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Mathematics

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Mathematics
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Mathematics (din nye Dagbamba ni boondi binsheli Laasabu malibu) nyɛla baŋsim bɔhimbu yaɣi shɛli din yihiri bee n kahigira ka lahi pu n pu sɔ bɔbili shɛŋa din yina ni di ti bo ninneesim n ti tabibi baŋdiba n ti pahi laasabu malibu maŋ maŋa. Mathematics mali la yaɣa pam ka di shɛŋa nye , number theory (din bɔhimbu jandi kalinli bihi), algebra (din bɔhimbu jandi binyɛra biɛhigu), geometry (din bɔhimbu jandi binyɛra nama, poli shɛŋa di ni tooi deegi ni di lɔɣa), analysis (din bɔhimbu jandi taɣibu taɣibu din niŋdi saha kam), n ti pahi set thory (din bɔhimbu jandiri bin muna ka di lahi nyɛ bin shɛli din laɣimdi bin yara di ni tu ni di be luɣi shɛli, dina n nyɛ daantaligu n ti mathematics bɔhimbu dagaadam).

Mathematics nyɛla din jandi binyɛra din ku tooi dii n nya buɣisibu n ti pahi di nahimbu biɛhigu puuni, Mathematics mali la daliri din ka daɣiri m maliwuhiri binyɛra nahingbana ni shaharanima, din dɔli zalisinima din be di puuni nam biehigu ni n yihiri laasabu bihi na, Laasabu bihi ŋɔ dina n nyɛ theorems bee axioms saha shɛŋa din gari ŋɔ nyɛla nahaingbana shɛŋa din leei pilibu shee n ti laasabu bɔhimbu ni baŋsim vihigu shee, [1]

Mathematics mali bukata pam n-ti natural science, engineering, medicine, finance, computer science, n-ti pahi social science. Mathematics nyɛla bɛ ni mali bin shɛli n niɣindi biɛhigu pam, mathematics daantaligi din nyɛ yɛlimaŋli kuli nyɛla di ni za di gama zuɣu ka bɛ zaŋ di maŋ n jali science tumtumsa la. Mathematics yaɣ'shɛga, kamani statistics mini game theory nyɛla din mini di applications kpini taba, ka bɛ pu li n niŋ applied mathematics bɔŋ ni. Yaɣ'shɛŋa mi biɛni ka di mini di application shɛli bɛ kpini taba ka bɛ pu dimi n niŋ pure mathematics bɔŋ ni, amaa nyaanga ha, di tooi mali nin nyabu applications.[2][3]

Di yi kana taarihi polo, mathematics dalirinim din ne n dɔya ka ka gbaliŋ nyɛ din daa tuugi yina Creek mathematics puuni, din niŋ bayana Euclid's Elements puuni.[4] Tum di piligu, bɛ daa pirigi la mathematics n-ti geometry mini arithmetics (natural numbers niɣimbu mini pirigibu) naɣila 16th mini 17th centuries la saha, algebra mini infinitesimal calculus ni daa yoli yina. Lala saha maa hali ni zuŋɔ, alizama di be mathematical innovations mini scientific discoveries sunsuuni chɛmi ka di zaa.zoora.[5] 19th century bahigu ha, mathematics kpabu yalimuɣisira daa chɛmi ka bɛ mali ka yooi di soya,[6] ka di chɛ ka be niŋ suhipiɛlli ni mathematics yaɣa maa kalinli pahibu ni di applications yaɣa. Saha ŋɔ mathematics din bɛ ti karim zona ni ŋɔ mali ̃Bɛ ni tooi lahi yɛli ni di nyɛla  equation shɛli din bɛ za di naba ayi zuɣu, ka nyɛ polynomial equations (din tahi algebraic geometry na).mathematics yaɣa din du puya n gari pihinu.[7][8]

[Areas of mathematics] mathematics yaɣa yaɣa

[mali niŋ | mali mi di yibu sheena n-niŋ]

poi ni ninneesim mathematics daa nyala di pirgi yaɣa ayi zuɣu:: arithmetic, din nye kalinli yaltɔɣa, ni geometry din nye binyara ŋmahangbana "shapes" yaltɔɣa.[9] "pseudoscience" yaɣ'shaŋa kaman "numerology" mini "astrology", daa nyala din ka walginsim ni mathematics.[10]

Di ni daa ti niŋ ka nina nehi yaɣa ayi daa nyala din pahi, laasabu malibu baŋsim yaɣili zuɣu, "Mathematical notation" dini n daa zan n ti "algebra" din nye n zaŋ "formulas"n tumdi tuma kaman kotomsi,bi yi ti a bohigu laasabu malibu ni ni (x + 2 = 5)"algebra" n nye din yan soŋ a ka tooi labisi lala bohigu maa. Din pahi n nye "calculus" din nye din pirigi buyi "differential calculus" mini "integral calculus", ka di mi wuhiri ti binyara ni taɣiri sham biehigu puuni. Lala yaɣa yaɣa ŋo arithmetic, geometry, algebra, ni calculus.[11] nyala din daa saɣi n ti ka beni hali ni "19th century" (zuŋo yuun'kobshii) naabu saha.Yaɣa kamani "celestial mechanics" mini "solid mechanics" daa lahi silimiinsi ni boli shaba "mathematicians" ni daa bohim shɛli amaa punpoŋo di pahila physics puuni..[12] Lala yaɣa ŋo daa kuli nyala din bohinda amaa ka daa lee bi za gama zuɣu m mali di konko yuli naɣila 17th century la saha.[13]

"19th century" (zuŋo yuun'kobshii) bahigu saha, mathematics kpabu yalimuɣisira daa chɛmi ka bɛ mali ka yoogi so'pala.[14][6] Yuuni 200 mathematics din be ti karim zona ni ŋɔ mali mathematics yaɣa din du puya n gari pihiyobu ni ata.[8] Lala yaɣa maa shɛga gɔhila yaɣ'kura maa, kamani di ni nyɛ yɛlimaŋli di yi kana number theory (dina n pa nyɛ puunpoŋo arithmetic di du maa) mini geometry. Di yaɣa din du maa shɛŋa gba mali la geometry di yuya puuni bee ka bɛ kuli kpuɣili kamani geometry  yaɣ'shɛli. Algebra mini calculus pa yaɣ'shɛli din du, amaa di pirigi mi n ti yaɣ'shɛga din dundu. Yaɣ'shɛga din pahipahi ka du daa laɣim la taba 20th century la saha bee ka bɛ daa yi bɛ kpuɣili ka di nyɛ mathematics kamani mathematical logic mini foundations.[7]

Number Theory

[mali niŋ | mali mi di yibu sheena n-niŋ]

Number theory daa pilimi ni numbanim niɣimbu, dina n nyɛ natural numbers (N) ka di daa ti zoogi n niŋ integers (Z) ni rational numbers (Q). Number theory ka bɛ daa na boondi arithmetic, amaa punpoŋo din ka bɛ pa mali n tiri  nambanim laasabu malibu.[15] Number theory pilli yila nadaa Babylon mini China nim sani. Number theory diba ayi shɛga din daa kuli niŋ bayana n nyɛ Euclid din nyɛ daadaa la saha Greece nim dini nti pahi Diophantus din nyɛ Alexandria nim dini la.[16] Saha ŋɔ number theory bohimbu din be suhini ka kutooi nya nyɛla din jɛndi Pierre de Fermat mini Leonhard Euler. Lala yaɣili maa daa nyɛla din nya nasara ka di nyɛla Adrien-Marie Legendre mini Carl Friedrich Gauss nuutimbu zuɣu.[17]

Niriba pam yalimi ni namba yɛli'muɣisira ni yan n nya faako, naɣila bɛ zaŋ la so'shɛli din du n tum tuma, din tooi yirina mathematics yaɣa gabbu ni na. Shɛhira din yi polo pam n nyɛ Fermat bahigu theorem maa. Lala biɛhiŋ ŋɔ maa nyɛla Pierre de Fermat ni daa sabi yuuni 1637, amaa yuuni 1994 ka Andrew Wiles daa wuhi di dihitabili, o daa zaŋla nɛma kamani scheme theory din be algebraic geometry ni la, category theory, n-ti pahi homological algebra.[18] Di shɛhira shɛli n nyɛ Goldbach biɛhiŋ la, din yɛli ni integer kam din mali jaa ka gari diba ayi nyɛla prime number diba ayi ni laɣim taba.[19]

Number theory laɣim la yaɣ'bobigu kamani analytic number theory, algebraic number theory, geometry of numbers (sodɔligu yaɣili), Diophantine analysis, nti pahi transcendence theory (yalli yaɣili).[7]

Geometry

[mali niŋ | mali mi di yibu sheena n-niŋ]

Geometry nyɛla mathematics wul'kura la zaɣ'yini. Di daa pilimi ni niriba sani shɛhiranim zaŋchaŋ shapes kamani lines, angles, ni circles polo, bɛ ni daa kpaŋsi binshɛli ka di nyɛla tiŋgbana galisim zahimbu mini tiŋgbanni mɛbu zuɣu, amaa ka di daa pirigi pirigi n pili yaɣ'pala.[20]

Nadaa la saha Greek n daa pili di daantali kpɛŋ din tiri dihitabili, ka di yara, ni yɛltɔɣili kam bɛ ni yɛli, ni di nyɛla yɛlimaŋli tuya kadi mali daliri. Shɛhira, ni di bɛ saɣi ni di saɣiti yalli din yɛra, ni binyɛra ayi waɣilim nyɛla dɛdɛ; di waɣilim nim maa tuya ni di mali daliri ka di yina ban namin pun niŋ li ka  mali di daliri nima (theorem) sani, nti pahi yɛltɔɣa din za di nyaanga. Lala yɛltɔɣa din za di nyaanga maa pala binshɛli din ni tooi lɛbi di daliri, dama di nyɛla bɛ maŋmansi dini(bɛ ni saɣi ti shɛli ka ka daliri), bɛɛ ka di pahi yɛltɔɣa shɛga din buɣisiri li (sokam ni saɣi ti shɛli ka ka bɛ daliri). Lala zalkpana ŋɔ maa nyɛla din nyɛ daantaliga n ti mathematics kam, bɛ daa yaligi lala yɛltɔɣa ŋɔ maa n ti geometry, ka Euclid daa laɣim lala yɛltɔɣa ŋɔ maa yuun kɔbisita yaanga pɔi ni anabi Isa kandina n niŋ o buku shɛli bɛ ni booni elements la ni.[21][22]

Euclidean geometry nyɛla bin shɛli din bɔhindi shapes mini di nɛma ni laɣimdi sham, ka di yina lines, planes ni circles polo. Di nyɛla din ka barilim (two dimentional figures) ni din mali barilim (three-dimenrional figures).[20]

Euclidean geometry nyɛla bɛ ni daa pili shɛli ka di ka soli taɣibu hali ni 17th century, René Descartes ni daa pili ti ni pa booni shɛli Cartesian coordinates la na. Ŋɔ maa nyɛla din chɛ ka taɣibu pam kana niriba tumtumsa mini tiɛha polo: Bɛ ni daa yi buɣisi real numbers ni di nyɛla boobu tarisi waɣilim maa (nyam number line), punpoŋo di pa  nyɛla din mali boobu tɔbu shɛɛ ka di za nambanim zaani. Algebra (din daa ti pa nyɛ calculus la) gba zaa ni tooi n mali geometric yɛlimuɣisira. Geometry daa ti pirigi mi n niŋ yaɣ'pala diba ayi: dina n nyɛ synthetic geometry, bɛ ni zaŋdi geometry soli din nyɛ din pali ni analytic geometry, din mi zaŋdi coordinates n tumdi tuma maa.[23]

Analytic geometry nyɛla din yɛra n kpari curves polo, di mini circles mini lines. Lala curves maa nyɛla graph of functions, differential geometry ni daa yina shɛli bɔhimbu ni la. Bɛ ni tooi lahi yɛli ni di nyɛla  equations shɛŋa din bɛ za di naba ayi zuɣu, ka nyɛ (din tahi algebraic geometry na). Analytic geometry lahi chɛmi ka Euclidean luɣi shɛŋa din du n gari yaɣi shɛŋa din mali puli.[20]

Etymology

[mali niŋ | mali mi di yibu sheena n-niŋ]

mathematics bachi nyɛla din yina Ancient Greek bachi máthēma ,ka di hurila Tɛmplet:Gloss, ka bin yihili shɛli and the derived mathēmatikḗ tékhnē (μαθηματικὴ τέχνη), ka di nyɛ Tɛmplet:Gloss. Di che silimiinsili saha shɛli din daa nŋɛ naabu sunsuuni siliminsili ka di yini French and Latin.[24]

[mali niŋ | mali mi di yibu sheena n-niŋ]

Di ŋmahanli, di nyɛla shikuru yini din yina di baa ayi puuni bin be Pythagoreanism ka boli mathēmatikoi (μαθηματικοί)—ka lala saha maa di daa yurimi ni "bohamba" ka di so "mathematicians" in punpɔŋɔ. Ka Pythagoreans daa nyɛla ban kana piligu ni bi constrain di lihigu zan ti bachi n-ti bohambu arithmetic ni geometry. Aristotle saha (384–322 BC) di hankali daa nŋɛ ya.[25] Latin mi ni silimiinsili ni, din pa di paala until 1700, mathematics bachi hurimi "astrology" ( beei saha shɛŋa "astronomy") ka di pa rather "mathematics"; ka gbunni tahira bela bela n-ti ti bin pa mili shɛli zaani gbaai 1500 to 1800. Taɣibu ŋɔ zaŋ la baŋsim taɣibu: di shahara n-nyɛ, Saint Augustine's ni a sorinima zom ka che mathematici, ka di wuhiri "astrologers", saha shɛŋa beni tahiri mi n-zaŋ ti bin shɛli din be niŋ zaŋ ti mathematicians.[26] Di zaha bobli bi ni boli silimiinsili labirimi ni ti Latin neuter zahabobli mathematica (Cicero), zaŋ doli Greek zahabobli ta mathēmatiká (τὰ μαθηματικά) ka di booni "bin sahakam mathematical", hali din nyɛ hankali zuli ni silimiinsili daa nŋɛla be ni suhi shɛli ka di nyɛla di anfaani mathematic(al) ka be zaanli ni bachi namda mathematics ni zaha palli, physics ni metaphysics soli ni daa naa, be ni daa nyɛ Greek nima sani.[27] English ni, bachi namda mathematics deeri zaha yini bachi niŋdili. Ka be maarili It is often maths[28] beei,North America, math.[29]

Ancient

[mali niŋ | mali mi di yibu sheena n-niŋ]
Babylon nima mathematical tablet bɛ ni bɔli shɛli Plimpton 322, din yuma chaŋ n ti lu yuuni 1800 pɔi ni Anabi Issah saha BC

Din lahi pahi ti n baŋdi a ni tooi kali binyɛra shɛm ti ni nyara, pɔi ni yali kura niriba ni tooi mali gbaabu zaŋ kpa kalibu bachi ŋa ka kalinli, kamani saha—dabisa, seasons, bee yuma.[30][31] shahira din be tɔm din galisi polo mathematics bi tooi yi polo naɣila zaŋ kpa 3000 BC, bo saha ka Babylonians mini Egyptians piligi zaŋ tum tuma arithmetic, algebra, ni geometry zaŋ ti farigu deebu ni tuma laɣ' kalindi bo Polo , zaŋ ti mebu ni malibu, mini zaŋ ti ban mi zuɣusaa yɛltɔɣa.[32] saha kura la saha mathematical sabbu din yina Mesopotamia mini Egypt zaa yina la 2000 ni ti ti 1800 BC.[33] pam daŋ bu sabbu bolibu Pythagorean dibaata ni so, zaŋ ti inference, the Pythagorean theorem nyami zaŋ ti pam ancient mini widespread mathematical concept zaɣ' bihi nyaaŋa arithmetic mini geometry. Di biɛla Babylonian mathematics din elementary arithmetic (laɣimbu, yihibu, multiplication, ni pirigibu) piligu yina polo din be archaeological record. The Babylonians gba kpe jishee dariza system ka tum a sexagesimal numeral system bi ni na zaŋdi shɛli n tumdi tuma zuŋɔ n-ti zahindi angles mini saha.[34] system|Hindunima–larbaawa numeral system]] ni di zalisi zaŋ ti di of its operations,di niŋ yi duniya ni zuŋɔ , ni gili chaŋ di ni taɣi di piligu millennium AD India puuni ni daa taɣaya ni ti Western duniya via Islamic mathematics.[35] ni yaɣ' shɛli lɛbigimsim of Indian mathematics be lala saha maa baŋbu ni approximation of sine ni cosine, ni daŋ yibu din be kalibu ŋa ku tooi kali.[36][37]]]

Ayi kana ti liɣi random population distribution (μ) balibu kam ni, samplingmean (x̄) maa lɛbigiri la Gaussian distribution ka di variance (σ) nyabu chani dɔli la central limit theorem of probability theory.[38]
Babylon nima mathematical tablet bɛ ni bɔli shɛli Plimpton 322, din yuma chaŋ n ti lu yuuni 1800  pɔi ni Anabi Issah saha BC
Anfooni din wuhiri kalinli bihi bɛ ni daa zaŋ shɛŋa n sabi Bakhshali manuscript gbaŋ, din yuma chaŋ n ti lu yuun kɔbga din pahi ayi pɔi ni Anabi Issah saha BC hali ti gbaai yuun kɔbga din pahi ayi AD
Anfooni din wuhiri yaɣili din bie al-Khwarizmi's Al-Jabr sabili ni
Anfooni din wuhiri mathematics baŋda ŋun yuli booni Carl Friedrich Gauss
Anfooni din wuhiri sigma (Σ) summation notation ni tumdi sham, di sabbu n ti pahi dibihi ni ga taba sham

Symbolic notation and terminology

[mali niŋ | mali mi di yibu sheena n-niŋ]
Anfooni din wuhiri sigma (Σ) summation notation ni tumdi sham, di sabbu n ti pahi dibihi ni ga taba sham

Mathematical notation nyɛla din yɛligi m be science puuni ni fitanima ni zaŋdi zaani complex concepts ni falinima m be concise, unambiguous, ni ni dede yaɣili. notation ŋɔ laɣimla of symbols zaŋ niŋ zaŋ zali operations, di nyɛla din bi yihi kalinli shɛŋa, zo' sima ni binshɛŋa din nyɛ mathematical binyɛra, ni n naa yi laɣimdiba ni niŋdi hankaya nima puuni ni formulas.[39] precisely pam, kalibunima ni mathematical shɛŋa niŋsim deera daa zaŋ ti by symbols bɔli yaɣa, din laɣim binshɛɣu kam Latin bee Greek bachinima, ni din gba pahi din be sabbu yaɣ' shɛli. Operation ni zosima zaa laɣimmi zaŋ ti symbols din gahim bee glyphs,[40] kamani + (zaŋ laɣim), × (multiplication), (integral), = (zaɣ' yini), ni < (din bi paai).[41] symbols ŋɔ maa zaa laɣimla ni zaŋ di tira zaligu din gahim ti hankaya puuni ni formulas.[42] zaɣa maŋli ni, hankaya ni formulas bi yirina di konko, amaa di paɣala yɛltɔɣa puuni punpɔŋɔ bala, where expressions ni tum di tuma bachi namda phrases ni formulas dema Mathematics malila yaligibu din mali yaa n taɣara broad range yaɣa shɛŋa din mali bohambu zaŋ kpa fali pirigibu din be yaɣa shɛŋa ni , idealized objects ni kamani bini kpɛri taba shɛm. Di biɛla rigorous baŋbu din yihi mebu din zani viɛnyɛla. An axiom bee postulate mathematical nyɛla statement din kpuɣiri di yɛlimaŋli ka ka din bɔri proof. If a mathematical statement ni nabi yihiri (bee din bi yihira), di nin nye conjecture. Zaŋ yi di series of rigorous nangbankpeeni kpuɣibudeductive reasoning, a statement yihiri di yɛlimaŋli lɛbila a theorem. theorem shɛli din gahim nye din tum maa maŋmaŋa prove theorem shɛli gba booni la lemma. A proven ni be yaɣ' shɛŋa ni pam laɣimbu ni ni nyɛ corollary.[43]

Numerous technical tooi nyɛ in mathematics n nyɛ neologisms, kamani polynomial ni homeomorphism.[44] technical shɛŋa terms nyɛla bachinima bala shɛŋa din yɔli ka tum be accurate gbunni ni may differ slightly n yina bi gbunni shɛli din yɔli puuni. Ŋmahinli, mathematics puuni, "bee" n-nyɛ "zaɣa yini, shɛli bee di zaa", ka, bal' shɛli din yɔli puuni, it is either ambiguous bee din wuhi"zaɣ' yini bee di shɛli amaa ka di zaa shɛli" (in mathematics, the latter n booni"din be kpaŋa bee"). Koobu, mathematical pam terms nyɛla bachi shɛli din yɔli din daa zaŋ niŋ bɛ ti pali ni nyɛ gbunni shɛli din pa yim.[45] din boŋɔ ni tooi taɣi din dɔni viɛnyɛla ni yɛlimaŋli mathematical assertions, amaa ni yina n-nyɛ yɛltɔɣa yɔli n ti niriba ban ka bɔri din wuhi ani nyɛ so. Ŋmahinli, " module kam zaa din nyɛ alaha nyɛla zaɣ' pati" ni " field nyɛla alwa

Kundivihira

[mali niŋ | mali mi di yibu sheena n-niŋ]
Isaac Newton
Gottfried Wilhelm von Leibniz
Anfooni din wuhiri mathematics baŋda ayi ban yuya booni Isaac Newton (ŋun bie nuzaa zuɣu) n ti pahi Gottfried Wilhelm Leibniz ŋun nam infinitesimal calculus.
Anfooni din wuhiri pendulum
The skin of this giantpufferfish exhibits a Turing pattern, which can be modeled by reaction–diffusion systems.
Fractal with a scaling symmetry and a central symmetry
The front side of the Fields Medal with an illustration of the Greek polymath Archimedes
  1. ↑ Hipólito, Inês Viegas (August 9–15, 2015). "Abstract Cognition and the Natureof Mathematical Proof". In Kanzian, Christian; Mitterer, Josef; Neges, Katharina (eds.). Realismus – Relativismus – Konstruktivismus: Beiträge des 38. Internationalen Wittgenstein Symposiums [Realism – Relativism – Constructivism: Contributions of the 38th International Wittgenstein Symposium] (PDF) (in German and English). 23. Kirchberg am Wechsel, Austria: Austrian Ludwig Wittgenstein Society. pp. 132–134. ISSN 1022-3398. OCLC 236026294. Archived (PDF) from the original on November 7, 2022. Retrieved January 17, 2024.(at ResearchGate Tɛmplet:Open access Archived Silimin gɔli November 5, 2022, at the Wayback Machine)
  2. ↑ Peterson, Ivars (1988). The mathematical tourist: snapshots of modern mathematics. New York: Freeman. ISBN 978-0-7167-1953-3.
  3. ↑ The Unreasonable Effectiveness of Mathematics in the Natural Sciences.
  4. ↑ Euclidís Elements, by far his most famous and important work, is a comprehensive collection of the mathematical knowledge discovered by the classical Greeks, and thus represents a mathematical history of the age just prior to Euclid and the development o.
  5. ↑ MathSciNet.
  6. 1 2 Kleiner, Israel (1991-12-01). "Rigor and Proof in Mathematics: A Historical Perspective". Mathematics Magazine 64 (5): 291–314. DOI:10.1080/0025570X.1991.11977625. ISSN 0025-570X.
  7. 1 2 3 Mihaljevic, Helena; Santamaría, Lucía (2024-06-17), "The gender publication gap in mathematics: A bibliometric analysis of zbMATH data", 90 Years of zbMATH, EMS Press, pp. 9–16, ISBN 978-3-98547-073-0, retrieved 2025-12-16
  8. 1 2 American Mathematical Society (ed.). Notices of the American Mathematical Society. Providence, R.I., etc.: American Mathematical Society. OCLC 1480366.
  9. ↑ Bell, E. T. (1992) [1945]. "General Prospectus". The Development of Mathematics (2nd ed.). Dover Publications. p. 3. ISBN 978-0-486-27239-9. LCCN 45010599. OCLC 523284. ... mathematics has come down to the present by the two main streams of number and form. The first carried along arithmetic and algebra, the second, geometry.
  10. ↑ Tiwari, Sarju (1992). "A Mirror of Civilization". Mathematics in History, Culture, Philosophy, and Science (1st ed.). New Delhi, India: Mittal Publications. p. 27. ISBN 978-81-7099-404-6. LCCN 92909575. OCLC 28115124. Retrieved November 13, 2025. It is unfortunate that two curses of mathematics--Numerology and Astrology were also born with it and have been more acceptable to the masses than mathematics itself.
  11. ↑ Restivo, Sal (1992). "Mathematics from the Ground Up". In Bunge, Mario (ed.). Mathematics in Society and History. Episteme. 20. Kluwer Academic Publishers. p. 14. ISBN 0-7923-1765-3. LCCN 25709270. OCLC 92013695.
  12. ↑ Musielak, Dora (2022). Leonhard Euler and the Foundations of Celestial Mechanics. History of Physics. Springer International Publishing. doi:10.1007/978-3-031-12322-1. eISSN 2730-7557. ISBN 978-3-031-12321-4. ISSN 2730-7549. OCLC 1332780664. S2CID 253240718 Check |s2cid= value (help).
  13. ↑ Biggs, N. L. (May 1979). "The roots of combinatorics". Historia Mathematica 6 (2): 109–136. DOI:10.1016/0315-0860(79)90074-0. ISSN 0315-0860.
  14. ↑ (2016-07-01) "{{{title}}}". DOI:10.55619/pc.gr.92012.
  15. ↑ Leveque, W. (1977). "Fundamentals of number theory".
  16. ↑ (1997-11-15) "The Queen of Mathematics : A Historically Motivated Guide to Number Theory" (in en). Taylor & Francis. DOI:10.1201/9781439864623/queen-mathematics-jay-goldman.
  17. ↑ Weil, André (2001). "Number Theory" (in en). SpringerLink. DOI:10.1007/978-0-8176-4571-7.
  18. ↑ Kleiner, Israel (2000-03-31). "From Fermat to Wiles: Fermat's Last Theorem Becomes a Theorem" (in en). Elemente der Mathematik 55 (1): 19–37. DOI:10.1007/pl00000079. ISSN 0013-6018.
  19. ↑ The Goldbach Conjecture | Series in Pure Mathematics (en). [[|Wp/azb/Digital object identifier|DOI]]:10.1142/5096#t=aboutBook.
  20. 1 2 3 Mathematics.
  21. ↑ Hilbert, David (1902). The Foundations of Geometry (in English). Open court publishing Company. ISBN 978-0-7222-3046-6.
  22. ↑ Hartshorne, Robin (2005-09-28). Geometry: Euclid and Beyond (in English). Springer Science & Business Media. ISBN 978-0-387-98650-0.
  23. ↑ History of analytic geometry | WorldCat.org (en).
  24. ↑ Tɛmplet:Multiref
  25. ↑ Perisho, Margaret W. (Spring 1965). "The Etymology of Mathematical Terms". Pi Mu Epsilon Journal 4 (2): 62–66. ISSN 0031-952X.
  26. ↑ Boas, Ralph P. (1995). "What Augustine Didn't Say About Mathematicians". In Alexanderson, Gerald L.; Mugler, Dale H. (eds.). Lion Hunting and Other Mathematical Pursuits: A Collection of Mathematics, Verse, and Stories. Mathematical Association of America. p. 257. ISBN 978-0-88385-323-8. LCCN 94078313. OCLC 633018890.
  27. ↑ The Oxford Dictionary of English Etymology, Oxford English Dictionary, sub "mathematics", "mathematic", "mathematics".
  28. ↑ Maths (Noun). Oxford University Press.
  29. ↑ Math (Noun³). Oxford University Press.
  30. ↑ See, for example, Wilder, Raymond L. Evolution of Mathematical Concepts; an Elementary Study. passim.
  31. ↑ Zaslavsky, Claudia (1999). Africa Counts: Number and Pattern in African Culture. Chicago Review Press. ISBN 978-1-61374-115-3. OCLC 843204342.
  32. ↑ Kline 1990, Chapter 1.
  33. ↑ Mesopotamia[dead link] pg 10. Retrieved June 1, 2024
  34. ↑ Boyer 1991, "Mesopotamia" pp. 24–27.
  35. ↑ Ore, Øystein (1988). [[[:Tɛmplet:GBurl]] Number Theory and Its History] Check |url= value (help). Courier Corporation. pp. 19–24. ISBN 978-0-486-65620-5. Retrieved November 14, 2022.
  36. ↑ Singh, A. N. (January 1936). "On the Use of Series in Hindu Mathematics". Osiris 1: 606–628. DOI:10.1086/368443.
  37. ↑ {{cite book | chapter=Use of series in India | last1=Kolachana | first1=A. | last2=Mahesh | first2=K. | last3=Ramasubramanian | first3=K. | title=Studies in Indian Mathematics and Astronomy | series=Sources and Studies in the History of Mathematics and Physical Sciences | pages=438–461 | publisher=Springer | publication-place=Singapore | isbn=978-981-13-7325-1 | year=2019 | doi=10.1007/978-981-13-7326

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    [[File:Rubik's cube.svg|thumb|alt=A shuffled 3x3 rubik's cube |The Rubik's Cube group is a concrete application of group theory.<ref>Joyner, David (2008). "The (legal) Rubik's Cube group". Adventures in Group Theory: Rubik's Cube, Merlin's Machine, and Other Mathematical Toys (2nd ed.). Johns Hopkins University Press. pp. 219–232. ISBN 978-0-8018-9012-3. LCCN 2008011322. OCLC 213765703.
  38. ↑ Rouaud, Mathieu (April 2017) [First published July 2013]. Probability, Statistics and Estimation (PDF). p. 10. Archived (PDF) from the original on October 9, 2022. Retrieved February 13, 2024.
  39. ↑ Wolfram, Stephan (October 2000). Mathematical Notation: Past and Future. MathML and Math on the Web: MathML International Conference 2000, Urbana Champaign, USA. Archived from the original on November 16, 2022. Retrieved February 3, 2024.
  40. ↑ (December 3, 2020) "Knowledge of Mathematical Symbols Goes Beyond Numbers". Journal of Numerical Cognition 6 (3): 322–354. DOI:10.5964/jnc.v6i3.293.
  41. ↑ AMS Style Guide. American Mathematical Society (October 2017).
  42. ↑ (2000) "Constituent Structure in Mathematical Expressions". Proceedings of the Annual Meeting of the Cognitive Science Society 22.
  43. ↑ Rossi, Richard J. (2006). Theorems, Corollaries, Lemmas, and Methods of Proof. Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts. John Wiley & Sons. pp. 1–14, 47–48. ISBN 978-0-470-04295-3. LCCN 2006041609. OCLC 64085024.
  44. ↑ Earliest Uses of Some Words of Mathematics. University of St. Andrews.
  45. ↑ Silver, Daniel S. (November–December 2017). "The New Language of Mathematics". The American Scientist 105: 364–371. DOI:10.1511/2017.105.6.364. ISSN 0003-0996.
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