figvra demonstrationem seqvitvr cvm (a,b)=2+i(a, b) = 2 + i, (c,d)=1+i(c, d) = 1 + i et (e,f)=1−i(e, f) = 1 - i, qvi ibi zz, ww et uu appellantvr: tria prodvcta qvae in associativitate et distribvtivitate intervenivnt ad idem pvnctvm dvcvnt, 4+2i4 + 2i.

Demonstratio

  1. (c,d)⋅(a,b)=(ca−db,cb+da)=(a,b)⋅(c,d)(c, d) \cdot (a, b) = (\resaltar{ca - db}, \resaltar{cb + da}) = (a, b) \cdot (c, d)

    Commvtativitas prodvcti in formvla legitvr: si (a,b)(a, b) cvm (c,d)(c, d) permvtatvr, nvlla componens mvtatvr, qvia ca=acca = ac, db=bddb = bd et cb+da=ad+bccb + da = ad + bc.

  2. [(a,b)⋅(c,d)]⋅(e,f)=(ac−bd,ad+bc)⋅(e,f)=(ace−bde−adf−bcf,acf−bdf+ade+bce)\left[ (a, b) \cdot (c, d) \right] \cdot (e, f) = \resaltar{(ac - bd, ad + bc)} \cdot (e, f) = \dato{p}{(ace - bde - adf - bcf, acf - bdf + ade + bce)}

    Ad associativitatem, primvm dvos priores factores mvltiplicamvs et evolvimvs.

  3. (a,b)⋅[(c,d)⋅(e,f)]=(a,b)⋅(ce−df,cf+de)=(ace−adf−bcf−bde,acf+ade+bce−bdf)(a, b) \cdot \left[ (c, d) \cdot (e, f) \right] = (a, b) \cdot \resaltar{(ce - df, cf + de)} = \resaltar{(ace - adf - bcf - bde, acf + ade + bce - bdf)}

    Nvnc primvm dvos postremos factores mvltiplicamvs: iidem octo termini prodevnt, alio ordine.

  4. (a,b)⋅(c+e,d+f)=(ac−bd+ae−bf,ad+bc+af+be)(a, b) \cdot (c + e, d + f) = (\resaltar{ac - bd} + \resaltar{ae - bf}, \resaltar{ad + bc} + \resaltar{af + be})

    Ad distribvtivitatem, qvamqve componentem evolvimvs et terminos rvrsvs colligimvs: eventvs est svmma ex (a,b)⋅(c,d)(a, b) \cdot (c, d) et (a,b)⋅(e,f)(a, b) \cdot (e, f).

  5. (a,b)⋅(1,0)=(a⋅1−b⋅0,a⋅0+b⋅1)=(a,b)(a, b) \cdot (1, 0) = (a \cdot 1 - b \cdot 0, a \cdot 0 + b \cdot 1) = \resaltar{(a, b)}

    Postremo, (1,0)(1, 0) est nevtrvm mvltiplicationis, et (−a,−b)(-a, -b) ad (a,b)(a, b) additvm dat (0,0)(0, 0).

Qvapropter cvm formis binomialibvs operamvr vt cvm expressionibvs algebraicis in littera ii, et pro qvoqve i2i^2 ponitvr −1-1. Praesertim prodvctvm formvlam recvperat qva definitvm est:

(a+bi)(c+di)=ac+adi+bci+bdi2=(ac−bd)+(ad+bc)i(a + bi)(c + di) = ac + adi + bci + bdi^2 = (ac - bd) + (ad + bc)i

Exempli gratia, (2+3i)(4−i)=8−2i+12i−3i2=11+10i(2 + 3i)(4 - i) = 8 - 2i + 12i - 3i^2 = 11 + 10i. Prodvcta notabilia, vt qvadratvm binomii vel differentia qvadratorvm, et binomivm Newtonianvm ex lectione de combinatoria in C\mathbb{C} sine mvtatione valent, qvia eorvm demonstrationes nihil nisi has proprietates adhibent. Qvod non servatvr est ordo: ex problemate solvto 5 lectionis de generibvs nvmerorvm, nvllvs in C\mathbb{C} ordo cvm operationibvs congrvens exsistit, ita vt inaeqvalitas inter complexos non reales sensv careat. Magnitvdo complexi modvlo eivs aestimabitvr.