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Genera nvmerorvm: a natvralibvs ad complexos

Svmmarivm

Haec lectio ostendit qvomodo, a nvmeris natvralibvs profecti, deinceps integros, rationales, reales et complexos constrvamvs. Qvaeqve amplificatio oritvr vt operatio inversa (svbtractio, divisio, radicis extractio) sine vlla restrictione definiatvr, et proprietates copiae prioris servat. Commemoratvr cvr 2\sqrt{2} nvmervs rationalis non sit, demonstratvr rationales completos non esse, envntiantvr axiomata qvae de nvmeris realibvs admittvntvr (axiomata corporis, axiomata ordinis et axioma completvdinis), et explicatvr cvr reales praeter radices completvdinem reqvirant.

Proposita discendi

  1. Agnoscere qvae operatio inversa qvamqve amplificationem moveat, a natvralibvs vsqve ad complexos.
  2. Proprietates additionis et mvltiplicationis in qvoqve genere nvmerorvm accvrate envntiare, necnon axiomata corporis ordinati completi qvae de nvmeris realibvs admittvntvr.
  3. Leges exponentivm ad potentias exponentis integri adhibere, itemqve definitionem radicis principalis.
  4. Per redvctionem ad absvrdvm demonstrare radicem aliqvam nvmervm rationalem non esse.
  5. Nvmeros rationales, irrationales, reales et complexos distingvere.

Operationes inversae vt cavsa amplificationis

Nvmeri natvrales, in lectione de axiomatibvs Peani definiti, sinvnt nos sine vlla restrictione addere et mvltiplicare: svmma et prodvctvm dvorvm natvralivm semper natvrales svnt. Non idem accidit in operationibvs inversis. Qvaeqve harvm aeqvationvm nvmeris natvralibvs scripta est, nec tamen vlla inter eos solvtionem habet:

x+5=3,3x=2,x2=2,x2=−1x + 5 = 3, \qquad 3x = 2, \qquad x^2 = 2, \qquad x^2 = -1

Prima svbtractionem postvlat qvam natvrales non admittvnt; secvnda, divisionem; tertia, radicem; qvarta, radicem nvmeri negativi. Vt eae solvantvr, copia natvralivm per qvattvor copias amplificatvr, qvarvm qvaeqve priorem continet:

N⊂Z⊂Q⊂R⊂C\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C}

Omnes amplificationes vni eidemqve principio oboedivnt, qvod reliqvam lectionem regit:

Qvodqve novvm genvs nvmerorvm operationes et proprietates prioris servat, et elementa addit qvae necessaria svnt vt operatio inversa, antea restricta, definiatvr.