Vidi fonton por Aritmetiko

[[Dosiero:Tables generales aritmetique MG 2108.jpg|eta|Aritmetikajn tabelojn por infanoj, [[Laŭzano]], 1835.]]
'''Aritmetiko''' estas branĉo de [[matematiko]], studanta la naturon kaj proprecojn de [[nombro]]j kaj speciale la proprecojn de la [[Operacio (matematiko)|tradiciaj operacioj]] kiujn oni faris pere de ili — nome [[adicio]], [[subtraho]], [[multipliko]] kaj [[divido]]. Ĝi inkluzivas ankaŭ la studon pri [[algoritmo]]j kaj [[operacioj per nombroj|operacioj]] super [[entjera nombro|entjeroj]] kaj [[frakcio]]j, [[kvociento]]j, [[proporcio]]j, [[procento]]j. La nomo devenas de la vorto {{Lang-el|ἀριθμός}} ''arithmos'', kiu signifas "nombron" en [[greka lingvo]].

La moderna aritmetiko ne okupiĝas nur pri elementaj [[kalkulo]]j, sed pri gravaj ecoj de entjeroj ([[primo|primeco]], [[plej granda komuna divizoro]], [[Faktoro|primaj faktoroj]] ktp.) kaj similaj ecoj de aliaj objektoj (ekzemple elementoj de [[Ringo-teorio|ringoj]]). Evoluo de Aritmetiko venigis al apartigo de novaj matematikaj branĉoj: [[Algebro]] kaj [[Nombroteorio]].

== Historio ==
La prahistorio de aritmetiko estas limigita al malgranda nombro de artifaktoj kiuj povas indiki la koncepton de adicio kaj subtraho, kies plej bone konata restaĵo estas la [[osto de Iŝango|Iŝanga osto]] el [[Demokratia Respubliko Kongo|centra Afriko]], kiu datas el iam inter 20,000 kaj 18,000 jaroj a.K., kvankam ties interpretado estas disputata.<ref>{{Citaĵo el libro|titolo=How Mathematics Happened: The First 50,000 Years|familia nomo=Rudman|persona nomo=Peter Strom|jaro=2007|eldoninto=Prometheus Books|ISBN=978-1-59102-477-4|paĝo=64}}</ref>

La plej fruaj verkitaj registroj indikis ke jam en [[Antikva Egipto]] kaj [[Babilono]] oni uzis ĉiujn operaciojn de elementa aritmetiko tiom frue kiom ĉe 2000 a.K. Tiuj artifaktoj ne ĉiam pruvas la specifan procezon uzitan por solvi numerajn problemojn, sed la karakteroj de la partikulara [[nombrosistemo]] forte influas la komplikecon de metodoj. La [[Hieroglifo|hieroglifa sistemo]] de la [[egiptaj ciferoj]], same kiel la postaj [[Romaj ciferoj]], descendis el la kalkulaj markoj uzitaj populare por kalkuladi. Ambaŭokaze, tiu deveno rezultis en [[Valoro (komputado)|valoroj]] kiuj uzis [[Dekuma nombrosistemo|dekuman]] [[Bazo (nombrosistemo)|bazon]], sed ne inkludis [[Pozicia nombrosistemo|pozician nombrosistemon]]. Kompleksa kalkulado pere de romaj ciferoj postulis la helpon de [[Multipliktabulo|kalkultabulo]] aŭ de [[abako]] por havi la rezultojn.

La plej fruaj nombrosistemoj kiuj inkludis [[Pozicia nombrosistemo|pozician nombrosistemon]] ne estis dekumaj, kiel la [[sesdekuma sistemo]] (kun bazo de 60) de la [[Babilonaj ciferoj]] kaj la [[dudekuma sistemo]] (kun bazo de 20) en kiu konsistis la [[Majaaj ciferoj|ciferoj]] de [[Majaoj]]. Pro tiu pozici-valora [[koncepto]], la kapablo reuzi la samajn ciferojn por diferencaj valoroj kontribuis al pli simplaj kaj pli efikaj metodoj de [[kalkulo]].

[[Dosiero:EuclidStatueOxford.jpg|eta|dekstra|Skulptaĵo reprezentanta [[Eŭklido]]n, Oxford University Museum.]]
La kontinua historia disvolvigo de la moderna aritmetiko startas ĉe la [[Helenisma Epoko]] de [[Antikva Grekio]], kvankam ĝi originiĝis multe pli poste ol la menciitaj ekzemploj de Babilonanoj kaj Egiptianoj. Antaŭ la verkoj de [[Eŭklido]] de ĉirkaŭ la jaro 300 a.K., Grekaj matematikistoj koincidis kun la filozofiaj kaj mistikismaj kredoj. Por ekzemplo, [[Nikomaĥo (matematikisto)|Nikomaĥo]] resumis la vidpunktojn de la pli frua [[Pitagorismo|Pitagorisma]] alproksimiĝo al nombroj, kaj ties rilataro de unu kun alia, en sia ''Enkonduko al Aritmetiko''.

La [[Grekaj ciferoj]] estis uzitaj de [[Arkimedo]], [[Diofanto]] kaj aliaj en [[pozicia notacio]] ne tre diferenca el la nuntempa. Mankis al la antikvaj grekoj simbolo por [[nulo]] ĝis la Helenisma Epoko, kaj ili uzis tri apartajn seriojn de simboloj kiel [[cifero]]j: nome unu serio por la lokoj de unuoj, alia por la lokoj de dekoj, kaj alia tria por la lokoj de centoj. Por la loko de miloj ili uzis la simbolojn de la unuoloko, kaj tiel plu. Ties aldona [[algoritmo]] estis sama al tiu nuntempa, kaj ties [[multiplika algoritmo]] estis nur tre malmulte diferenca. Ties [[Longa divido|longa divida]] algoritmo estis la sama, kaj la algoritmo por kalkuli [[Kvadrata radiko|kvadratajn radikojn]], populare uzita ĝis tiom ĵuse kiom ĝis la 20a jarcento, estis konata jam de Arkimedo, kiu eble inventis ĝin. Li fakte preferis ĝin al la metodo de [[Herono]] de sinsekva alproksimiĝo ĉar, kalkulita, [[cifero]] ne ŝanĝas, kaj la [[Kvadrata radiko|kvadrataj radikoj]] de [[Perfekta kvadrato|perfektaj kvadratoj]], kiaj {{nombro|7485696}}, finas tuj kiel {{nombro|2736}}. Ĉe ciferoj kun [[Frakcio (matematiko)|frakcia]] parto, kiel {{nombro|546.934}}, ili uzis negativajn [[Potenco (matematiko)|potencojn]] de 60 anstataŭ negativajn potencojn de 10 por la frakcia parto, nome {{nombro|0.934}}.<ref>''The Works of Archimedes'', Chapter IV, ''Arithmetic in Archimedes'', eldonita de T.L. Heath, Dover Publications Inc, New York, 2002.</ref>

La [[Antikva Ĉinio|antikvaj ĉinoj]] progresis en aritmetikaj studoj el la [[Dinastio Ŝang]] kaj pluis tra la [[Dinastio Tang]], el bazaj ciferoj al antaŭenirinta [[algebro]]. La antikvaj ĉinoj uzis [[Pozicia notacio|pozician notacion]] similan al tiu de la grekoj. Ĉar ankaŭ al ili mankis simbolo por [[nulo]], ili havis unu serion de simboloj por la loko de unuoj, kaj duan serion por la loko de dekoj. Por por la loko de centoj, ili reuzis la simbolojn de la loko de unuoj, kaj tiel plu. Iliaj simboloj estis bazitaj sur la antikvaj [[kalkulbastono]]j. Estas komplika demando determini precize kiam la ĉinoj ekkalkulis per pozicia reprezentado, sed ĝi certe okazis definitive antaŭ la jaro 400 a.K.<ref>Joseph Needham, ''Science and Civilization in China'', Vol. 3, p. 9, Cambridge University Press, 1959.</ref> La antikvaj ĉinoj estis la unuaj kiuj klare malkovris, komprenis, kaj aplikis [[Negativa nombro|negativajn nombrojn]] kiel oni klarigas en la verko ''Naŭ ĉapitroj pri la matematika arto'' (''Ĝiuĵang Suanŝu''), kiu estis verkita de [[Liu Hui]].

La laŭgrada disvolvigo de la [[Arabaj ciferoj|Hind–arabaj ciferoj]] sendepende de la difinita loko-valora koncepto kaj [[pozicia notacio]], kiu kombinis la pli simplajn metodojn por kalkuloj per dekuma bazo kaj la uzadon de cifero reprezentante [[nulo]]n. Tio ebligis ke la sistemo konsiste reprezentu kaj grandajn kaj malgrandajn [[entjero]]jn. Tiu alproksimiĝo finfine anstataŭis ĉiujn aliajn sistemojn. En la komenco de la 6a jarcento p.K., la hindia matematikisto [[Aryabhata]] aligis ekzistintan version de tiu sistemo en sia verko, kaj eksperimentis per diversaj notacioj. En la 7a jarcento, [[Brahmagupta]] establis la uzadon de 0 (nulo) kiel aparta cifero kaj determinis la rezultojn por multiplikado, divido, sumo kaj subtraho de nulo kaj de ĉiuj aliaj ciferoj, escepte ĉe la rezulto de divido per nulo. Liatempa, la siriaka episkopo Severo Seboĥt (650) diris, "Hindianoj posedas kalkulmetodon kiu ne estas sufiĉe laŭdita. Ties racia sistemo de matematiko, aŭ ties kalkulmetodo. Mi celas la sistemon kiu uzas naŭ simbolojn."<ref>Reference: Revue de l'Orient Chretien de François Nau pp.327-338. (1929)</ref> Ankaŭ la [[araboj]] lernis tiun novan metodon kaj nomis ĝin ''ĥesab'' (la arablingva radiko por kalkulo).

[[Dosiero:Leibniz Stepped Reckoner.png|eta|200ra|La [[Radokalkulilo]] de Leibniz estis la unua kalkulilo kiu povis plenumi la kvar aritmetikajn operaciojn.]]
Kvankam la ''[[Codex Vigilanus]]'' priskribis fruan formon de [[Arabaj ciferoj]] (sen 0) ĉirkaŭ la jaro 976, [[Leonardo da Pisa|Leonardo de Pisa]] ([[Fibonacci]]) estis ĉefa respondeco pri la etendo de ties uzado tra Eŭropo post la publikigo de sia libro ''[[Liber Abaci]]'' en 1202. Li verkis, "La metodo de hindianoj ({{Lang-la|Modus Indoram}}) surpasas ajnan konatan metodon por kalkuli. Ĝi estas mirinda metodo. Ili faras siajn kalkulojn uzante naŭ ciferojn kaj simbolon por [[nulo]]".<ref>Reference: Sigler, L., "Fibonacci's Liber Abaci", Springer, 2003.</ref>

En la [[Mezepoko]], aritmetiko estis unu el la [[sep liberaj artoj]] instruataj en [[universitato]]j.

La florado de [[algebro]] en la [[Islama Orepoko|mezepoka]] [[Islama mondo]] kaj en la [[Renesanco|renesanca]] [[Eŭropo]] estis elkreskaĵo de la enorma simpligo de kalkulado pere de dekuma notacio.

Variaj tipoj de iloj estis inventitaj kaj amplekse uzataj por helpi en la tasko de nombra kalkulado. Antaŭ la [[Renesanco]], ili estis variaj tipoj de [[abako]]j. Pli ĵusaj ekzemploj estas [[glitkalkulilo]]j, [[Abako (grafikaĵo)|nomografoj]] kaj [[Mekanika kalkulilo|mekanikaj kalkuliloj]], kiaj la [[paskala kalkulilo]]. Nuntempe, ili estis anstataŭitaj de elektronikaj [[kalkulilo]]j kaj [[komputilo]]j.

== Aritmetikaj operacioj ==
La bazaj aritmetikaj operacioj estas [[adicio]], [[subtraho]], [[multipliko]] kaj [[divido]]. Ĉiuj el ili estas du[[argumento|argumenta]] operacio, t.e. ĝi ricevas du [[nombro]]jn kaj finkalkule eldonas alian trian. Kvankam oni inkludas ankaŭ pliajn antaŭenirintajn operaciojn, kiel manipulado de [[procento]]j, [[Kvadrata radiko|kvadrataj radikoj]], [[Potenco (matematiko)|eksponentado]], kaj [[logaritmo|logaritmaj funkcioj]]. Aritmetiko estas plenumita laŭ ordo de operacioj.

En matematiko, la bazaj duargumentaj operacioj estas la jenaj:

=== Adicio ===
{{Ĉefartikolo|Adicio}}
[[File:Arithmetic operations.svg|thumb|alt=Diagram of symbols of arithmetic operations|La ĉefaj aritmetikaj operacioj estas adicio, subtraho, multipliko kaj divido.]]
Adicio estas operacio por trovi la sumon de nombroj aŭ kvantoj. La signo de adicio estas + (plus). Se oni adicias 5 kaj 3, oni ricevos 8. 5 kaj 3 estas la adiciatoj, 8 estas la sum-valoro. Precize, oni nomu la termon '5 + 3' sum-termon aŭ sum-esprimon, sed fak-mallonge ofte ankaŭ nur sumon.

La tradicia maniero koncerne la skribforman adicion - ekz-e de la naturaj nombroj 295 kaj 107 ([[dekuma sistemo|dekume prezentitaj]]) - okazus jene:

<math> {\begin{matrix}
                \ &2_{\ } &9_{\ } &5 \\
                 +&1_{\color{Red}1}    &0_{\color{Red}1}    &7
\end{matrix}\over
\begin{matrix}
            \quad &4_{\ } &0_{\ } &2
\end{matrix}} </math>

La kalkulado reduktiĝas al relative facila (elementa) sumado de unu-[[arabaj ciferoj|cifera]]j [[nombro]]j. Unu-cifera nombro de unuoj de la sum-valoro estas skribenda sub la sum-streko, kaj la unucifera nombro de dekoj estas ŝovenda maldekstren super la strekon. Tiu restaĵo estas la ''ŝovenda nombro'' ([[w:de:Übertrag|de:Übertrag]]).
La ruĝaj ciferoj sur la bildo estas tia ''ŝovenda nombro'' (de dekoj), kiu estas ricevita post for-apartigo de la dekstraflanka cifero (nombro de unuoj).

=== Subtraho ===
{{Ĉefartikolo|Subtraho}}
Subtraho estas operacio por trovi la diferencon inter unua nombro (aŭ kvanto) kaj dua nombro (aŭ kvanto), en tiu ordo; la signo de subtraho estas − (minus). Ekz. ĉe la subtraho: 9−6=3 oni diras, ke 9 estas la malpliigato, 6 estas la subtrahanto, 3 estas la diferenco. Ĝia [[kontraŭa operacio]] estas la adicio.

=== Multipliko ===
{{Ĉefartikolo|Multipliko}}
En matematiko, multipliko aŭ [[obligo]] estas duargumenta operacio. Ĝi povas esti aplikata al diversaj objektoj. Argumentoj de multipliko estas [[faktoro]]j kaj la rezulto estas [[produto]].

Multipliko povas esti skribata per kelkaj manieroj. Ekzemple multipliko de 5 kaj 2:

:5×2
:5·2
:5*2

La [[asterisko]] (*) estas ofte uzata en komputiloj ĉar ĝi estas simbolo sur ĉiu klavaro, sed ĝi estas malofte uzata kiam oni skribas permane. Ĉi tiu uzado devenis en la [[programlingvo]] [[Fortrano]].

Ofte, multipliko estas subkomprenata sen uzo de aparta signo. Ĉi tiu estas normo en [[algebro]], ekzemple:
:5''x''
:''xy''

Ĉi tiu potenciale povas misigi se nomoj de variabloj povas havi pli ol unu literon. Ĉi tiu skribmaniero estas ne uzata inter nombroj sola: 52 ne estas 5 × 2. Kodo de la multiplika signo × estas [[deksesuma]] D7 en [[unikodo]], en [[HTML]] ĝi povas esti skribata kiel ×.

=== Divido ===
{{Ĉefartikolo|Divido}}
En matematiko, aparte en elementa aritmetiko, '''divido''' estas aritmetika operacio kiu estas la inverso de [[multipliko]].

Specife, se <math>c \times b = a</math>
kie ''b'' estas ne [[nulo]], tiam
:<math>\frac ab = c</math>
kio estas, ''a'' dividita per ''b'' egalas al ''c''. Oni povas pli koncize legi kiel "a sur b egala al c".

Ekzemple, <math>\frac 63 = 2</math> ĉar <math>2 \times 3 = 6\,</math>.

En la pli supre esprimo, ''a'' estas nomita la '''dividato''' aŭ '''numeratoro''', ''b'' la '''dividanto''' aŭ '''denominatoro''' kaj ''c'' la '''kvociento'''.

[[Divido per nulo]] (do kiam la dividanto estas nulo) estas kutime ne difinita.

Divido estas plej ofte montrita per skribado de la ''dividato'' super la ''dividanto'' kun horizontalo inter ili. Ekzemple, ''a'' dividita per ''b'' estas skribata kiel <math>\frac ab</math>.

Maniero esprimi dividon tute sur unu linio estas skribi la ''dividaton'', tiam [[oblikvo]]n, tiam la ''dividanton'', tiamaniere: <math>a/b\,</math>.
Ĉi tio estas la kutima maniero precizigi dividon en plej multaj komputilaj [[Programlingvo|programadaj lingvoj]] ĉar ĝi povas facile esti tajpita kiel simpla vico de signoj.

Presa variado kiu estas meze inter ĉi tiuj du formoj uzas [[oblikvo]]n sed altigas la dividaton, kaj malaltigas la dividanton: <sup>a</sup>⁄<sub>b</sub>

Iu ajn el tiuj formoj povas ordinare elmontri [[Frakcio (matematiko)|frakcion]]. Frakcio estas divida esprimo kie kaj dividato kaj dividanto estas [[Entjero|entjeroj]] (kvankam tipe nomitaj la ''numeratoro'' kaj ''denominatoro'' respektive), kaj ne estas [[implico]], ke la divido bezonas esti plue pritaksita.

== Nombroteorio ==
{{Ĉefartikolo|Nombroteorio}}
'''Nombroteorio''' estas branĉo de [[matematiko]], dediĉita al la studo de proprecoj de [[entjera nombro|entjeroj]] kaj ĝiaj ĝeneraligoj (ekz. [[Algebra entjero|algebraj entjeroj]]). La demandoj pri la [[plej granda komuna divizoro]], la [[plej malgranda komuna oblo]], [[Malkomponado (informadiko)|malkomponado]] je [[primo]]j, prezento de [[naturaj nombroj|natura nombro]] en iu certa formo, ĝia [[dividebleco]] kaj aliaj temoj estas studobjektoj de la nombroteorio. Ĝi inkluzivas ankaŭ: [[Teorio de komparo|teorion de komparoj]], [[diofanta ekvacio|diofantaj ekvacioj]], [[ĉena frakcio]], diofantaj alproksimiĝoj, [[Transcenda nombro|transcendaj ekvacioj]] k.a. Ekde la [[1980-aj jaroj]] nombroteorio trovis surprizajn aplikojn en [[ĉifrado]] ([[kriptografio]]); ĝi ebligis la unuajn [[Nesimetria ĉifro|nesimetriajn ĉifrojn]]. En speciala literaturo oni ofte trovas ankaŭ sinonimajn terminojn – '''Teorio de Nombroj''' aŭ '''Teorio pri Nombroj'''.

== Aritmetikaj konceptoj ==
* [[Resto]]
* [[Fundamenta teoremo de aritmetiko]]
* [[aroj de nombroj]]
* [[nombrosistemo]]
* [[dekuma nombrosistemo]]
* [[aritmetika meznombro]]
* [[aritmetika vico]]

== Operacioj ==
* [[adicio]]
* [[subtraho]]
* [[multipliko]]
* [[divido]]
* [[procento]]
* [[kvadrata radiko]]
* [[potenco (matematiko)|potenco]]
* [[logaritmo]]

== Vidu ankaŭ ==
* [[Algebro]]
* [[Geometrio]]
* [[Kalkulo]]
* [[Matematiko]]
* [[Trigonometrio]]

== Notoj ==
{{Referencoj}}

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* {{cite book |last1=Quintero |first1=Ana Helvia |last2=Rosario |first2=Hector |title=Math Makes Sense!: A Constructivist Approach To The Teaching And Learning Of Mathematics |date=2016 |publisher=World Scientific |isbn=978-1-78326-866-5 |url=https://books.google.com/books?id=YAXyCwAAQBAJ&pg=PA74 |language=en}}
* {{cite book |author1-link=  |last1=Rajan |first1=Hridesh |title=An Experiential Introduction to Principles of Programming Languages |date=2022 |publisher=MIT Press |isbn=978-0-262-36243-6 |url=https://books.google.com/books?id=OCE6EAAAQBAJ&pg=PA17 |language=en}}
* {{cite book |last1=Reilly |first1=Norman R. |title=Introduction to Applied Algebraic Systems |date=2009 |publisher=Oxford University Press |isbn=978-0-19-970992-2 |url=https://books.google.com/books?id=q33he4hOlKcC&pg=PA75 |language=en}}
* {{cite book |last1=Reitano |first1=Robert R. |title=Introduction to Quantitative Finance: A Math Tool Kit |date=2010 |publisher=MIT Press |isbn=978-0-262-01369-7 |url=https://books.google.com/books?id=JYX6AQAAQBAJ&pg=PA42 |language=en}}
* {{cite book |last1=Resnick |first1=L. B. |last2=Ford |first2=W. W. |title=Psychology of Mathematics for Instruction |date=2012 |publisher=Routledge |isbn=978-1-136-55759-0 |url=https://books.google.com/books?id=xj-j8pw2HN8C&pg=PA110 |language=en}}
* {{cite book |last1=Reynolds |first1=Barbara E. |editor1-last=Selin |editor1-first=Helaine |title=Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures |date=12a de Marto 2008 |publisher=Springer Science & Business Media |isbn=978-1-4020-4559-2 |url=https://books.google.com/books?id=kt9DIY1g9HYC&pg=PA1 |language=en |chapter=Abacus}}
* {{cite book |last1=Riesel |first1=Hans |title=Prime Numbers and Computer Methods for Factorization |date=2012 |publisher=Springer Science & Business Media |isbn=978-1-4612-0251-6 |url=https://books.google.com/books?id=ITvaBwAAQBAJ&pg=PA2 |language=en}}
* {{cite book |last1=Rising |first1=Gerald R. |last2=Matthews |first2=James R. |last3=Schoaff |first3=Eileen |last4=Matthew |first4=Judith |title=About Mathematics |date=2021 |publisher=Linus Learning |isbn=978-1-60797-892-3 |url=https://books.google.com/books?id=hjVZEAAAQBAJ&pg=PA110 |language=en}}
* {{cite book |last1=Robbins |first1=Neville |title=Beginning Number Theory |date=2006 |publisher=Jones & Bartlett Learning |isbn=978-0-7637-3768-9 |url=https://books.google.com/books?id=TtLMrKDsDuIC&pg=PR12-IA1 |language=en}}
* {{cite book |last1=Rodda |first1=Harvey J. E. |last2=Little |first2=Max A. |title=Understanding Mathematical and Statistical Techniques in Hydrology: An Examples-based Approach |date=2015 |publisher=John Wiley & Sons |isbn=978-1-119-07659-9 |url=https://books.google.com/books?id=cb_dCgAAQBAJ&pg=PA7 |language=en}}
* {{cite book |last1=Roe |first1=John |last2=deForest |first2=Russ |last3=Jamshidi |first3=Sara |title=Mathematics for Sustainability |date=2018 |publisher=Springer |isbn=978-3-319-76660-7 |url=https://books.google.com/books?id=3ppYDwAAQBAJ&pg=PA24 |language=en}}
* {{cite book |last1=Romanowski |first1=Perry |editor1-last=Lerner |editor1-first=Brenda Wilmoth |editor2-last=Lerner |editor2-first=K. Lee |title=The Gale Encyclopedia of Science |date=2008 |publisher=Thompson Gale |isbn=978-1-4144-2877-2 |edition=4th |chapter=Arithmetic |url=https://www.encyclopedia.com/science-and-technology/mathematics/mathematics/arithmetic}}
* {{cite book |last1=Rooney |first1=Anne |title=Think Like a Mathematician |date=2021 |publisher=The Rosen Publishing Group |isbn=978-1-4994-7092-5 |url=https://books.google.com/books?id=WnVeEAAAQBAJ&pg=PA34 |language=en}}
* {{cite book |last1=Rossi |first1=Richard J. |title=Theorems, Corollaries, Lemmas, and Methods of Proof |date=2011 |publisher=John Wiley & Sons |isbn=978-1-118-03057-8 |url=https://books.google.com/books?id=kSwVGbBtel8C&pg=PA101 |language=en}}
* {{cite book |last1=Ruthven |first1=Kenneth |chapter=12. Calculators in the Mathematics Curriculum: The Scope of Personal Computational Technology |editor-last1=Bishop |editor-first1=Alan |editor-last2=Clements |editor-first2=M. A. (Ken) |editor-last3=Keitel-Kreidt |editor-first3=Christine |editor-last4=Kilpatrick |editor-first4=Jeremy |editor-last5=Laborde |editor-first5=Colette |title=International Handbook of Mathematics Education |date=2012 |publisher=Springer Science & Business Media |isbn=978-94-009-1465-0 |chapter-url=https://books.google.com/books?id=RjZzAgAAQBAJ&pg=PA435 |language=en}}
* {{cite book |last1=Sally |first1=Judith D. |last2=Sally (Jr.) |first2=Paul J. |title=Integers, Fractions, and Arithmetic: A Guide for Teachers |date=2012 |publisher=American Mathematical Soc. |isbn=978-0-8218-8798-1 |url=https://books.google.com/books?id=Ntjq07-FA_IC&pg=PA3 |language=en}}
* {{cite book |last1=Seaman |first1=William |last2=Rossler |first2=Otto E. |last3=Burgin |first3=Mark |title=Chaos, Information, And The Future Of Physics: The Seaman-rossler Dialogue With Information Perspectives By Burgin And Seaman |date=2023 |publisher=World Scientific |isbn=978-981-12-7138-0 |url=https://books.google.com/books?id=213PEAAAQBAJ&pg=PA226 |language=en}}
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* {{cite book |last1=Smith |first1=David E. |title=History of Mathematics |date=1958 |publisher=Courier Corporation |isbn=978-0-486-20430-7 |url=https://books.google.com/books?id=uTytJGnTf1kC&pg=PA7 |language=en}}
* {{cite book |last1=Smyth |first1=William |title=Elementary Algebra: For Schools and Academies |date=1864 |publisher=Bailey and Noyes |url=https://books.google.com/books?id=BqQZAAAAYAAJ&pg=PA55 |language=en |oclc=3901163143}}
* {{cite book |last1=Sophian |first1=Catherine |title=The Origins of Mathematical Knowledge in Childhood |date=2017 |publisher=Routledge |isbn=978-1-351-54175-6 |url=https://books.google.com/books?id=8FYPEAAAQBAJ&pg=PA84 |language=en}}
* {{cite book |last1=Sperling |first1=Abraham |last2=Stuart |first2=Monroe |title=Mathematics |publisher=Elsevier Science |isbn=978-0-7506-0405-5 |date=1981 }}
* {{cite book |last1=Stakhov |first1=Alexey |title=Mathematics Of Harmony As A New Interdisciplinary Direction And 'Golden' Paradigm Of Modern Science - Volume 2: Algorithmic Measurement Theory, Fibonacci And Golden Arithmetic's And Ternary Mirror-symmetrical Arithmetic |date=2020 |publisher=World Scientific |isbn=978-981-12-1348-9 |url=https://books.google.com/books?id=Fkn9DwAAQBAJ&pg=PA73 |language=en}}
* {{cite book |last1=Sternberg |first1=Robert J. |last2=Ben-Zeev |first2=Talia |title=The Nature of Mathematical Thinking |date=12an de Oktobro 2012 |publisher=Routledge |isbn=978-1-136-48750-7 |url=https://books.google.com/books?id=q7F777rDl1AC&pg=PA95 |language=en}}
* {{cite book |last1=Stevenson |first1=Angus |last2=Waite |first2=Maurice |title=Concise Oxford English Dictionary: Luxury Edition |date=2011 |publisher=OUP Oxford |isbn=978-0-19-960111-0 |url=https://books.google.com/books?id=sYScAQAAQBAJ&pg=PA70 |language=en}}
* {{cite book |last1=Stewart |first1=David E. |title=Numerical Analysis: A Graduate Course |date=2022 |publisher=Springer Nature |isbn=978-3-031-08121-7 |url=https://books.google.com/books?id=twafEAAAQBAJ&pg=PA26 |language=en}}
* {{cite book |last1=Strathern |first1=Paul |title=Turing And The Computer |date=31 October 2012 |publisher=Random House |isbn=978-1-4481-0656-1 |url=https://books.google.com/books?id=NJm7Z5iQE24C&pg=PP9 |language=en}}
* {{cite book |last1=Swanson |first1=Irena |title=Introduction To Analysis With Complex Numbers |date=18a de Februaro 2021 |publisher=World Scientific |isbn=978-981-12-2587-1 |url=https://books.google.com/books?id=cHshEAAAQBAJ&pg=PA107 |language=en}}
* {{cite book |last1=Swartzlander |first1=Earl E. |editor1-last=Oklobdzija |editor1-first=Vojin G. |title=Digital Design and Fabrication |date=2017 |publisher=CRC Press |isbn=978-0-8493-8604-6 |chapter-url=https://books.google.com/books?id=VOnyWUUUj04C&pg=SA11-PA19 |language=en |chapter=High-Speed Computer Arithmetic}}
* {{cite book |last1=Tarasov |first1=Vasily |title=Quantum Mechanics of Non-Hamiltonian and Dissipative Systems |date=2008 |publisher=Elsevier |isbn=978-0-08-055971-1 |url=https://books.google.com/books?id=pHK11tfdE3QC&pg=PA57 |language=en}}
* {{cite book |last1=Taylor |first1=Joseph L. |title=Foundations of Analysis |date=2012 |publisher=American Mathematical Soc. |isbn=978-0-8218-8984-8 |url=https://books.google.com/books?id=0ky2W1loV-QC&pg=PA8 |language=en}}
* {{cite book |last1=Thiam |first1=Thierno |last2=Rochon |first2=Gilbert |title=Sustainability, Emerging Technologies, and Pan-Africanism |date=2019 |publisher=Springer Nature |isbn=978-3-030-22180-5 |url=https://books.google.com/books?id=EWSsDwAAQBAJ&pg=PA164 |language=en}}
* {{cite book |last1=Tiles |first1=Mary |editor1-last=Irvine |editor1-first=Andrew D. |title=Philosophy of Mathematics |date=2009 |publisher=Elsevier |isbn=978-0-08-093058-9 |chapter-url=https://books.google.com/books?id=mbn35b2ghgkC&pg=PA243 |language=en |chapter=A Kantian Perspective on the Philosophy of Mathematics}}
* {{cite book |last1=Uspenskii |first1=V. A. |last2=Semenov |first2=A. L. |chapter=Solvable and Unsolvable Algorithmic Problems |editor-last1=Tabachnikov |editor-first1=Serge |title=Kvant Selecta: Combinatorics, I: Combinatorics, I |date=2001 |publisher=American Mathematical Soc. |isbn=978-0-8218-2171-8 |chapter-url=https://books.google.com/books?id=yID37VIW-t8C&pg=PA113 |language=en}}
* {{cite book |last1=Vaccaro |first1=Alfredo |last2=Pepiciello |first2=Antonio |title=Affine Arithmetic-Based Methods for Uncertain Power System Analysis |date=2022 |publisher=Elsevier |isbn=978-0-323-90503-9 |url=https://books.google.com/books?id=tZxBEAAAQBAJ&pg=PA9 |language=en}}
* {{cite book |last1=Verschaffel |first1=Lieven |last2=Torbeyns |first2=Joke |last3=De Smedt |first3=Bert |editor1-last=Seel |editor1-first=Norbert M. |title=Encyclopedia of the Sciences of Learning |date=2011 |publisher=Springer Science & Business Media |isbn=978-1-4419-1427-9 |chapter-url=https://books.google.com/books?id=xZuSxo4JxoAC&pg=PA2177 |language=en |chapter=Mental Arithmetic}}
* {{cite web |author1=Victoria Department of Education Staff |title=Numeracy for All Learners |url=https://www.education.vic.gov.au/school/teachers/teachingresources/discipline/maths/Pages/numeracy-for-all-learners.aspx |website=www.education.vic.gov.au |access-date=22 November 2023 |language=en |date=2023 }}
* {{cite web |last1=Vinogradov |first1=A. I. |title=Algebraic Number Theory |url=https://encyclopediaofmath.org/wiki/Algebraic_number_theory |website=Encyclopedia of Mathematics |publisher=Springer |access-date=23 October 2023 |date=2019 }}
* {{cite book |last1=Vullo |first1=Vincenzo |title=Gears: Volume 3: A Concise History |date=2020 |publisher=Springer Nature |isbn=978-3-030-40164-1 |url=https://books.google.com/books?id=g67QDwAAQBAJ&pg=PA140 |language=en}}
* {{cite book |last1=Waite |first1=Maurice |title=Pocket Oxford English Dictionary |date=2013 |publisher=OUP Oxford |isbn=978-0-19-966615-7 |url=https://books.google.com/books?id=xqKcAQAAQBAJ&pg=PA42 |language=en}}
* {{cite book |last1=Wallis |first1=W. D. |title=A Beginner's Guide to Discrete Mathematics |date=2011 |publisher=Springer Science & Business Media |isbn=978-0-8176-8286-6 |url=https://books.google.com/books?id=18W4_LJ5bL0C&pg=PA303 |language=en}}
* {{cite book |last1=Wallis |first1=W. D. |title=A Beginner's Guide to Discrete Mathematics |date=2013 |publisher=Springer Science & Business Media |isbn=978-1-4757-3826-1 |url=https://books.google.com/books?id=ONgRBwAAQBAJ&pg=PA20 |language=en}}
* {{cite book |last1=Wang |first1=Hao |title=A Logical Journey: From Gödel to Philosophy |date=1997 |publisher=MIT Press |isbn=978-0-262-26125-8 |url=https://books.google.com/books?id=pckvCy6L_ocC&pg=PA334 |language=en}}
* {{cite book |last1=Ward |first1=J. P. |title=Quaternions and Cayley Numbers: Algebra and Applications |date=2012 |publisher=Springer Science & Business Media |isbn=978-94-011-5768-1 |url=https://books.google.com/books?id=LVDvCAAAQBAJ&pg=PA55 |language=en}}
* {{cite book |last1=Wedell |first1=Moritz |editor1-last=Classen |editor1-first=Albrecht |title=Handbook of Medieval Culture. Volume 2 |date=2015 |publisher=Walter de Gruyter GmbH & Co KG |isbn=978-3-11-037763-7 |chapter-url=https://books.google.com/books?id=Uh5pCgAAQBAJ&pg=PA1235 |language=en |chapter=Numbers}}
* {{cite book |last1=Weil |first1=André |title=Number Theory: An Approach Through History From Hammurapi to Legendre |date=2009 |publisher=Springer Science & Business Media |isbn=978-0-8176-4571-7 |url=https://books.google.com/books?id=Ar7gBwAAQBAJ&pg=PR9 |language=en}}
* {{cite web |last1=Weir |first1=Alan |title=Formalism in the Philosophy of Mathematics |url=https://plato.stanford.edu/entries/formalism-mathematics/ |website=The Stanford Encyclopedia of Philosophy |publisher=Metaphysics Research Lab, Stanford University |access-date=22 November 2023 |date=2022 }}
* {{cite book |last1=Wheater |first1=Carolyn |title=Algebra I |date=2015 |publisher=Dorling Kindersley Limited |isbn=978-0-241-88779-0 |url=https://books.google.com/books?id=Q7R3EAAAQBAJ&pg=PP19 |language=en}}
* {{cite book |last1=Wilson |first1=Robin |title=Number Theory: A Very Short Introduction |date=2020 |publisher=Oxford University Press |isbn=978-0-19-879809-5 |url=https://books.google.com/books?id=fcDgDwAAQBAJ&pg=PA1 |language=en}}
* {{cite book |last1=Wright |first1=Robert J. |last2=Ellemor-Collins |first2=David |last3=Tabor |first3=Pamela D. |title=Developing Number Knowledge: Assessment, Teaching and Intervention with 7-11 Year Olds |date=2011 |publisher=SAGE |isbn=978-1-4462-8927-3 |url=https://books.google.com/books?id=3yqdEAAAQBAJ&pg=PA136 |language=en}}
* {{cite book |last1=Xu |first1=Zhiwei |last2=Zhang |first2=Jialin |title=Computational Thinking: A Perspective on Computer Science |date=1a de Januaro 2022 |publisher=Springer Nature |isbn=978-981-16-3848-0 |url=https://books.google.com/books?id=s2RXEAAAQBAJ&pg=PA121 |language=en}}
* {{cite book |last1=Yadin |first1=Aharon |title=Computer Systems Architecture |date=2016 |publisher=CRC Press |isbn=978-1-315-35592-4 |url=https://books.google.com/books?id=KzeLDQAAQBAJ&pg=PT24 |language=en}}
* {{cite book |last1=Yan |first1=Song Y. |title=Number Theory for Computing |date=2002 |publisher=Springer Science & Business Media |isbn=978-3-642-07710-4 |language=en}}
* {{cite book |last1=Yan |first1=Song Y. |title=Computational Number Theory and Modern Cryptography |date=2013a |publisher=John Wiley & Sons |isbn=978-1-118-18858-3 |url=https://books.google.com/books?id=74oBi4ys0UUC&pg=PA15 |language=en}}
* {{cite book |last1=Young |first1=Cynthia Y.|author1-link= |title=Precalculus |date=2010 |publisher=John Wiley & Sons |isbn=978-0-471-75684-2 |url=https://books.google.com/books?id=9HRLAn326zEC&pg=RA1-PA994 |language=en}}
* {{cite book |last1=Young |first1=Cynthia Y.|title=Algebra and Trigonometry |date=2021 |publisher=John Wiley & Sons |isbn=978-1-119-77830-1 |url=https://books.google.com/books?id=hpVFEAAAQBAJ&pg=RA1-PA4 |language=en}}
* {{cite book |last1=Zhang |first1=G. |title=Logic of Domains |date=6 December 2012 |publisher=Springer Science & Business Media |isbn=978-1-4612-0445-9 |url=https://books.google.com/books?id=GRTSBwAAQBAJ&pg=PA130 |language=en}}

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