Ad argvmentvm transi
Topos Uranos
Latine

Radices ordinis n nvmeri complexi et radices vnitatis

Svmmarivm

Haec lectio in nvmeris complexis aeqvationem zn=wz^n = w solvit. Forma polari et formvla Moivreana demonstrat omnem nvmervm complexvm a nihilo diversvm exacte nn radices nn-tas habere, qvae modvlvm commvnem habent et argvmenta intervallo 2πn\frac{2\pi}{n} separata, ita vt polygonvm regvlare circa originem centratvm efficiant. Deinde radices vnitatis tractat: svnt potentiae vnivs earvm, mvltiplicantvr sicvt residva divisionis per nn addvntvr, potentiae earvm exponentis mm svmmam nn avt 00 habent provt nn nvmervm mm dividit necne, et polynomivm zn−1z^n - 1 in factores resolvvnt. Per maximvm commvnem divisorem radices primitivas definit, qvarvm potentiae omnes radices percvrrvnt, et, radicibvs conivgatis consociatis, factorizationem realem polynomiorvm xn−1x^n - 1 et xn+1x^n + 1 in factores lineares et qvadraticos invenit, qva valor cos⁡2π5\cos \frac{2\pi}{5} itervm apparet.

Proposita discendi

  1. Radices nn nn-tas nvmeri complexi a nihilo diversi forma polari invenire, ostendere eas exacte nn esse, easqve vt vertices polygoni regvlaris describere.
  2. Cvm radicibvs vnitatis vt potentiis vnivs earvm operari, svmmas potentiarvm earvm compvtare et polynomivm zn−1z^n - 1 in factores resolvere.
  3. Per maximvm commvnem divisorem decernere qvae radices vnitatis primitivae sint, et criterivm identitate Bezovtiana demonstrare.
  4. Polynomia xn−1x^n - 1 et xn+1x^n + 1 in realibvs in factores resolvere radicibvs conivgatis consociatis, et radices vnitatis ad problemata de distantiis, de valoribvs trigonometricis et de aeqvationibvs adhibere.

Radices nn-tae nvmeri complexi

In lectione cvrsvs Algebra et Geometria I de nvmeris complexis aeqvatio z3=1z^3 = 1 solvta est differentia cvborvm in factores resolvta, et nvntiatvm est radices indicis nn cvivsvis nvmeri complexi formam polarem postvlatvras esse. Methodvs factorizationis generalis non fit: aeqvationis z5=1+iz^5 = 1 + i nvllvs factor in conspectv est. Forma polaris avtem, in lectione de forma polari nvmerorvm complexorvm tractata, aeqvationem in dvas condiciones reales convertit, alteram inter modvlos, alteram inter argvmenta, qvia, per formvlam in lectione de formvla Moivreana demonstratam, nvmervm complexvm ad potentiam nn elevare est modvlvm eivs ad nn elevare et argvmentvm eivs per nn mvltiplicare.

Sint ww nvmervs complexvs et nn nvmervs natvralis. Nvmervs complexvs zz est radix nn-ta ipsivs ww cvm zn=wz^n = w. Radices secvndae radices qvadratae vocantvr, et tertiae radices cvbicae.

Solvs nvmervs complexvs cvivs potentia nn-ta est 00 est ipse 00, qvia prodvctvm nvmerorvm complexorvm non evanescit nisi aliqvis eivs factor evanescit; casvs igitvr notabilis est w≠0w \neq 0. Tvm ww forma polari scribitvr, w=R(cos⁡φ+isin⁡φ)w = R(\cos \varphi + i \sin \varphi), cvm R=∣w∣>0R = \abs{w} > 0 et φ\varphi qvovis eivs argvmento. Signvm Rn\sqrt[n]{R} sensvm servat qvem habet in lectione cvrsvs I de potentiis exponentis rationalis et radicalibvs: designat radicem principalem nvmeri realis positivi RR, vnicvm nvmervm realem positivvm cvivs potentia nn-ta est RR. Radicale reservatvr radici principali nvmeri realis non negativi et, vt in illa lectione, radici reali indicis imparis nvmeri realis negativi, vt −83=−2\sqrt[3]{-8} = -2; alii nvmero complexo adhibitvm, tantvm cvm expressa admonitione adhibebitvr id qvamvis eivs radicem designare, qvia, vt mox videbitvr, nvlla ex eivs nn radicibvs a ceteris distingvitvr, et vnam ex convento eligere leges radicalivm violat: si radix qvadrata ipsivs −1-1 nvmervs ii vocaretvr, prodvctvm hvivs radicis per se ipsam esset −1-1, non radix qvadrata ipsivs (−1)(−1)=1(-1)(-1) = 1.

Postremo adhibetvr theorema de aeqvalitate in forma polari, in eadem lectione demonstratvm: dvo nvmeri complexi a nihilo diversi aeqvales svnt si et solvm si evndem modvlvm habent et argvmenta eorvm mvltiplo integro ipsivs 2π2\pi differvnt.

TheoremaRadices nn-tae

Sint w=R(cos⁡φ+isin⁡φ)w = R(\cos \varphi + i \sin \varphi) nvmervs complexvs a nihilo diversvs, cvm R=∣w∣R = \abs{w}, et nn nvmervs natvralis. Aeqvatio zn=wz^n = w exacte nn solvtiones diversas habet, qvae svnt nvmeri complexi zkz_k infra scripti, pro k=0,1,…,n−1k = 0, 1, \ldots, n - 1. Omnes modvlvm Rn\sqrt[n]{R} habent, et argvmentvm cvivsqve argvmentvm praecedentis excedit intervallo 2πn\frac{2\pi}{n}.

zk=Rn(cos⁡φ+2kπn+isin⁡φ+2kπn)z_k = \sqrt[n]{R} \left( \cos \frac{\varphi + 2k\pi}{n} + i \sin \frac{\varphi + 2k\pi}{n} \right)

Solvtio forma polari qvaeritvr et formae polares ipsorvm znz^n et ww aeqvantvr. Figvra demonstrationem seqvitvr cvm w=−8=8(cos⁡π+isin⁡π)w = -8 = 8(\cos \pi + i \sin \pi) et n=3n = 3, qvorvm radices cvbicae modvlvm 22 habent.

Demonstratio

  1. z=r(cos⁡θ+isin⁡θ)→zn=rn(cos⁡nθ+isin⁡nθ)z = r(\cos \theta + i \sin \theta) \rightarrow \resaltar{z^n} = \dato{m}{\resaltar{r^n (\cos n\theta + i \sin n\theta)}}

    Solvtio nvlla esse non potest, qvia 0n=0≠w0^n = 0 \neq w. Eam forma polari scribimvs, cvm r>0r > 0, et ad potentiam nn formvla Moivreana elevamvs.

  2. zn=w↔rn=R∧nθ=φ+2kπ,k∈Z\resaltar{z^n = w} \leftrightarrow \dato{c}{\resaltar{r^n = R \wedge n\theta = \varphi + 2k\pi}}, \quad k \in \mathbb{Z}

    Per theorema de aeqvalitate in forma polari, aeqvatio zn=wz^n = w dvabvs condicionibvs realibvs aeqvivalet: modvli aeqvales svnt et argvmenta mvltiplo integro ipsivs 2π2\pi differvnt.

  3. r=Rn,θk=φ+2kπn,k∈Zr = \resaltar{\sqrt[n]{R}}, \qquad \dato{t}{\theta_k = \resaltar{\frac{\varphi + 2k\pi}{n}}}, \quad k \in \mathbb{Z}

    Nvmervs realis positivvs cvivs potentia nn-ta est RR vnicvs est, radix principalis, et argvmentvm nvmero integro kk determinatvr. Vicissim omnis nvmervs complexvs hoc modvlo et hoc argvmento solvtio est, qvia ambas condiciones explet. Cvm w=−8w = -8 et n=3n = 3, fivnt r=2r = 2 et θ=π+2kπ3\theta = \frac{\pi + 2k\pi}{3}.

  4. θqn+j=φ+2(qn+j)πn=θj+2qπ→zqn+j=zj\theta_{qn + j} = \resaltar{\frac{\varphi + 2(qn + j)\pi}{n}} = \resaltar{\theta_j + 2q\pi} \rightarrow z_{qn + j} = z_j

    Nvmeri integri kk infiniti svnt, solvtiones avtem non. Per divisionem cvm residvo, omnis nvmervs integer scribitvr vt k=qn+jk = qn + j, cvm qq integro et 0≤j≤n−10 \leq j \leq n - 1; argvmentvm θk\theta_k tvm excedit θj\theta_j mvltiplo integro ipsivs 2π2\pi, et dvae solvtiones congrvvnt. In figvra, z3=z0z_3 = z_0.

  5. 0<θl−θj=2(l−j)πn<2π→zl≠zj0 < \theta_l - \theta_j = \resaltar{\frac{2(l - j)\pi}{n}} < 2\pi \rightarrow \resaltar{z_l \neq z_j}

    Solvtiones z0,z1,…,zn−1z_0, z_1, \ldots, z_{n - 1} inter se diversae svnt: si 0≤j<l≤n−10 \leq j < l \leq n - 1, argvmenta earvm differvnt nvmero stricte inter 00 et 2π2\pi posito, qvi mvltiplvm integrvm ipsivs 2π2\pi non est. Svnt igitvr exacte nn solvtiones. Cvm w=−8w = -8, svnt 1+i31 + i\sqrt{3}, −2-2 et 1−i31 - i\sqrt{3}.

In vsv, prima radix argvmentvm φn\frac{\varphi}{n} habet, et qvaeqve ceterarvm obtinetvr addendo 2πn\frac{2\pi}{n} argvmento praecedentis. Argvmentvm φ\varphi ipsivs ww alio, φ+2π\varphi + 2\pi, mvtare copiam radicvm non mvtat, solvm earvm nvmerationem: qvae antea erat z1z_1 nvnc vocatvr z0z_0. Pro n=2n = 2, argvmenta dvarvm radicvm differvnt intervallo π\pi, ita vt oppositae sint, z1=−z0z_1 = -z_0; est dvplex signvm qvod in radice qvadrata nvmeri complexi apparet, in lectione cvrsvs I de nvmeris complexis partibvs realibvs et imaginariis aeqvatis compvtata. Nvmervs radicvm praeterea congrvit cvm eo qvod de polynomiis scitvr: per lemma de radicibvs diversis ex lectione cvrsvs I de algebra polynomiorvm, cvivs demonstratio sine mvtatione in complexis valet, polynomivm zn−wz^n - w, gradvs nn, plvres qvam nn radices diversas habere non potest; theorema ostendit id omnes habere.

Polygonvm regvlare radicvm

Radices nn evndem modvlvm habent, ita vt in circvmferentia centro 00 et radio Rn\sqrt[n]{R} iaceant, et argvmenta earvm aeqvaliter distant. Theorema qvod seqvitvr hanc dispositionem per prodvctvm nvmerorvm complexorvm exprimit, qvod in forma polari modvlos mvltiplicat et argvmenta addit, vt in lectione de forma polari demonstratvm est.

TheoremaPolygonvm radicvm

Sint ww nvmervs complexvs a nihilo diversvs, cvm R=∣w∣R = \abs{w}, et n≥3n \geq 3 nvmervs natvralis. Qvaeqve radix nn-ta ipsivs ww ex praecedenti obtinetvr eam per nvmervm ω\omega infra scriptvm mvltiplicando, qvod idem est atqve eam angvlo 2πn\frac{2\pi}{n} circa originem convertere; radices nn svnt vertices polygoni regvlaris nn latervm circvmferentiae centro 00 et radio Rn\sqrt[n]{R} inscripti, et latvs eivs metitvr 2Rnsin⁡πn2\sqrt[n]{R} \sin \frac{\pi}{n}.

zk+1=zk ω,ω=cos⁡2πn+isin⁡2πnz_{k + 1} = z_k \, \omega, \qquad \omega = \cos \frac{2\pi}{n} + i \sin \frac{2\pi}{n}

Figvra demonstrationem seqvitvr cvm w=64iw = 64i, cvivs radices cvbicae, qvartae et sextae in circvmferentiis radiorvm 643=4\sqrt[3]{64} = 4, 644=22\sqrt[4]{64} = 2\sqrt{2} et 646=2\sqrt[6]{64} = 2 iacent.

Demonstratio

  1. zk ω=Rn(cos⁡(θk+2πn)+isin⁡(θk+2πn))=zk+1z_k \, \omega = \sqrt[n]{R} \left( \cos \resaltar{\left( \theta_k + \frac{2\pi}{n} \right)} + i \sin \left( \theta_k + \frac{2\pi}{n} \right) \right) = \dato{g}{\resaltar{z_{k + 1}}}

    Qvoniam ∣ω∣=1\abs{\omega} = 1, prodvctvm zkωz_k \omega modvlvm ipsivs zkz_k habet et argvmentvm eivs avctvm intervallo 2πn\frac{2\pi}{n}: est radix qvae proxime seqvitvr. Per indvctionem in kk, qvaeqve radix est prima per potentiam ipsivs ω\omega mvltiplicata, zk=z0ωkz_k = z_0 \omega^k.

  2. ∣uω−vω∣=∣(u−v)ω∣=∣u−v∣∣ω∣=∣u−v∣\abs{u\omega - v\omega} = \abs{\resaltar{(u - v)\omega}} = \abs{u - v} \abs{\omega} = \dato{d}{\resaltar{\abs{u - v}}}

    Mvltiplicare per ω\omega est igitvr angvlo 2πn\frac{2\pi}{n} circa originem convertere, et haec conversio distantias servat, qvia modvlvs prodvcti est prodvctvm modvlorvm.

  3. ∣zk+2−zk+1∣=∣zk+1ω−zkω∣=∣zk+1−zk∣\abs{z_{k + 2} - z_{k + 1}} = \abs{\resaltar{z_{k + 1} \omega - z_k \omega}} = \resaltar{\abs{z_{k + 1} - z_k}}

    Conversio qvodqve latvs polygoni in latvs qvod proxime seqvitvr transfert, ita vt omnia latera aeqvalia sint.

  4. zn−1ω=z0ωn−1ω=z0ωn=z0z_{n - 1} \omega = z_0 \resaltar{\omega^{n - 1} \omega} = z_0 \omega^n = \resaltar{z_0}

    Polygonvm clavditvr, qvia ωn=1\omega^n = 1: conversio vltimvm verticem in primvm transfert. Vertices nn in circvmferentia iacent et latera aeqvalia svnt. Praeterea conversio triangvlvm verticibvs zkz_k, zk+1z_{k + 1} et zk+2z_{k + 2} in triangvlvm verticibvs zk+1z_{k + 1}, zk+2z_{k + 2} et zk+3z_{k + 3} transfert tria latera servans, et per theorema cosinvvm angvlvs triangvli tribvs lateribvs eivs determinatvr: angvli polygoni, qvi svnt angvli horvm triangvlorvm ad verticem medivm, aeqvales svnt, et polygonvm regvlare est.

  5. ∣z1−z0∣=∣z0∣∣ω−1∣=2Rnsin⁡πn\abs{z_1 - z_0} = \abs{z_0} \resaltar{\abs{\omega - 1}} = \resaltar{2\sqrt[n]{R} \sin \frac{\pi}{n}}

    Latvs compvtatvr in triangvlo verticibvs 00, z0z_0 et z1z_1. Per formvlam angvli dimidii, ∣ω−1∣2=2−2cos⁡2πn=4sin⁡2πn\abs{\omega - 1}^2 = 2 - 2\cos \frac{2\pi}{n} = 4\sin^2 \frac{\pi}{n}, et sinvs ipsivs πn\frac{\pi}{n} positivvs est. Pro radicibvs sextis ipsivs 64i64i, latvs valet 2⋅2⋅12=22 \cdot 2 \cdot \frac{1}{2} = 2, aeqvale radio, vt in omni hexagono regvlari.

Pro n=2n = 2, polygonvm in diametrvm degenerat, cvivs extrema svnt dvae radices oppositae, et pro n=1n = 1 vna sola radix est, ipse ww. Qvod radices polygonvm regvlare efficivnt explicat cvr nvlla a ceteris distingvatvr: conversio angvlo 2πn\frac{2\pi}{n} eas permvtat figvra non mvtata.