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Nvmeri rationales: fractiones earvmqve operationes

Svmmarivm

Haec lectio nvmeros rationales ex integris constrvit. Fractionem definit vt par integrorvm cvivs alter terminvs non nvllvs est, demonstrat aeqvalitatem prodvctorvm decvssatorvm se gerere vt aeqvalitatem, svmmamqve et prodvctvm non pendere a fractionibvs electis ad qvemqve nvmervm repraesentandvm, et ostendit omnem nvmervm rationalem habere fractionem irredvcibilem et vnam tantvm. Deinde tractat calcvlvm fractionvm (denominatorem commvnem, simplificationem ante prodvctvm, fractiones compositas et errores freqventissimos) et ordinem nvmerorvm rationalivm cvm dvabvs eivs proprietatibvs propriis: densitate et proprietate Archimedea.

Proposita discendi

  1. Nvmeros rationales definire vt fractiones integrorvm qvae per aeqvalitatem prodvctorvm decvssatorvm identificantvr, et demonstrare svmmam et prodvctvm bene definita esse.
  2. Exsistentiam et vnicitatem fractionis irredvcibilis demonstrare, eaqve vti ad simplificandvm et ad decernendvm vtrvm dvae fractiones evndem nvmervm repraesentent.
  3. Cvm fractionibvs efficaciter operari: ad denominatorem commvnem per minimvm commvne mvltiplvm redvcere, ante mvltiplicationem simplificare, fractiones compositas solvere et errores vsitatos agnoscere.
  4. Nvmeros rationales comparare et in demonstrationibvs proprietates ordinis, densitatem et proprietatem Archimedeam adhibere.

Fractiones earvmqve aeqvivalentia

In lectione de generibvs nvmerorvm nvmeri rationales vt fractiones integrorvm denominatore non nvllo exhibiti svnt, et monitvm est fractionem non posse vt divisionem definiri, qvia divisio id ipsvm est qvod integri non permittvnt. Haec lectio constrvctionem ibi nvntiatam perficit. Eivs initivm svnt integri lectionis de nvmeris integris, cvm svmma, prodvcto, ordine svo et proprietate qva assidve vtemvr: prodvctvm dvorvm integrorvm non nvllorvm non nvllvm est. Inde seqvitvr factorem non nvllvm cancellari posse, qvia si ab=acab = ac cvm a≠0a \neq 0, tvm a(b−c)=0a(b - c) = 0, ideoqve b−c=0b - c = 0.

Definitio (fractio). Fractio est par ordinatvm (p,q)(p, q) integrorvm cvm q≠0q \neq 0, qvod scribitvr pq\frac{p}{q}; integer pp eivs nvmerator est et qq eivs denominator. Fractiones pq\frac{p}{q} et rs\frac{r}{s} aeqvivalentes dicvntvr cvm prodvcta earvm decvssata congrvvnt, id est cvm ps=qrps = qr.

Sic 23\frac{2}{3} et 46\frac{4}{6} aeqvivalentes svnt, qvia 2⋅6=3⋅42 \cdot 6 = 3 \cdot 4, cvm 23\frac{2}{3} et 34\frac{3}{4} non sint, qvia 2⋅4≠3⋅32 \cdot 4 \neq 3 \cdot 3. Definitio non est arbitraria. Si fractiones qvotientes esse debent, aeqvalitas pq=rs\frac{p}{q} = \frac{r}{s}, vtroqve membro per qsqs mvltiplicato, aeqvalitati ps=qrps = qr aeqvivalere debet; definitio hanc conseqventiam pro initio svmit, qvae sola prodvcta integrorvm continet ideoqve sensvm habet anteqvam divisio exsistat. Vt avtem aeqvivalentia vicem aeqvalitatis gerere possit, se gerere debet vt illa: relatio aeqvivalentiae esse debet, eo sensv qvo in lectione de copiis, relationibvs et fvnctionibvs.

TheoremaAeqvivalentia fractionvm

Aeqvivalentia fractionvm reflexiva, symmetrica et transitiva est: omnis fractio sibi ipsi aeqvivalens est; si pq\frac{p}{q} aeqvivalens est fractioni rs\frac{r}{s}, tvm rs\frac{r}{s} aeqvivalens est fractioni pq\frac{p}{q}; et si pq\frac{p}{q} aeqvivalens est fractioni rs\frac{r}{s} et rs\frac{r}{s} fractioni tu\frac{t}{u}, tvm pq\frac{p}{q} aeqvivalens est fractioni tu\frac{t}{u}. Dvae priores proprietates ex commvtativitate prodvcti seqvvntvr; tertia factorem cancellandvm postvlat, et sola est in qva condicio intervenit vt denominatores non nvlli sint. Figvra pq\frac{p}{q} depingit vt pvnctvm coordinatarvm (q,p)(q, p): fractiones aeqvivalentes cvm origine in directvm positae apparent.

Demonstratio

  1. p⋅q=q⋅p\resaltar{p \cdot q} = \resaltar{q \cdot p}

    Reflexivitas: ex commvtativitate prodvcti prodvcta decvssata fractionis secvm ipsa congrvvnt.

  2. ps=qr→rq=sp\resaltar{ps = qr} \rightarrow \resaltar{rq = sp}

    Symmetria: si ps=qrps = qr, commvtativitas prodvcti et symmetria aeqvalitatis eandem condicionem dant, mvneribvs fractionvm permvtatis.

  3. ps=qr,ru=st\dato{h1}{ps = qr}, \quad \dato{h2}{ru = st}

    Transitivitas: a dvabvs hypothesibvs proficiscimvr, et ad pu=qtpu = qt pervenire volvmvs. In figvra, 23\frac{2}{3}, 46\frac{4}{6} et 69\frac{6}{9}.

  4. (ps)u=q(ru)=q(st)(ps)u = q(\resaltar{ru}) = q(\resaltar{st})

    Primam aeqvalitatem per uu mvltiplicamvs et pro ruru ponimvs stst, secvndvm alteram.

  5. s(pu)=s(qt)→s(pu−qt)=0\resaltar{s(pu) = s(qt)} \rightarrow \resaltar{s(pu - qt) = 0}

    Ex associativitate et commvtativitate factor ss in vtroqve membro apparet; omnia in primvm transferimvs evmqve vt factorem commvnem extrahimvs.

  6. pu−qt=0→pu=qt\resaltar{pu - qt = 0} \rightarrow \resaltar{pu = qt}

    Cvm s≠0s \neq 0 et prodvctvm dvorvm integrorvm non nvllorvm non nvllvm sit, alter factor nihil est: fractiones extremae aeqvivalentes svnt.

Qvae si ita svnt, ad aeqvivalentiam fractionvm valent eventvs generales lectionis de copiis, relationibvs et fvnctionibvs, qvos repetere non oportet: ex theoremate de aeqvivalentiis et partitionibvs classes fractionvm aeqvivalentivm partitionem copiae fractionvm efficivnt, ita vt qvaeqve fractio ad vnam tantvm classem pertineat et dvae classes diversae nvllam fractionem commvnem habeant. Qvaeqve classis nvmervs est.

Definitio (nvmervs rationalis). Nvmervs rationalis est classis aeqvivalentiae fractionis, id est copia omnivm fractionvm ei aeqvivalentivm, et qvaelibet earvm evm repraesentat; in notatione lectionis de copiis, relationibvs et fvnctionibvs, nvmervs rationalis a pq\frac{p}{q} repraesentatvs est classis C(pq)C\left( \frac{p}{q} \right), et hic simpliciter pq\frac{p}{q} scribitvr. Copia nvmerorvm rationalivm, qvae est copia qvotiens fractionvm per earvm aeqvivalentiam, per Q\mathbb{Q} notatvr. Scribitvr pq=rs\frac{p}{q} = \frac{r}{s} cvm ambae fractiones evndem nvmervm rationalem repraesentant, id est cvm aeqvivalentes svnt, vt affirmat lemma de classibvs aeqvivalentiae ibi demonstratvm.
pq=rs↔ps=qr\frac{p}{q} = \frac{r}{s} \leftrightarrow ps = qr

Qvisqve integer pp cvm nvmero rationali a p1\frac{p}{1} repraesentato identificatvr. Identificatio integros diversos non confvndit, qvia p1=r1\frac{p}{1} = \frac{r}{1} idem valet atqve p⋅1=1⋅rp \cdot 1 = 1 \cdot r, id est p=rp = r; hoc sensv Z⊆Q\mathbb{Z} \subseteq \mathbb{Q}. Postea confirmabitvr operationes nvmerorvm rationalivm, ad integros adhibitas, vsitatas reddere. Regvla practica qva saepissime vtimvr vt fractionem mvtemvs nvmero non mvtato haec est.

LemmaAmplificatio et simplificatio

Pro omnibvs integris pp, qq et kk, cvm q≠0q \neq 0 et k≠0k \neq 0, valet kpkq=pq\frac{kp}{kq} = \frac{p}{q}. Praesertim −p−q=pq\frac{-p}{-q} = \frac{p}{q}, ita vt omnis nvmervs rationalis per fractionem denominatoris natvralis repraesentetvr. A sinistra ad dextram lecta, aeqvalitas factorem commvnem simplificat; a dextra ad sinistram lecta, fractionem amplificat. Figvra eivs sensvm ostendit: si qvaeqve particvla vnitatis in kk particvlas aeqvales dividitvr, eadem portio particvlis kk vicibvs plvribvs describitvr.

Demonstratio

  1. pq=23\frac{p}{q} = \frac{2}{3}

    In figvra, 23\frac{2}{3} vnitatem in q=3q = 3 particvlas aeqvales dividit et p=2p = 2 ex eis svmit.

  2. kpkq=2⋅22⋅3=46\frac{kp}{kq} = \frac{\resaltar{2 \cdot 2}}{\resaltar{2 \cdot 3}} = \resaltar{\frac{4}{6}}

    Si qvaeqve particvla in k=2k = 2 particvlas aeqvales dividitvr, vnitas in kqkq particvlas divisa manet, et portio svmpta kpkp ex eis continet.

  3. 4⋅24⋅3=812\frac{\resaltar{4 \cdot 2}}{\resaltar{4 \cdot 3}} = \resaltar{\frac{8}{12}}

    Cvm k=4k = 4, eadem portio dvodecim particvlis describitvr.

  4. (kp)⋅q=k(pq)=(kq)⋅p(kp) \cdot q = \resaltar{k(pq)} = \resaltar{(kq) \cdot p}

    Aeqvivalentia a pictvra non pendet: prodvcta decvssata congrvvnt ex associativitate et commvtativitate prodvcti. Condicio k≠0k \neq 0 id solvm praestat vt kqkq denominator admissibilis sit.

  5. −p−q=pq,2−3=−23\frac{-p}{-q} = \frac{p}{q}, \quad \frac{2}{-3} = \frac{\resaltar{-2}}{\resaltar{3}}

    Cvm k=−1k = -1 mvtatio signi vtrivsqve termini obtinetvr; si denominator negativvs est, haec mvtatio evm natvralem reddit.