Systemata nvmerationis positionalis

Svmmarivm

Haec lectio notationem positionalem tractat, in qva valor cvivsqve digiti a loco qvem obtinet pendet. Demonstrat omnem nvmervm natvralem in qvalibet basi $b \geq 2$ vno tantvm modo scribi, ratione qvae in dividendo per basim itervm atqve itervm consistit, et docet qvomodo nvmeri inter basim $10$ et bases $2$ , $8$ et $16$ convertantvr, atqve inter has vltimas digitis binariis conivngendis. Ostendit additionem et mvltiplicationem in qvalibet basi iisdem regvlis ac in basi $10$ compvtari, exponit cvr criteria divisibilitatis valeant eaqve ad alias bases extendit, et nota de historia nvmerationis deqve vsv basis $2$ in compvtatris conclvditvr.

Proposita discendi

Notationem positionalem interpretari et valorem nvmeri in qvalibet basi scripti compvtare.
Exsistentiam et vnicitatem repraesentationis in basi $b$ demonstrare, eamqve divisionibvs continvatis invenire.
Nvmeros inter bases $2$ , $8$ , $10$ et $16$ convertere, et in basi $2$ addere et mvltiplicare.
Criteria divisibilitatis per $b - 1$ et per $b + 1$ in basi $b$ ivstificare, eaqve in basi $10$ et in aliis basibvs adhibere.

Notatio positionalis

Cvm $2025$ scribimvs, non qvattvor digiti addvntvr, sed qvisqve per potentiam ipsivs $10$ mvltiplicatvr qvae a loco eivs pendet: prior $2$ dvo milia valet; alter, dvas decades. Digitvs $0$ significat nvllas esse centvrias et, locvm svvm obtinendo, ceteros in loco sibi debito retinet.

2025 = 2 \cdot 10^3 + 0 \cdot 10^2 + 2 \cdot 10 + 5

Hoc principivm, qvod notatio positionalis appellatvr, a nvmero $10$ non pendet. Si $b \geq 2$ nvmervs natvralis est, qvi basis appellatvr, digiti in basi $b$ svnt nvmeri $0, 1, \ldots, b - 1$ , et series digitorvm $a_k, a_{k-1}, \ldots, a_1, a_0$ , cvm $a_k \neq 0$ , hvnc nvmervm repraesentat:

n = a_k b^k + a_{k-1} b^{k-1} + \cdots + a_1 b + a_0

Digiti alivs ivxta alivm scribvntvr, a sinistra ad dextram, et basis vt index inferior indicatvr: exempli gratia, $3412_5$ est nvmervs qvi in basi $5$ digitis $3$ , $4$ , $1$ et $2$ scribitvr, et legitvr « $3412$ in basi $5$ ». Dicitvr $a_i$ locvm $i$ obtinere, a dextra nvmeratvm ab $0$ incipiendo. Sine indice, nvmervs in basi $10$ scriptvs est. In basi $2$ dvo tantvm digiti svnt, $0$ et $1$ , qvi bit appellantvr; in basi $16$ sedecim reqvirvntvr, et decem digitis vsitatis litterae A, B, C, D, E et F addvntvr, qvae $10$ , $11$ , $12$ , $13$ , $14$ et $15$ valent.

Iam vero valor nvmeri in basi $b$ scripti compvtari potest $b$ vt factorem commvnem itervm atqve itervm extrahendo, qvo potentiae compvtandae vitantvr: haec est regvla Horneri.

LemmaRegvla Horneri

Pro omni $k$ et omnibvs nvmeris $a_0, a_1, \ldots, a_k$ , svmma prodvctorvm $a_i b^i$ obtinetvr a $a_k$ incipiendo et $k$ vicibvs operationem iterando qva per $b$ mvltiplicatvr et coefficiens proximvs additvr.

a_k b^k + a_{k-1} b^{k-1} + \cdots + a_1 b + a_0 = (\cdots((a_k b + a_{k-1}) b + a_{k-2}) \cdots) b + a_0

Demonstratio est indvctio in $k$ qvae sola distribvtivitate vtitvr. Expressio est valor in $x = b$ polynomii $a_k x^k + \cdots + a_1 x + a_0$ , et lectio de algebra polynomiorvm ad eam redibit.

Demonstratio

$a_0 = a_0$
Gradvs initialis: pro $k = 0$ , ambo membra ad $a_0$ redigvntvr, qvia regvla nvllam operationem efficit.
$\dato{h}{c_k b^k + \cdots + c_1 b + c_0 = (\cdots(c_k b + c_{k-1}) \cdots) b + c_0}$
Hypothesis indvctionis: regvla valet pro omni svmma $k + 1$ terminorvm, coefficientibvs qvibvslibet, qvae vsqve ad potentiam $b^k$ pertinet.
$a_{k+1} b^{k+1} + \cdots + a_1 b + a_0 = \resaltar{(a_{k+1} b^k + \cdots + a_2 b + a_1)} \, b + a_0$
Sit svmma qvae vsqve ad $b^{k+1}$ pertinet. Per distribvtivitatem, $b$ vt factor commvnis ex omnibvs terminis praeter vltimvm extrahitvr.
$a_{k+1} b^{k+1} + \cdots + a_1 b + a_0 = \resaltar{((\cdots(a_{k+1} b + a_k) \cdots) b + a_1)} \, b + a_0$
Parenthesis formam hypothesis indvctionis habet, cvm coefficientibvs $c_i = a_{i+1}$ ; regvla scribitvr, et eam per $b$ mvltiplicando et $a_0$ addendo regvla pro $k + 1$ obtinetvr. Per principivm indvctionis, pro omni $k$ valet.

ExemplvmValor nvmeri in basi 5 scripti

Nvmervm compvtare qvi in basi $5$ scribitvr $3412_5$ , dvobvs modis: valores locorvm addendo et regvla Horneri.

Demonstratio

$3412_5 = \resaltar{3} \cdot 5^3 + \resaltar{4} \cdot 5^2 + \resaltar{1} \cdot 5 + \resaltar{2}$
Qvisqve digitvs per potentiam ipsivs $5$ mvltiplicatvr qvae loco eivs respondet: $5^3$ , $5^2$ , $5$ et $1$ .
$3 \cdot \resaltar{125} + 4 \cdot \resaltar{25} + 1 \cdot 5 + 2 = \resaltar{375} + \resaltar{100} + 5 + 2 = \dato{v}{482}$
Potentias et prodvcta compvtamvs, et addimvs.
$3412_5 = \resaltar{((3 \cdot 5 + 4) \cdot 5 + 1)} \cdot 5 + 2$
Regvla Horneri $5$ vt factorem commvnem ex tribvs terminis primis extrahimvs, deinde ex dvobvs primis parenthesis.
$\resaltar{3 \cdot 5 + 4} = 19, \quad \resaltar{19 \cdot 5 + 1} = 96, \quad \resaltar{96 \cdot 5 + 2} = \resaltar{482}$
Ab interiore ad exteriora compvtatvr: qvisqve exitvs per basim mvltiplicatvr eiqve digitvs proximvs additvr. Idem valor obtinetvr, sine vlla potentia.

In reliqva parte hvivs lectionis

Exsistentia et vnicitas repraesentationis
Bases 2, 8 et 16
Operationes in basi 2
Criteria divisibilitatis
Divisores basis
Divisibilitas per $b - 1$ et per $b + 1$
Nota historica et ad vsvm spectans
Problemata solvta
Nvmervs dvplex svi inversi
Digiti permvtati
Nvmeri palindromi pari digitorvm nvmero
Mvltiplicare per dvplicationes
Pondera Bacheti
Problemata proposita
Digiti potentiae sine compvtatione
Inscriptio interretis
A basi 3 ad basim 9
Divisibilitas per 7, per 11 et per 13
Dvpla in dvabvs basibvs