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Expressio decimalis nvmerorvm rationalivm

Svmmarivm

Haec lectio scriptvram decimalem nvmerorvm rationalivm perscrvtatvr. Ostendit digitos qvotientis ex residvis divisionis longae nasci, valorem expressionis decimalis infinitae per intervalla inclvsa et proprietatem Archimedeam definit, et tria theoremata demonstrat qvae eam describvnt: fractio irredvcibilis expressionem finitam habet si et solvm si denominator eivs nvllos factores primos praeter 22 et 55 habet; expressio cvivsvis nvmeri rationalis finita avt periodica est, periodo cvivs longitvdo denominatore minor est; et vicissim omnis expressio periodica nvmervm rationalem repraesentat, cvivs fractio generatrix svmma progressionis geometricae obtinetvr. Inde seqvvntvr aeqvalitas inter 0,999… et 11, descriptio irrationalivm per expressiones non periodicas et, postremo, notatio scientifica, digiti significantes et margo erroris qvem rotvndatio committit.

Proposita discendi

  1. Expressionem decimalem fractionis divisione longa obtinere et, residvis adhibitis, rationem reddere cvr qvisqve digitvs inter 00 et 99 sit et cvr expressio ad fractionem accedat.
  2. Ex factorizatione denominatoris diivdicare vtrvm fractio expressionem finitam an periodicam habeat, et longitvdinem periodi eivs margine coercere.
  3. Fractionem generatricem expressionis periodicae invenire et aeqvalitates, qvalis est ea qvae inter 0,999… et 11 intercedit, accvrate confirmare.
  4. Nvmeros notatione scientifica scribere, eos ad datvm digitorvm significantivm nvmervm rotvndare et errores absolvtvm ac relativvm qvi committvntvr margine coercere.

Divisio longa et digiti decimales

In lectione de nvmeris rationalibvs fractiones earvmqve operationes tractatae svnt; in lectione de systematibvs nvmerationis positionalis, scriptvra nvmerorvm natvralivm per digitos. Expressio decimalis vtrvmqve conivngit: notationem positionalem ad dextram virgvlae extendit, potentiis negativis nvmeri 1010 adhibitis. Sic nvmervs qvi 0,375 scribitvr ex definitione svmma est decimarvm, centesimarvm et millesimarvm:

310+7102+5103=375103=38\frac{3}{10} + \frac{7}{10^2} + \frac{5}{10^3} = \frac{375}{10^3} = \frac{3}{8}

Generatim expressio decimalis finita constat ex parte integra NN et ex finito digitorvm nvmero c1,c2,…,cnc_1, c_2, \ldots, c_n, qvorvm qvisqve inter 00 et 99 est, et nvmervm qvi seqvitvr repraesentat:

N+c110+c2102+⋯+cn10n=N+∑k=1nck10kN + \frac{c_1}{10} + \frac{c_2}{10^2} + \cdots + \frac{c_n}{10^n} = N + \sum_{k = 1}^{n} \frac{c_k}{10^k}

Cvm signvm nvmeri rationalis seorsvm scribatvr, satis est nvmeros rationales positivos considerare. Ad expressionem fractionis pq\frac{p}{q} nvmerorvm natvralivm inveniendam adhibetvr divisio longa, qvae nihil alivd est qvam divisio cvm residvo iterata: primvm dividitvr pp per qq, qvod partem integram NN et residvvm r0r_0 dat; deinde qvaqve vice mvltiplicatvr per 1010 residvvm postremvm idqve dividitvr per qq. Qvotiens hvivs divisionis est digitvs seqvens, et residvvm eivs vicem qvae proxime seqvitvr alit:

p=qN+r0,10rk−1=qck+rk,0≤rk<qp = qN + r_0, \qquad 10 r_{k - 1} = q c_k + r_k, \qquad 0 \le r_k < q
ExemplvmDivisio longa nvmeri 3 per 7

Digitos fractionis 37\frac{3}{7} compvtamvs. Qvaqve vice scribvntvr dividendvs, qvi decies residvvm privs aeqvat, digitvs qvi obtinetvr et residvvm novvm. Figvra scalam divisionvm ostendit et, svpra, digitos qvi accvmvlantvr.

Demonstratio

  1. 10⋅3=30=7⋅4+210 \cdot \dato{r0}{3} = 30 = 7 \cdot \resaltar{4} + \resaltar{2}

    Cvm 3<73 < 7, pars integra est 00 et residvvm primvm est ipse nvmerator, r0=3r_0 = 3. Residvvm per 1010 mvltiplicamvs et per 77 dividimvs: digitvs decimarvm est 44.

  2. 10⋅2=20=7⋅2+610 \cdot 2 = 20 = 7 \cdot \resaltar{2} + \resaltar{6}

    Residvo 22 digitvs centesimarvm obtinetvr.

  3. 10⋅6=60=7⋅8+410 \cdot 6 = 60 = 7 \cdot \resaltar{8} + \resaltar{4}

    Residvo 66, digitvs millesimarvm.

  4. 10⋅4=40=7⋅5+510 \cdot 4 = 40 = 7 \cdot \resaltar{5} + \resaltar{5}

    Residvo 44, digitvs qvartvs.

  5. 10⋅5=50=7⋅7+110 \cdot 5 = 50 = 7 \cdot \resaltar{7} + \resaltar{1}

    Residvo 55, qvintvs.

  6. 10⋅1=10=7⋅1+310 \cdot 1 = 10 = 7 \cdot \resaltar{1} + \resaltar{3}

    Residvo 11, sextvs; et residvvm novvm est 33, idem a qvo incepimvs.

  7. 37=428571106+1106⋅37\frac{3}{7} = \frac{428571}{10^6} + \frac{1}{10^6} \cdot \resaltar{\frac{3}{7}}

    Hinc divisio per singvlas vices iteratvr: qvod dividendvm restat, scilicet 37\frac{3}{7} mvltiplicatvm per 1106\frac{1}{10^6}, est idem qvotiens sex locis promotvs. Digiti agmen efficivnt qvod sine fine repetitvr.

Expressio fractionis 37\frac{3}{7} est igitvr 0,428571428571…, periodo 428571. Exemplvm totam lectionem qvasi in semine continet: digiti ex residvis orivntvr, residva divisore minora svnt et, simvl ac vnvm repetitvr, omnia qvae seqvvntvr repetvntvr. Primvm qvod confirmare oportet est digitos sic obtentos veros esse digitos et ad fractionem qvam proxime libet accedere.

LemmaDigiti divisionis longae

Sint pp et qq nvmeri natvrales, et sint NN, r0r_0 et, pro qvovis nvmero natvrali kk, ckc_k et rkr_k nvmeri qvos divisio longa nvmeri pp per qq praebet. Tvm qvodqve ckc_k digitvs est, inter 00 et 99, et pro omni nvmero natvrali nn valet aeqvalitas seqvens, in qva addendvm postremvm non negativvm est et minvs qvam 110n\frac{1}{10^n}.

pq=N+∑k=1nck10k+rnq⋅10n\frac{p}{q} = N + \sum_{k = 1}^{n} \frac{c_k}{10^k} + \frac{r_n}{q \cdot 10^n}

Demonstratio per indvctionem in nn procedit. Figvra argvmentvm comitatvr cvm 37\frac{3}{7}: qvisqve digitvs novvs vnvm ex decem intervallis minoribvs intervalli prioris eligit, qvod amplificatvr vt conspiciatvr.

Demonstratio

  1. qck≤10rk−1<10q→0≤ck≤9q c_k \le 10 r_{k - 1} < 10q \rightarrow 0 \le c_k \le 9

    Qvodqve ckc_k digitvs est: cvm residvvm privs minvs sit qvam qq, dividendvs 10rk−110 r_{k - 1} minor est qvam 10q10q, et qvotiens minor est qvam 1010.

  2. pq=N+r0q\dato{base}{\frac{p}{q} = N + \frac{r_0}{q}}

    Pro n=0n = 0, sine digitis, aeqvalitas est prima divisio cvm residvo, per qq divisa.

  3. rnq⋅10n=10rnq⋅10n+1=qcn+1+rn+1q⋅10n+1=cn+110n+1+rn+1q⋅10n+1\dato{paso}{\frac{r_n}{q \cdot 10^n} = \frac{\resaltar{10 r_n}}{q \cdot 10^{n + 1}} = \frac{\resaltar{q c_{n + 1} + r_{n + 1}}}{q \cdot 10^{n + 1}} = \frac{c_{n + 1}}{10^{n + 1}} + \frac{r_{n + 1}}{q \cdot 10^{n + 1}}}

    Gradvs indvctivvs: addendvm postremvm divisione seqventi, 10rn=qcn+1+rn+110 r_n = q c_{n + 1} + r_{n + 1}, rescribitvr et in dvas fractiones dividitvr.

  4. pq=N+∑k=1n+1ck10k+rn+1q⋅10n+1\frac{p}{q} = N + \sum_{k = 1}^{n + 1} \frac{c_k}{10^k} + \frac{r_{n + 1}}{q \cdot 10^{n + 1}}

    Svbstitvendo in aeqvalitatem ordinis nn obtinetvr aeqvalitas ordinis n+1n + 1; per indvctionem aeqvalitas pro omni nn valet.

  5. 0≤pq−Tn=rnq⋅10n<110n0 \le \frac{p}{q} - T_n = \frac{\resaltar{r_n}}{q \cdot 10^n} < \frac{1}{10^n}

    Cvm 0≤rn<q0 \le r_n < q, addendvm postremvm non negativvm est et minvs qvam 110n\frac{1}{10^n}: fractio minvs qvam 110n\frac{1}{10^n} distat a sva trvncatione ordinis nn, nvmero TnT_n qvi ex parte integra et nn primis digitis constat.

Lemma dicit pq\frac{p}{q} esse in intervallo qvod in TnT_n incipit et longitvdinem 110n\frac{1}{10^n} habet, extremo sinistro inclvso et dextro exclvso; haec intervalla, qvae in lectione de recta reali diligenter tractabvntvr, sic scribvntvr:

Tn=N+∑k=1nck10k,pq∈[Tn,Tn+110n)T_n = N + \sum_{k = 1}^{n} \frac{c_k}{10^k}, \qquad \frac{p}{q} \in \ico{T_n}{T_n + \frac{1}{10^n}}