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Lamba

Gyara hanyoyin mahaɗar
Daga Wikipedia, Insakulofidiya ta kyauta.
Rukunin lambobi masu rikitarwa
lissafi
Bayanai
Ƙaramin ɓangare na mathematical object (en) Fassara
Bangare na set of numbers (en) Fassara
Has characteristic (en) Fassara type of number (en) Fassara
Manifestation of (en) Fassara quantity (en) Fassara

Lamba abu ne na lissafi da ake amfani da shi don ƙidaya, auna, da kuma lakabi. Misalai na asali sune lambobi na halitta 1, 2, 3, 4, da sauransu.[1] Ana iya wakilta lambobi cikin harshe tare da kalmomin lamba. Ƙari a duniya, ana iya wakilta lambobi ɗaya ta hanyar alamomi, da ake kira lambobi ; misali, "5" lamba ce da ke wakiltar lamba biyar. Kamar yadda kawai ƙananan adadin alamomin za a iya haddace su, ƙananan lambobi yawanci ana tsara su a cikin tsarin lambobi, wanda hanya ce mai tsari don wakiltar kowace lamba. Mafi yawan tsarin lambobi shine tsarin lambobi na Hindu-Larabci, wanda ke ba da izinin wakilcin kowane lamba ta amfani da haɗuwa da alamomin ƙididdiga guda goma, da ake kira lambobi.[2] Baya ga amfani da su wajen kirgawa da aunawa, ana amfani da lambobi sau da yawa don lakabi (kamar yadda suke da lambobin tarho ), don yin oda (kamar yadda tare da lambobin serial ), da kuma lambobin (kamar yadda suke da ISBNs). A cikin amfanin gama-gari, lamba ba ta bambanta a fili da lambar da take wakilta ba.

A cikin ilimin lissafi, an tsawaita ra'ayin lamba a cikin ƙarni don haɗawa da sifili (0),[3] lambobi masu karancin daraja,[4] lambobi masu ma'ana kamar rabi ɗaya. , lambobi na ainihi kamar tushen murabba'in 2 da π,[5] da lambobi masu rikitarwa waɗanda ke tsawaita ainihin lambobi tare da tushen murabba'i na−1 (da haɗe-haɗe tare da lambobi na ainihi ta ƙara ko rage yawan adadinsa).[6] Ana yin ƙididdigewa tare da lambobi tare da ayyukan lissafi, wanda aka fi sani da ƙari shine ƙari, raguwa, ninkawa, rarrabawa, da ƙari. Nazarin su ko amfani da su ana kiran su lissafi, kalma wanda kuma yana iya komawa zuwa ka'idar lamba, nazarin kaddarorin lambobi.

Bayan amfaninsu na amfani, lambobi suna da mahimmancin al'adu a duk faɗin duniya.[7][8] Misali, a cikin al'ummar Yamma, ana ɗaukar lamba 13 a matsayin rashin sa'a, kuma "miliyan" na iya nuna "mai yawa" maimakon ainihin adadi.[9] Ko da yake yanzu ana ɗaukarsa azaman pseudoscience, imani da mahimmancin sufi na lambobi, wanda aka sani da numerology, ya mamaye tunanin da da na da. Ƙididdigar ƙididdiga ta yi tasiri sosai ga ci gaban ilimin lissafi na Girkanci, yana ƙarfafa binciken matsalolin da yawa a ka'idar lamba waɗanda har yanzu suna da sha'awa a yau.[9]

A cikin karni na 19, masu ilmin lissafi sun fara haɓaka ƙididdiga daban-daban waɗanda ke raba wasu kaddarorin lambobi, kuma ana iya ganin su suna faɗaɗa ra'ayi. Daga cikin na farko akwai lambobin hypercomplex, wanda ya ƙunshi nau'i-nau'i daban-daban ko gyare-gyare na tsarin lambobi masu rikitarwa. A cikin ilimin lissafi na zamani, ana ɗaukar tsarin lambobi masu mahimmanci misalai na musamman na ƙarin tsarin algebra gaba ɗaya kamar zobba da filayen, kuma aikace-aikacen kalmar "lambar" lamari ne na al'ada, ba tare da mahimmancin mahimmanci ba.[10]

Tarihi

[gyara sashe | gyara masomin]

Farkon anfani da lambobi

[gyara sashe | gyara masomin]

An gano kasusuwa da sauran kayan tarihi tare da yanke alamomin da mutane da yawa suka yi imani da cewa alamomi ne.[11] Wataƙila an yi amfani da waɗannan alamomin don kirga lokacin da suka wuce, kamar adadin kwanaki, zagayowar wata ko adana bayanai na adadi, kamar na dabbobi.

Tsarin ƙididdigewa ba shi da ra'ayi na ƙimar wuri (kamar yadda yake a cikin ƙayyadaddun ƙima), wanda ke iyakance wakilcin manyan lambobi. Duk da haka, ana ɗaukar tsarin ƙididdigewa a matsayin nau'in tsarin ƙididdiga na farko.

Tsarin da aka sani na farko tare da ƙimar wuri shine tsarin Mesopotamiya tushe na 60 (c. 3400 BC) kuma sanannen tushe na tsarin 10 na farko yana zuwa 3100 BC a Masar.[12]

Numerals

[gyara sashe | gyara masomin]

Ya kamata a bambanta lambobi daga numerals, alamomin da ake amfani da su don wakiltar lambobi. Masarawa sun ƙirƙiro tsarin ƙididdiga na farko, kuma Girkawa sun bi taswirar ƙidayar su akan haruffan Ionian da Doric. Lambobin Roman, tsarin da ya yi amfani da haɗin haruffa daga haruffan Roman, ya kasance mafi rinjaye a Turai har zuwa yaduwar tsarin lambobi mafi girma na Hindu-Larabci a kusa da ƙarshen karni na 14, kuma tsarin lambobi na Hindu-Larabci ya kasance mafi yawan tsarin wakilci don wakiltar. lambobi a duniya a yau.[13] Makullin tasirin tsarin shine alamar sifili, wanda tsoffin masana lissafin Indiya suka haɓaka a kusa da 500 AD.

Sifili

[gyara sashe | gyara masomin]

Sanin farko da aka rubuta amfani da kwanakin sifili zuwa AD 628, kuma ya bayyana a cikin Brāhmasphuṭasiddhānta, babban aikin masanin lissafin Indiya Brahmagupta. Ya yi magani 0 a matsayin lamba kuma sun tattauna ayyukan da suka haɗa da shi, gami da rarraba.[14] A wannan lokacin (7th karni) manufar ta kai ga Cambodia a matsayin lambobin Khmer, kuma bayanai sun nuna ra'ayin daga baya ya yadu zuwa kasar Sin da kasashen musulmi.

Lambar 605 a cikin lambobin Khmer, daga rubutu daga 683 AD. Fara amfani da sifili azaman adadi na goma.

Brāhmasphuṭasiddhānta na Brahmagupta shine littafi na farko da ya ambaci sifili a matsayin lamba, don haka Brahmagupta yawanci ana ɗaukarsa a matsayin farkon wanda ya tsara manufar sifili. Ya ba da ka'idojin amfani da sifili tare da lambobi marasa kyau da masu kyau, kamar "sifili da lambar tabbatacce ita ce lamba mai kyau, kuma lambar mara kyau da sifili ita ce mummunan lamba." Brāhmasphuṭasiddhānta shine rubutun farko da aka sani don ɗaukar sifili azaman lamba a kansa, maimakon a matsayin kawai lamba mai riƙewa a wakiltar wata lamba kamar yadda Babila suka yi ko kuma a matsayin alama don ƙarancin yawa kamar yadda Ptolemy ya yi kuma Romawa.

Amfani da 0 a matsayin lamba ya kamata a bambanta daga amfani da shi azaman adadin ma'auni a tsarin ƙimar wuri . An yi amfani da litattafai na dā da yawa 0. Nassosin Babila da na Masar sun yi amfani da shi. Masarawa sun yi amfani da kalmar nfr don nuna sifili ma'auni a cikin lissafin shiga biyu. Rubutun Indiya sun yi amfani da kalmar Sanskrit Shunye ko shunya don komawa ga manufar banza . A cikin rubutun lissafi wannan kalma sau da yawa tana nufin lamba sifili.[15] A cikin irin wannan yanayin, Pāṇini (ƙarni na 5 BC) ya yi amfani da ma'aikacin null (sifili) a cikin Ashtadhyayi, misali na farko na nahawu na algebra don harshen Sanskrit (kuma duba Pingala ).

Akwai sauran amfani da sifili kafin Brahmagupta, kodayake takaddun ba su cika kamar yadda yake a cikin Brāhmasphuṭasiddhānta .

Bayanai sun nuna cewa Girkawan zamanin da ba su da tabbas game da matsayin 0 a matsayin lamba: sun tambayi kansu "ta yaya 'babu' zai zama wani abu?" haifar da ban sha'awa falsafa da kuma, ta hanyar Medieval zamani, addini muhawara game da yanayi da wanzuwar 0 da vacuum. Paradoxes na Zeno na Elea sun dogara da wani sashi akan fassarar rashin tabbas na 0. (Tsohon Helenawa ma sun yi tambaya ko 1 ya kasance lamba.)

Marigayi Olmec na kudancin tsakiyar Mexico sun fara amfani da alamar sifili, harsashi glyph, a cikin Sabuwar Duniya, mai yiwuwa a 4th century BC amma tabbas ta hanyar 40. BC, wanda ya zama wani muhimmin sashi na lambobi na Maya da kalandar Maya. Mayan lissafin da aka yi amfani da tushe 4 da gindi 5 an rubuta azaman tushe 20. George I. Sánchez a cikin 1961 ya ba da rahoton tushe 4 ,baza 5 "yatsa" abacus.

A shekara ta 130 AD, Ptolemy, wanda Hipparchus da Babila suka rinjaye shi, yana amfani da alamar 0 (ƙaramin da'ira mai tsayi mai tsayi) a cikin tsarin lambobi na jima'i in ba haka ba ta amfani da lambobin haruffa na Helenanci. Domin an yi amfani da shi shi kaɗai, ba a matsayin mai riƙe da wuri kawai ba, wannan sifili na Hellenistic shine farkon rubuce-rubucen amfani da sifilin gaskiya a cikin Tsohuwar Duniya. A cikin rubuce-rubucen Byzantine daga baya na Syntaxis Mathematica ( Almagest ), sifilin Hellenistic ya rikiɗe zuwa harafin Helenanci Omicron (in ba haka ba ma'ana). 70).

An yi amfani da wani sifili na gaskiya a cikin tebur tare da lambobin Roman ta 525 (wanda Dionysius Exigus yayi amfani da shi na farko), amma a matsayin kalma, nulla . ma'ana ba komai, ba a matsayin alama ba. Lokacin da aka samar 0 a matsayin saura, nihil , kuma ba ma'anar komai ba, an yi amfani da shi. Wadannan sifilai na tsakiya duk masu kwamfutoci na gaba (masu ƙididdigewa na Easter ) sun yi amfani da su. An yi amfani da keɓantaccen amfani da farkon su, N, a cikin tebur na lambobi na Romawa ta Bede ko abokin aiki kusan 725, alamar sifili na gaskiya.

Negative numbers

[gyara sashe | gyara masomin]

An gano maƙasudin ra'ayi na negative numbers tun a 100-50 BC a China. Babi tara akan fasahar lissafi sun ƙunshi hanyoyin gano wuraren ƙididdiga; An yi amfani da jajayen sanduna don nuna ƙididdiga masu kyau, baƙar fata ga korau.[16] Magana ta farko a cikin aikin yammacin duniya shine a karni na 3 AD a Girka. Diophantus yayi magana akan ma'auni daidai da 4x + 20 = 0 (maganin ba shi da kyau) a cikin Arithmetica, yana mai cewa lissafin ya ba da sakamako na negative number.

Masana ilimin lissafi na Turai, a mafi yawan lokuta, sun yi tsayayya da ra'ayi na lambobi mara kyau har zuwa karni na 17, kodayake Fibonacci ya ba da izinin magance matsalolin kudi a cikin matsalolin kudi inda za a iya fassara su a matsayin bashi (babi na 13 na Liber Abaci, 1202) kuma daga baya a matsayin hasara (a cikin Flos). René Descartes ya kira su tushen ƙarya yayin da suke girma cikin algebra polynomials duk da haka ya sami hanyar musanya tushen gaskiya da tushen ƙarya kuma. A lokaci guda kuma, Sinawa suna nuna munanan lambobi ta hanyar zana bugun jini ta hanyar dama-mafi mafi yawan lambobi marasa sifili na madaidaicin lamba ta daidai.[17] Farkon amfani da lambobi mara kyau a cikin aikin Turai shine Nicolas Chuquet a cikin ƙarni na 15. Ya yi amfani da su a matsayin masu magana, amma ya kira su da "lambobi marasa hankali".

Rational numbers

[gyara sashe | gyara masomin]

Wataƙila manufar lambobi masu ɓarna sun samo asali ne tun kafin tarihi. Masarawa na da sun yi amfani da bayanin juzu'in su na Masar don lambobi masu ma'ana a cikin rubutun lissafi kamar Rhind Mathematical Papyrus da Papyrus Kahun. Masana lissafin Girka na gargajiya da na Indiya sun yi nazari kan ka'idar lambobi masu ma'ana, a matsayin wani ɓangare na babban binciken ka'idar lamba.[18] Mafi sanannun waɗannan su ne abubuwan Euclid, waɗanda suka yi kusan 300 BC. Daga cikin rubutun Indiya, wanda ya fi dacewa shine Sthananga Sutra, wanda kuma ya shafi ka'idar lamba a matsayin wani ɓangare na nazarin lissafi na gaba ɗaya.

Irrational numbers

[gyara sashe | gyara masomin]

Amfanin farko na lambobi marasa ma'ana shine a cikin Sulba Sutras na Indiya wanda aka haɗa tsakanin 800 da 500 BC.[19] Hippasus ya gano lambobi marasa ma'ana yayin ƙoƙarin wakiltar tushen murabba'in 2 azaman juzu'i. Duk da haka, Pythagoras ya gaskanta da cikar lambobi, kuma ya kasa yarda da wanzuwar lambobi marasa ma'ana. Ba zai iya karyata wanzuwarsu ta hanyar tunani ba, amma ya kasa yarda da lambobi marasa ma'ana, don haka, ana zarginsa kuma akai-akai akai-akai, ya yanke wa Hippasus hukuncin kisa ta hanyar nutsewa, don hana yada wannan labari mai ban tsoro.[20]

Ƙarni na 16 ya kawo karɓuwar Turai ta ƙarshe na lambobi mara kyau da na ɓarna. Ya zuwa karni na 17, masu ilmin lissafi gabaɗaya sun yi amfani da juzu'i na ƙima tare da bayanin zamani. Sai dai, sai a karni na 19, masanan lissafi suka raba marasa hankali zuwa algebra da sassa masu wuce gona da iri, kuma sun sake yin nazarin kimiyya na rashin hankali. Ya kasance kusan kwance tun Euclid. A cikin 1872, an gabatar da buga ka'idodin Karl Weierstrass (da almajirinsa E. Kossak), Eduard Heine, [21] Georg Cantor, [22] da Richard Dedekind.[23] A cikin 1869, Charles Méray ya ɗauki matsayi ɗaya na tashi kamar Heine, amma ka'idar gabaɗaya ana magana da ita zuwa shekara ta 1872. Hanyar Weierstrass gabaɗaya ta Salvatore Pincherle (1880), kuma Dedekind's ya sami ƙarin shahara ta hanyar aikin marubucin daga baya (1888) da Tanneryment (18 ta Paul). Weierstrass, Cantor, da Heine sun kafa ra'ayoyinsu akan jerin marasa iyaka, yayin da Dedekind ya samo nasa akan ra'ayin yanke (Schnitt) a cikin tsarin lambobi na ainihi, yana raba duk lambobi masu ma'ana zuwa ƙungiyoyi biyu suna da takamaiman halaye. Batun ya sami gudummawa daga baya a hannun Weierstrass, Kronecker, [24] da Méray.

Transcendental numbers and reals

[gyara sashe | gyara masomin]

Kasancewar lambobi masu wucewa[25] Liouville ne ya fara kafa shi (1844, 1851). Hermite ya tabbatar a cikin 1873 cewa e transcendental ne kuma Lindemann ya tabbatar a cikin 1882 cewa π ya wuce gona da iri. A ƙarshe, Cantor ya nuna cewa saitin duk ainihin lambobi ba su da iyaka amma saitin duk lambobin algebra ba su da iyaka, don haka akwai adadi mara iyaka na lambobi masu wucewa.

Rabe rabe na ainihi

[gyara sashe | gyara masomin]

Ana iya rarraba lambobi zuwa saiti, da ake kira saitin lamba ko tsarin lamba, kamar lambobi na halitta da ainihin lambobi. Babban tsarin lamba sune kamar haka:

Main number systems
Symbol Name Examples/Explanation
Natural numbers 0, 1, 2, 3, 4, 5, ... or 1, 2, 3, 4, 5, ...

or are sometimes used.

Integers ..., −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, ...
Rational numbers Samfuri:Sfrac where a and b are integers and b is not 0
Real numbers The limit of a convergent sequence of rational numbers
Complex numbers a + bi where a and b are real numbers and i is a formal square root of −1

Kowane ɗayan waɗannan tsarin lambobi yanki ne na na gaba. Don haka, alal misali, lamba ta hankali ita ma lamba ce ta gaske, kuma kowane lamba ta gaske ma lamba ce mai rikitarwa. Ana iya bayyana wannan ta alama kamar

.

Lambobi na halitta

[gyara sashe | gyara masomin]
The natural numbers, starting with 1

Lambobin da aka fi sani su ne lambobi na halitta (wani lokaci ana kiran lambobi gabaɗaya ko lambobi): 1, 2, 3, da sauransu. A al'ada, jerin lambobi na dabi'a sun fara da 1 (0 ba a la'akari da lamba ga tsohuwar Helenawa ba.) Duk da haka, a cikin karni na 19, saitin theorists da sauran mathematicians sun fara ciki har da 0 (kardinality na komai a cikin komai, watau 0 abubuwa, inda 0 shine mafi ƙarancin lambar katin) a cikin saitin lambobi na halitta.[26] A yau, masu ilimin lissafi daban-daban suna amfani da kalmar don kwatanta saiti biyu, gami da 0 ko a'a.

A cikin tsarin lambobi 10 na tushe, a kusan amfani da duniya a yau don ayyukan lissafin lissafi, ana rubuta alamomin lambobi na halitta ta amfani da lambobi goma: 0, 1, 2, 3, 4, 5, 6, 7, 8, da 9. Radix ko tushe shine adadin lambobi na musamman na lambobi, ciki har da sifili, cewa tsarin lambobi yana amfani da lambobi (don tsarin 0 don wakiltar lambobi). A cikin wannan tsarin tushe guda 10, lambobi mafi dama na lambar dabi'a yana da darajar wuri 1, kuma kowane lambobi yana da darajar wuri sau goma fiye da wurin darajar lambobi zuwa dama.

A cikin ka'idar saiti, wacce ke da ikon yin aiki azaman tushen axiomatic don lissafin zamani,[27] lambobi na halitta ana iya wakilta su da nau'ikan saiti daidai. Misali, ana iya wakilta lamba 3 azaman ajin duk saitin da ke da ainihin abubuwa uku. A madadin, a cikin Peano Arithmetic, lambar 3 tana wakiltar sss0, inda s shine aikin "majiyi" (watau 3 shine magaji na uku na 0). Yawancin wakilci daban-daban suna yiwuwa; duk abin da ake buƙata don wakiltar 3 a hukumance shine rubuta wata alama ko tsarin alamomi sau uku.

Integers

[gyara sashe | gyara masomin]

An ayyana maƙasudin madaidaicin lamba a matsayin lamba da ke samar da 0 lokacin da aka ƙara ta zuwa madaidaicin intiger. Yawancin lambobi marasa kyau ana rubuta su da wata alama mara kyau (alamar ragi). Misali, ana rubuta ma'anar 7 -7, da 7 + (-7) = 0. Lokacin da aka haɗa saitin lambobi marasa kyau tare da saitin lambobi na halitta (ciki har da 0), ana bayyana sakamakon a matsayin saitin lamba, Z kuma an rubuta {\displaystyle \ mathbb {Z} }{\displaystyle \mathbb {Z}}. Anan harafin Z ya fito daga Jamusanci 'lamba' Zahl. Saitin intigers yana samar da zobe tare da ƙari da haɓaka ayyuka.

Lambobin dabi'a suna samar da juzu'i na lamba. Kamar yadda babu wani ma'auni na gama gari don haɗawa ko a'a na sifili a cikin lambobi na halitta, lambobi na halitta waɗanda ba tare da sifili ba ana kiransu da ƙima masu inganci, kuma lambobin dabi'a tare da sifili ana kiran su azaman integers mara kyau.

Manazarta

[gyara sashe | gyara masomin]
  1. ↑ "number, n." OED Online (in Turanci). Oxford University Press. Archived from the original on 2018-10-04. Retrieved 2017-05-16.
  2. ↑ "numeral, adj. and n." OED Online. Oxford University Press. Archived from the original on 2022-07-30. Retrieved 2017-05-16.
  3. ↑ Matson, John. "The Origin of Zero". Scientific American (in Turanci). Archived from the original on 2017-08-26. Retrieved 2017-05-16.
  4. ↑ Hodgkin, Luke (2005-06-02). A History of Mathematics: From Mesopotamia to Modernity (in Turanci). OUP Oxford. pp. 85–88. ISBN 978-0-19-152383-0. Archived from the original on 2019-02-04. Retrieved 2017-05-16.
  5. ↑ Mathematics across cultures : the history of non-western mathematics. Dordrecht: Kluwer Academic. 2000. pp. 410–411. ISBN 1-4020-0260-2.
  6. ↑ Descartes, René (1954) [1637]. La Géométrie: The Geometry of René Descartes with a facsimile of the first edition. Dover Publications. ISBN 0-486-60068-8. Retrieved 20 April 2011.
  7. ↑ Gilsdorf, Thomas E. (2012). Introduction to cultural mathematics : with case studies in the Otomies and the Incas. Hoboken, N.J.: Wiley. ISBN 978-1-118-19416-4. OCLC 793103475.
  8. ↑ Restivo, Sal P. (1992). Mathematics in society and history : sociological inquiries. Dordrecht. ISBN 978-94-011-2944-2. OCLC 883391697.
  9. ↑ 9.0 9.1 Ore, Øystein (1988). Number theory and its history. New York: Dover. ISBN 0-486-65620-9. OCLC 17413345.
  10. ↑ Gouvêa, Fernando Q. The Princeton Companion to Mathematics, Chapter II.1, "The Origins of Modern Mathematics", p. 82. Princeton University Press, September 28, 2008. Samfuri:Isbn. "Today, it is no longer that easy to decide what counts as a 'number.' The objects from the original sequence of 'integer, rational, real, and complex' are certainly numbers, but so are the p-adics. The quaternions are rarely referred to as 'numbers,' on the other hand, though they can be used to coordinatize certain mathematical notions."
  11. ↑ Marshack, Alexander (1971). The roots of civilization; the cognitive beginnings of man's first art, symbol, and notation ([1st ed.] ed.). New York: McGraw-Hill. ISBN 0-07-040535-2. OCLC 257105.
  12. ↑ "Egyptian Mathematical Papyri – Mathematicians of the African Diaspora". Math.buffalo.edu. Archived from the original on 2015-04-07. Retrieved 2012-01-30.
  13. ↑ Chrisomalis, Stephen (2003-09-01). "The Egyptian origin of the Greek alphabetic numerals". Antiquity. 77 (297): 485–96. doi:10.1017/S0003598X00092541. ISSN 0003-598X. S2CID 160523072.
  14. ↑ Aczel, Amir D. (2015-05-07). "My Quest to Find the First Zero". TIME (in Turanci). Retrieved 2025-02-15.
  15. ↑ "Historia Matematica Mailing List Archive: Re: [HM] The Zero Story: a question". Sunsite.utk.edu. 1999-04-26. Archived from the original on 2012-01-12. Retrieved 2012-01-30.
  16. ↑ Staszkow, Ronald; Robert Bradshaw (2004). The Mathematical Palette (3rd ed.). Brooks Cole. p. 41. ISBN 0-534-40365-4.
  17. ↑ Smith, David Eugene (1958). History of Modern Mathematics. Dover Publications. p. 259. ISBN 0-486-20429-4.
  18. ↑ "Classical Greek culture (article)". Khan Academy (in Turanci). Archived from the original on 2022-05-04. Retrieved 2022-05-04.
  19. ↑ Selin, Helaine, ed. (2000). Mathematics across cultures: the history of non-Western mathematics. Kluwer Academic Publishers. p. 451. ISBN 0-7923-6481-3.
  20. ↑ Bernard Frischer (1984). "Horace and the Monuments: A New Interpretation of the Archytas Ode". In D.R. Shackleton Bailey (ed.). Harvard Studies in Classical Philology. Harvard University Press. p. 83. ISBN 0-674-37935-7.
  21. ↑ Eduard Heine, "Die Elemente der Functionenlehre", [Crelle's] Journal für die reine und angewandte Mathematik, No. 74 (1872): 172–188.
  22. ↑ Georg Cantor, "Ueber unendliche, lineare Punktmannichfaltigkeiten", pt. 5, Mathematische Annalen, 21, 4 (1883‑12): 545–591.
  23. ↑ Richard Dedekind, Stetigkeit & irrationale Zahlen Archived 2021-07-09 at the Wayback Machine (Braunschweig: Friedrich Vieweg & Sohn, 1872). Subsequently published in: ———, Gesammelte mathematische Werke, ed. Robert Fricke, Emmy Noether & Öystein Ore (Braunschweig: Friedrich Vieweg & Sohn, 1932), vol. 3, pp. 315–334.
  24. ↑ L. Kronecker, "Ueber den Zahlbegriff", [Crelle's] Journal für die reine und angewandte Mathematik, No. 101 (1887): 337–355.
  25. ↑ Bogomolny, A. "What's a number?". Interactive Mathematics Miscellany and Puzzles. Archived from the original on 23 September 2010. Retrieved 11 July 2010.
  26. ↑ "natural number". Merriam-Webster.com. Merriam-Webster. Archived from the original on 13 December 2019. Retrieved 4 October 2014.
  27. ↑ Suppes, Patrick (1972). Axiomatic Set Theory. Courier Dover Publications. p. 1. ISBN 0-486-61630-4.
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