Algebra Tarixi – History Of Algebra

“Algebra” so’zi Arabcha الljbr so’zi al-jabrva bu O’rta asr Fors matematikasi tomonidan 830 yilda yozilgan risoladan kelib chiqadi, Muhammad ibn Muso al-Xuvrizmi arabcha nomi, Kitob al-muḫtṣar fī Hisob al-gabr va-l-muqobala, deb tarjima qilish mumkin Tugatish va muvozanatlash bo’yicha hisoblash bo’yicha ixcham kitob. Ning tizimli echimi uchun berilgan risola chiziqli va kvadrat tenglamalar. Bir tarixga ko’ra, “[i] t faqat qanday shartlarda ekanligiga aniq emas al-jabr va muqobala degan ma’noni anglatadi, ammo odatdagi talqin oldingi tarjimada nazarda tutilganga o’xshashdir. “Al-jabr” so’zi, ehtimol “tiklash” yoki “tugatish” kabi bir narsani anglatar edi va ayirilgan atamalarni tenglamaning boshqa tomoniga ko’chirishga ishora qiladi; “muqobala” so’zi “qisqartirish” yoki “muvozanatlash” degan ma’noni anglatadi, ya’ni tenglamaning qarama-qarshi tomonlarida o’xshash atamalarni bekor qilish. Al-Xorazmiy davridan ancha vaqt o’tgach Ispaniyada arablarning ta’siri Don Kixot, bu erda “algebrista” so’zi suyak o’rnatuvchi, ya’ni “tiklovchi” uchun ishlatiladi. ” [1] Ushbu atama al-Xorazmiy tomonidan o’zi kiritgan operatsiyalarni tavsiflash uchun ishlatiladi “kamaytirish “va” muvozanatlash “, olib tashlangan atamalarni tenglamaning boshqa tomoniga ko’chirishni nazarda tutadi, ya’ni tenglamaning qarama-qarshi tomonlarida o’xshash atamalarni bekor qilish. [2]

Algebra bosqichlari

Shuningdek qarang: Algebra xronologiyasi

Algebraik ifoda

Algebra har doim hamma joyda matematikada mavjud bo’lgan simvolizmdan foydalana olmadi; o’rniga, u uchta aniq bosqichdan o’tdi. Ramziy algebra rivojlanish bosqichlari taxminan quyidagicha: [3]

Ritorik algebra, unda tenglamalar to’liq jumlalar bilan yoziladi. Masalan, ning ritorik shakli x + 1 = 2 – bu “narsa plyus bitta ikkiga teng” yoki ehtimol “narsa plyus 1 2 ga teng”. Ritorik algebra birinchi bo’lib qadimgi tomonidan ishlab chiqilgan Bobilliklar va XVI asrgacha hukmron bo’lib qoldi.

Sinxronlashtirilgan algebra, unda ba’zi bir simvolizm ishlatiladi, ammo unda ramziy algebraning barcha xususiyatlari mavjud emas. Masalan, ayirishning tenglamaning bir tomonida faqat bir marta ishlatilishi mumkin bo’lgan cheklov bo’lishi mumkin, bu esa ramziy algebra bilan bog’liq emas. Sinxronlashtirilgan algebraik ifoda birinchi bo’lib paydo bo’ldi Diofant ‘ Arifmetika (Milodiy 3-asr), undan keyin Braxmagupta “s Braxma Sfuta Siddxanta (7-asr).

Simvolik algebra, unda to’liq ramziylik ishlatiladi. Bunga dastlabki qadamlarni bir nechta odamlarning ishlarida ko’rish mumkin Islom matematiklari kabi Ibn al-Banna (13-14 asrlar) va al-Kalasadi (15-asr), garchi to’liq ramziy algebra tomonidan ishlab chiqilgan bo’lsa ham François Viette (16-asr). Keyinchalik, Rene Dekart (17-asr) zamonaviy yozuvlarni joriy qildi (masalan, dan foydalanish x—pastga qarang ) va geometriyada yuzaga keladigan masalalarni algebra bo’yicha ifodalash va echish mumkinligini ko’rsatdi (Dekart geometriyasi ).

Algebrada simvolizmdan foydalanish yoki etishmasligi kabi bir xil ahamiyatga ega bo’lib, ko’rib chiqilgan tenglamalar darajasi edi. Kvadrat tenglamalar dastlabki algebrada muhim rol o’ynagan; va tarixning aksariyat qismida, zamonaviy davrning boshlanishigacha barcha kvadrat tenglamalar uchta toifadan biriga tegishli deb tasniflangan.

Bu erda p va q musbat.Bu trixotomiya formaning kvadratik tenglamalari tufayli yuzaga keladi x 2 + p x + q = 0 , p va q musbat, ijobiy ildizlarga ega emas. [4]

Ramziy algebraning ritorik va sinxoplangan bosqichlari orasida a geometrik konstruktiv algebra klassik tomonidan ishlab chiqilgan Yunoncha va Vedik hind matematiklari unda geometriya orqali algebraik tenglamalar echilgan. Masalan, shaklning tenglamasi x 2 = A maydon kvadratining tomonini topish orqali hal qilindi A.

Kontseptual bosqichlar

Algebraik g’oyalarni ifodalashning uch bosqichidan tashqari, ba’zi mualliflar algebra rivojlanishidagi to’rtta kontseptual bosqichni ifoda o’zgarishi bilan birga sodir bo’lganligini tan oldilar. Ushbu to’rt bosqich quyidagicha edi: [5] [ birlamchi bo’lmagan manba kerak ]

Geometrik bosqich, bu erda algebra tushunchalari asosan geometrik. Bu tarixdan boshlanadi Bobilliklar va bilan davom etdi Yunonlar, va keyinchalik tomonidan qayta tiklandi Omar Xayyom.
Statik tenglamalarni echish bosqichi, bu erda aniq munosabatlarni qondiradigan raqamlarni topish maqsadi. Geometrik algebradan uzoqlashish boshlangan Diofant va Braxmagupta, lekin algebra qat’iy ravishda statik tenglamalarni echish bosqichiga o’tmadi Al-Xorazmiy algebraik masalalarni echish uchun umumlashtirilgan algoritmik jarayonlarni joriy etdi.
Dinamik funktsiya bosqichi, bu erda harakat asosiy g’oyadir. A g’oyasi funktsiya bilan paydo bo’lishni boshladi Sharaf al-Din at-Tsī, lekin algebra qat’iy ravishda dinamik funktsiya bosqichiga o’tmadi Gotfrid Leybnits.
Mavhum bosqich, bu erda matematik tuzilish markaziy rol o’ynaydi. Mavhum algebra asosan 19 va 20 asrlarning mahsulidir.

Bobil

Shuningdek qarang: Bobil matematikasi
The Plimpton 322 planshet.

Algebraning kelib chiqishi qadimgi davrlarda kuzatilishi mumkin Bobilliklar, [ iqtibos kerak ] ularning ritorik algebraik tenglamalarini echishda ularga katta yordam beradigan pozitsion sanoq tizimini ishlab chiqqan. Bobilliklar aniq echimlarga emas, balki taxminlarga qiziqishgan va shuning uchun ular odatda foydalanar edilar chiziqli interpolatsiya taxminiy oraliq qiymatlarga. [6] Eng taniqli planshetlardan biri bu Plimpton 322 planshet, miloddan avvalgi 1900–1600 yillarda yaratilgan bo’lib, unda jadval berilgan Pifagor uch marta va yunon matematikasidan oldingi eng zamonaviy matematikani ifodalaydi. [7]

Bobil algebrasi o’sha davrdagi Misr algebrasiga qaraganda ancha rivojlangan edi; Misrliklar asosan chiziqli tenglamalar bilan shug’ullangan bo’lsa, bobilliklar kvadratik va kubik tenglamalarga ko’proq e’tibor berishgan. [6] Bobilliklar egiluvchan algebraik operatsiyalarni ishlab chiqdilar, ular yordamida tengliklarga teng qo’shib, tenglamaning ikkala tomonini o’xshash miqdorlar bilan ko’paytirdilar, shu bilan kasrlar va omillarni yo’q qildilar. [6] Ular faktoringning ko’plab oddiy shakllari bilan tanish edilar, [6] musbat ildizlari bo’lgan uch davrli kvadrat tenglamalar, [8] va ko’plab kub tenglamalar [9] garchi ular umumiy kubik tenglamani kamaytira olishganmi yoki yo’qligi noma’lum bo’lsa ham. [9]

Qadimgi Misr

Ning bir qismi Rind Papirus.
Shuningdek qarang: Misr matematikasi

Qadimgi Misr algebrasi asosan chiziqli tenglamalar bilan shug’ullangan, bobilliklar bu tenglamalarni juda oddiy deb topgan va matematikani misrliklarga qaraganda yuqori darajada rivojlantirgan. [6]

Rhind Papyrus, shuningdek Ahmes Papirus deb nomlanuvchi qadimgi Misr papirusi v. Miloddan avvalgi 1650 yilda Ahmes tomonidan yozilgan, u miloddan avvalgi 2000-1800 yillarda yozgan avvalgi asaridan ko’chirgan. [10] Bu tarixchilarga ma’lum bo’lgan eng keng qadimiy Misr matematik hujjati. [11] Rind papirusida shaklning chiziqli tenglamalari mavjud bo’lgan muammolar mavjud x + a x = b va x + a x + b x = v hal qilinadi, qaerda a, bva v ma’lum va x, “aha” yoki uyum deb nomlangan, noma’lum. [12] Ushbu echimlar “yolg’on pozitsiya usuli” yordamida amalga oshirilgan bo’lishi mumkin, yoki ehtimol regula falsi, bu erda avval ma’lum bir qiymat tenglamaning chap tomoniga almashtiriladi, keyin kerakli arifmetik hisob-kitoblar amalga oshiriladi, uchinchidan, natija tenglamaning o’ng tomoniga taqqoslanadi va nihoyat to’g’ri javob topilgan nisbatlar. Ba’zi bir muammolarda muallif uning echimini “tekshiradi” va shu bilan ma’lum bo’lgan eng sodda dalillardan birini yozadi. [12]

Yunon matematikasi

Shuningdek qarang: Yunon matematikasi

Ning saqlanib qolgan eng qadimgi qismlaridan biri Evklid “s Elementlar, Oksirinxusda topilgan va milodiy 100 yilga to’g’ri keladi (P. Oksi. 29 ). Diagramma II kitob, 5-taklif bilan birga keladi. [13]

Ba’zan, deb da’vo qilishadi Yunonlar algebra yo’q edi, ammo bu noto’g’ri. [14] Vaqtiga kelib Aflotun, Yunon matematikasi tubdan o’zgargan edi. Yunonlar a geometrik algebra bu erda atamalar geometrik ob’ektlar tomonlari bilan ifodalangan, [15] odatda ular bilan bog’liq bo’lgan harflar bo’lgan chiziqlar, [16] va bu yangi algebra shakli bilan ular “maydonlarni qo’llash” deb nomlanuvchi o’zlari ixtiro qilgan jarayon yordamida tenglamalarga echim topishga muvaffaq bo’lishdi. [15] “Maydonlarni qo’llash” geometrik algebraning faqat bir qismidir va u to’liq qamrab olingan Evklid “s Elementlar.

Geometrik algebraga ax = bc chiziqli tenglamani echish misol bo’la oladi. Qadimgi yunonlar bu tenglamani a: b va c: x nisbatlari orasidagi tenglik sifatida emas, balki maydonlarning tengligi sifatida ko’rib chiqish orqali hal qilishadi. Yunonlar yunonlari b va c uzunlikdagi to’rtburchakni yasaydilar, so’ngra to’rtburchaklar tomonlarini a uzunlikgacha kengaytiradilar va nihoyat ular to’rtburchakning echimi bo’lgan tomonini topish uchun kengaytirilgan to’rtburchakni to’ldiradilar. [15]

Timaridaning gullab-yashnashi

Iamblichus yilda Kirish arifmatika buni aytadi Timaridalar (miloddan avvalgi 400 yil – miloddan avvalgi 350 yil) bir vaqtning o’zida chiziqli tenglamalar bilan ishlagan. [17] Xususan, u “Timaridasning gullab-yashnashi” yoki “Timaridas gullari” deb nomlangan o’sha paytdagi mashhur qoidani yaratdi, unda quyidagilar ta’kidlanadi:

Agar yig’indisi bo’lsa n miqdorlar, shuningdek, ma’lum bir miqdorni o’z ichiga olgan har bir juftning yig’indisi berilgan bo’lsa, unda bu ma’lum miqdor ushbu juftlarning yig’indisi va birinchi berilgan summa orasidagi farqning 1 / (n – 2) ga teng. [18]

Evklidning isboti Elementlar chiziqli segment berilganida, uning tomonlaridan biri sifatida segmentni o’z ichiga olgan teng qirrali uchburchak mavjud.

yoki zamonaviy tushunchadan foydalangan holda quyidagi tizimning echimi n chiziqli tenglamalar n noma’lum, [17]

x = ( m 1 + m 2 + . . . + m n − 1 ) − s n − 2 = ( ∑ men = 1 n − 1 m men ) − s n − 2 + m_ + . + m_ ) – s> > = ^ m_ ) – s> >>

Iamblichus ushbu shaklda bo’lmagan ba’zi bir chiziqli tenglamalar tizimining ushbu shaklga qanday joylashishini tasvirlab beradi. [17]

Iskandariya evklidi

Ellinizm matematikasi Evklid tafsilotlar geometrik algebra.

Evklid (Yunoncha: Εὐκλείδης ) edi a Yunoncha gullab-yashnagan matematik Iskandariya, Misr, deyarli aniq hukmronligi davrida Ptolomey I (Miloddan avvalgi 323-283). [19] [20] Uning tug’ilgan yili ham, joyi ham yo’q [19] aniqlanmagan va uning o’lim holatlari.

Evklid “ning otasi” deb qaraladi geometriya “. Uning Elementlar eng muvaffaqiyatli hisoblanadi darslik ichida matematika tarixi. [19] U tarixdagi eng taniqli matematiklardan biri bo’lsa-da, unga tegishli yangi kashfiyotlar mavjud emas, aksincha u o’zining buyuk tushuntirish qobiliyatlari bilan yodda qolgan. [21] The Elementlar ba’zan o’ylanganidek, barcha yunon matematik bilimlarining hozirgi kungacha to’plami emas, aksincha bu unga boshlang’ich kirishdir. [22]

Elementlar

Yunonlarning geometrik ishi Evklidnikidir Elementlar, muayyan muammolarni hal qilishdan tashqari formulalarni umumiy tenglamalar bayon qilish va echish tizimlariga umumlashtirish uchun asos yaratdi.

II kitob Elementlar Evklid davrida geometrik algebrani bajarish uchun juda muhim bo’lgan o’n to’rtta taklifni o’z ichiga oladi. Ushbu takliflar va ularning natijalari bizning zamonaviy ramziy algebra va trigonometriyamizning geometrik ekvivalentlari. [14] Bugungi kunda, zamonaviy ramziy algebradan foydalanib, biz belgilarga ma’lum va noma’lum kattaliklarni (ya’ni raqamlarni) ko’rsatamiz va keyin ularga algebraik amallarni qo’llaymiz. Evklid davrida kattaliklarni chiziqli segmentlar deb hisoblashgan, natijada geometriya aksiomalari yoki teoremalari yordamida natijalar chiqarilgandi. [14]

Qo’shish va ko’paytirishning ko’plab asosiy qonunlari kiritilgan yoki geometrik ravishda isbotlangan Elementlar. Masalan, II kitobning 1-taklifida shunday deyilgan:

Agar ikkita to’g’ri chiziq bo’lsa va ulardan biri har qanday sonli bo’laklarga bo’linsa, ikkita to’g’ri chiziq tarkibidagi to’rtburchak kesilmagan to’g’ri chiziq va segmentlarning har biriga to’g’ri keladigan to’rtburchaklar bilan tengdir.

Ammo bu geometrik versiyadan boshqa narsa emas (chapda) tarqatuvchi qonun, a ( b + v + d ) = a b + a v + a d ; V va VII kitoblarda Elementlar The kommutativ va assotsiativ ko’paytirish qonunlari namoyish etildi. [14]

Ko’pgina asosiy tenglamalar geometrik jihatdan ham isbotlangan. Masalan, II kitobdagi 5-taklif buni tasdiqlaydi a 2 − b 2 = ( a + b ) ( a − b ) , [23] va II kitobdagi 4-taklif buni isbotlaydi ( a + b ) 2 = a 2 + 2 a b + b 2 . [14]

Bundan tashqari, ko’plab tenglamalarga berilgan geometrik echimlar ham mavjud. Masalan, II kitobning 6-taklifi kvadrat tenglamaning echimini beradi bolta + x 2 = b 2 va II kitobning 11-taklifi echimini topadi bolta + x 2 = a 2 . [24]

Ma’lumotlar

Ma’lumotlar Evklid tomonidan Iskandariya maktablarida foydalanish uchun yozilgan va bu kitobning dastlabki oltita kitobiga sherik sifatida ishlatilishi kerak edi. Elementlar. Kitobda o’n besh ta’rif va to’qson beshta bayon mavjud bo’lib, ulardan algebraik qoidalar yoki formulalar sifatida xizmat qiladigan yigirmaga yaqin bayonotlar mavjud. [25] Ushbu bayonotlarning ba’zilari kvadrat tenglamalar echimlarining geometrik ekvivalentlari. [25] Masalan; misol uchun, Ma’lumotlar tenglamalarning echimlarini o’z ichiga oladi dx 2 – qo’shimchalar + b 2 v = 0 va tanish bo’lgan Bobil tenglamasi xy = a 2 , x ± y = b . [25]

Konus kesimlari

A konus bo’limi konusning tekislik bilan kesishishidan kelib chiqadigan egri chiziq. Konusning uchta asosiy turi mavjud: ellipslar (shu jumladan doiralar ), parabolalar va giperbolalar. Konus kesimlari tomonidan kashf etilgan deb tanilgan Menaechmus [26] (miloddan avvalgi 380 yil – miloddan avvalgi 320 yil) va konus kesimlari bilan ishlash o’zlarining tenglamalari bilan ishlashga teng bo’lganligi sababli, ular kubik tenglamalar va boshqa yuqori darajadagi tenglamalarga teng geometrik rollarni o’ynashgan.

Menaechmus parabolada y tenglamani bilar edi 2 = lx ushlab turadi, qaerda l doimiy deb nomlanadi latus rektum, garchi u ikkita noma’lumdagi har qanday tenglama egri chiziqni belgilashini bilmasa ham. [27] U konusning bu xususiyatlarini va boshqalarni ham aniqlagan. Ushbu ma’lumotdan foydalanib, endi muammosiga echim topish mumkin edi kubning takrorlanishi ikkita parabola kesishgan nuqtalar uchun, kubik tenglamani echishga teng echim. [27]

Biz xabar beramiz Evtocius u kub tenglamani echishda foydalangan usuli tufayli edi Dionisodorus (Miloddan avvalgi 250 – Miloddan avvalgi 190). Dionysodorus kubni to’rtburchaklar kesmasi yordamida hal qildi giperbola va a parabola. Bu muammo bilan bog’liq edi Arximed ‘ Sfera va silindrda. Konus kesimlari ming yillar davomida yunon, keyinchalik islom va evropalik matematiklar tomonidan o’rganilib, ishlatilishi kerak edi. Jumladan Perga Apollonius mashhur Koniklar boshqa mavzular qatorida konus bo’limi bilan shug’ullanadi.

Xitoy

Shuningdek qarang: Xitoy matematikasi

Xitoy matematikasi miloddan avvalgi kamida 300 yilga to’g’ri keladi Zhoubi Suanjing, odatda Xitoyning eng qadimiy matematik hujjatlaridan biri hisoblanadi. [28]

Matematik san’at bo’yicha to’qqiz bob

Matematik san’at bo’yicha to’qqiz bob

Chiu-chang suan-shu yoki Matematik san’atning to’qqiz boblari Miloddan avvalgi 250 yillarda yozilgan, barcha xitoy matematik kitoblari orasida eng nufuzli kitoblardan biri bo’lib, u 246 ta masaladan iborat. Sakkizinchi bob musbat va manfiy sonlar yordamida aniqlangan va aniqlanmagan bir vaqtning o’zida chiziqli tenglamalarni echish bilan shug’ullanadi, bitta muammo beshta noma’lumda to’rtta tenglamani echish bilan bog’liq. [28]

Doira o’lchovlarining dengiz oynasi

Ts’e-yuan xay-ching, yoki Doira o’lchovlarining dengiz oynasi, tomonidan yozilgan 170 ga yaqin muammolar to’plami Li Zhi (yoki Li Ye) (milodiy 1192 – 1279). U foydalangan fan fa, yoki Horner usuli, oltitaga teng darajadagi tenglamalarni echish uchun, garchi u o’zining tenglamalarni echish usulini ta’riflamagan bo’lsa. [29]

To’qqiz qismda matematik risola

Shu-shu chiu-chang, yoki To’qqiz qismda matematik risola, boy hokim va vazir tomonidan yozilgan Ch’in Chiu-shao (taxminan 1202 yil – 1261 yil) va bir vaqtning o’zida mosliklarni hal qilish usuli ixtirosi bilan endi nomlangan Xitoyning qolgan teoremasi, bu xitoycha noaniq tahlilning eng yuqori nuqtasini belgilaydi. [29]

Sehrli kvadratchalar

Yang Xui (Paskal) uchburchagi, qadimgi xitoyliklar tomonidan tasvirlangan novda raqamlari.

Eng qadimgi sehrli kvadratlar Xitoyda paydo bo’lgan. [30] Yilda To’qqiz bob muallif chiziqli tenglamalarning koeffitsientlari va doimiy shartlarini sehrli kvadratga (ya’ni matritsaga) joylashtirish va sehrli kvadrat ustida ustunlarni kamaytirish amallarini bajarish bilan bir vaqtning o’zida chiziqli tenglamalar tizimini hal qiladi. [30] Uchdan kattaroq tartibdagi eng qadimgi sehrli kvadratlarga tegishli Yang Xui (taxminan 1261 – 1275 y.), ular o’nga qadar bo’lgan tartibli sehrli kvadratchalar bilan ishladilar. [31]

To’rt elementning qimmatli oynasi

Ssy-yüan yü-chien《四元玉鑒》, yoki To’rt elementning qimmatli ko’zgusi, tomonidan yozilgan Chu Shih-chie 1303 yilda va u Xitoy algebrasining rivojlanish cho’qqisiga chiqdi. The to’rt element, osmon, er, odam va materiya deb nomlangan bo’lib, uning algebraik tenglamalarida to’rtta noma’lum miqdorni ifodalagan. The Ssy-yüan yü-chien bir vaqtning o’zida tenglamalar va o’n to’rtdan yuqori darajadagi tenglamalar bilan shug’ullanadi. Muallif usulidan foydalanadi fan fa, bugun chaqirildi Horner usuli, bu tenglamalarni echish uchun. [32]

The Qimmat oyna arifmetik uchburchakning diagrammasi bilan ochiladi (Paskal uchburchagi ) dumaloq nol belgisidan foydalangan holda, Chu Shih-chie buning uchun kreditni rad etadi. Xuddi shunday uchburchak Yang Xuyning ishida ham paydo bo’ladi, ammo nol belgisi yo’q. [33]

Da isbotsiz berilgan ko’plab yig’indilar qatori tenglamalari mavjud Qimmat oyna. Xulosa seriyasining bir nechtasi: [33]

1 2 + 2 2 + 3 2 + ⋯ + n 2 = n ( n + 1 ) ( 2 n + 1 ) 3 ! 1 + 8 + 30 + 80 + ⋯ + n 2 ( n + 1 ) ( n + 2 ) 3 ! = n ( n + 1 ) ( n + 2 ) ( n + 3 ) ( 4 n + 1 ) 5 ! = >

Diofant

Shuningdek qarang: Diofant tenglamasi

Diophantus ‘ning 1621 yilgi nashrining muqovasi Arifmetika, tarjima qilingan Lotin tomonidan Klod Gaspard Bachet de Meziriac.

Diofant edi a Ellistik v yashagan matematik. Milodiy 250 y., Ammo bu sananing noaniqligi shunchalik katta ediki, u bir asrdan ko’proq vaqt o’tishi mumkin. U yozganligi bilan tanilgan Arifmetika, dastlab o’n uchta kitob bo’lgan, ammo ulardan faqat dastlabki oltitasi saqlanib qolgan risola. [34] Arifmetika an’anaviy yunon matematikasi bilan juda oz o’xshashligi bor, chunki u geometrik usullardan ajralgan va Bobil matematikasidan farq qiladi, chunki Diofant asosan oddiy taxminlar o’rniga aniq va noaniq aniq echimlar bilan shug’ullanadi. [35]

Odatda Diofant tenglamasining echilishi mumkin yoki yo’qligini aniqlash juda qiyin. Diofant hatto kvadrat tenglamaning ikkita echimi bo’lishi mumkinligini anglaganligini ko’rsatadigan biron bir dalil yo’q. Shuningdek, u bir vaqtning o’zida kvadratik tenglamalarni ko’rib chiqdi. [36] Shuningdek, Diophantusning barcha echimlaridan hech qanday umumiy usulni olish mumkin emas. [37]

Yilda Arifmetika, Diophantus birinchi bo’lib noma’lum raqamlar uchun belgilarni, shuningdek raqamlarning kuchlari, munosabatlar va operatsiyalarning qisqartmalaridan foydalanadi; [35] shu tariqa u hozirda ma’lum bo’lgan narsadan foydalangan sinxronlashtirilgan algebra. Diofantin bilan sinxronlashtirilgan algebra va zamonaviy algebraik yozuvlarning asosiy farqi shundaki, birinchisida operatsiyalar, munosabatlar va eksponentlar uchun maxsus belgilar yo’q edi. [38] Masalan, biz nima yozar edik

x 3 − 2 x 2 + 10 x − 1 = 5

Diofant buni shunday yozgan bo’lar edi

Κ Υ a̅ς ̅ ⫛ Δ Υ β̅ Μ a̅ b Μ ε̅

bu erda belgilar quyidagilarni ifodalaydi: [39] [40]

Belgilar	Vakillik
a̅	1ni ifodalaydi
β̅	2 ni ifodalaydi
ε̅	5 ni ifodalaydi
̅	10 ni ifodalaydi
ς	noma’lum miqdorni anglatadi (ya’ni o’zgaruvchi)
ἴσ	(qisqacha ςoς ) “teng” degan ma’noni anglatadi
⫛	unga ergashgan narsaning ayirilishini anglatadi ἴσ
Μ	o’zgaruvchining nolinchi kuchini anglatadi (ya’ni doimiy atama)
Δ Υ	yunon tilidan olingan o’zgaruvchining ikkinchi kuchini ifodalaydi gámíς , kuch yoki kuch degan ma’noni anglatadi
Κ Υ	yunon tilidan olingan o’zgaruvchining uchinchi kuchini ifodalaydi choς , kub degan ma’noni anglatadi
Δ Υ Δ	o’zgaruvchining to’rtinchi kuchini ifodalaydi
ΔΚ Υ	o’zgaruvchining beshinchi kuchini anglatadi
Κ Υ Κ	o’zgaruvchining oltinchi kuchini anglatadi

Koeffitsientlar o’zgaruvchilardan keyin keladi va bu qo’shimcha atamalarning yonma-yon joylashishi bilan ifodalanadi. Diofantning sinxronlashtirilgan tenglamasini zamonaviy ramziy tenglamaga belgi-belgi tarjimasi quyidagicha bo’ladi: [39]

x 3 1 x 10 − x 2 2 x 0 1 = x 0 5 1 10- > 2 > 1 = > 5>

va aniqlashtirish uchun, agar zamonaviy qavs va plyus ishlatilsa, yuqoridagi tenglama quyidagicha yozilishi mumkin: [39]

( x 3 1 + x 10 ) − ( x 2 2 + x 0 1 ) = x 0 5 1+ 10) – (> 2+ > 1) = > 5>

Arifmetika 150 ga yaqin aniq sonlar bilan echilgan muammolarning to’plami bo’lib, postulatsion rivojlanish mavjud emas va umumiy usul aniq tushuntirilmagan, ammo usulning umumiyligi nazarda tutilgan bo’lishi mumkin va tenglamalarning barcha echimlarini topishga urinish yo’q. [35] Arifmetika tarkibida bir nechta noma’lum kattaliklarga tegishli echilgan masalalar mavjud bo’lib, ular iloji bo’lsa, noma’lum miqdorlarni faqat bittasi bilan ifodalash orqali hal etiladi. [35] Arifmetika shuningdek, identifikatorlardan foydalanadi: [41]

( a 2 + b 2 ) ( v 2 + d 2 )	= ( a v + d b ) 2 + ( b v − a d ) 2
= ( a d + b v ) 2 + ( a v − b d ) 2

Hindiston

Shuningdek qarang: Hind matematikasi

Hind matematiklari sanoq sistemalarini o’rganishda faol edilar. Eng qadimgi Hind matematikasi hujjatlar miloddan avvalgi I ming yillikning o’rtalariga (miloddan avvalgi VI asr) tegishli. [42]

Hind matematikasida takrorlanadigan mavzular, boshqalar qatori, aniqlanadigan va noaniq chiziqli va kvadrat tenglamalar, oddiy menzuratsiya va Pifagor uchliklari. [43]

Aryabhata

Aryabhata (476–550) muallifi hind matematikasi Aryabhatiya. Unda u qoidalarni berdi, [44]

1 2 + 2 2 + ⋯ + n 2 = n ( n + 1 ) ( 2 n + 1 ) 6 dan yuqori

1 3 + 2 3 + ⋯ + n 3 = ( 1 + 2 + ⋯ + n ) 2

Braxma Sfuta Siddxanta

Braxmagupta (fl. 628) muallif bo’lgan hind matematikasi Braxma Sfuta Siddxanta. Braxmagupta o’z ishida musbat va manfiy ildizlar uchun umumiy kvadratik tenglamani hal qiladi. [45] Noaniq tahlilda Brahmagupta Pifagor uchliklarini beradi m , 1 2 ( m 2 n − n ) -n)> , 1 2 ( m 2 n + n ) + n)> , ammo bu Brahmagupta tanish bo’lishi mumkin bo’lgan eski Bobil qoidalarining o’zgartirilgan shakli. [46] U birinchi bo’lib chiziqli Diofant tenglamasiga umumiy echimni berdi ax + by = c, bu erda a, b va c butun sonlardir. Brahmagupta aniqlanmagan tenglamaga faqat bitta echim bergan Diophantusdan farqli o’laroq barchasi butun sonli echimlar; Ammo Brahmagupta Diofant singari ba’zi bir misollardan foydalanganligi, ba’zi tarixchilarning Brahmagupta asariga yoki hech bo’lmaganda Bobilning umumiy manbasiga yunoncha ta’sir o’tkazish imkoniyatini ko’rib chiqishiga olib keldi. [47]

Diofant algebrasi singari, Braxmagupta algebrasi ham sinxronlashtirildi. Qo’shish raqamlarni yonma-yon qo’yish, subtrahend ustiga nuqta qo’yish orqali ayirish va dividend ostiga bo’linuvchini dividend ostiga qo’yish orqali bo’linish bilan belgilandi. Ko’paytirish, evolyutsiya va noma’lum miqdorlar tegishli atamalarning qisqartirishlari bilan ifodalangan. [47] Yunonistonning ushbu sinxopatsiyaga ta’siri darajasi, agar mavjud bo’lsa, ma’lum emas va yunon va hind sinxopatsiyasi Bobilning umumiy manbasidan kelib chiqishi mumkin. [47]

Bskara II

Bskara II (1114 – taxminan 1185) – 12-asrning etakchi matematikasi. Algebrada u umumiy echimini bergan Pell tenglamasi. [47] U muallifi Lilavati va Vija-Ganitaaniqlangan va noaniq chiziqli va kvadrat tenglamalar va Pifagor uchliklari bilan bog’liq muammolarni o’z ichiga olgan [43] va u aniq va taxminiy gaplarni ajrata olmaydi. [48] Muammolari ko’p Lilavati va Vija-Ganita boshqa hind manbalaridan olingan va shuning uchun Bhaskara noaniq tahlil bilan shug’ullanishda eng yaxshisidir. [48]

Bxaskara nomlarning boshlang’ich belgilaridan ranglar uchun noma’lum o’zgaruvchilarning ramzi sifatida foydalanadi. Masalan, biz bugun qanday yozar edik

( − x − 1 ) + ( 2 x − 8 ) = x − 9

Bxaskara shunday deb yozgan bo’lar edi

. _ . yo 1 ru 1 . yo 2 ru 8 . Jami yo 1 ru 9

qayerda yo uchun so’zning birinchi bo’g’inini bildiradi qorava ru so’zidan olingan turlari. Raqamlar ustidagi nuqta olib tashlashni bildiradi.

Islom olami

Dan sahifa Tugatish va muvozanatlash bo’yicha hisoblash bo’yicha ixcham kitob.
Shuningdek qarang: Islom matematikasi

I asr Islomiy Arab imperiyasi deyarli hech qanday ilmiy yoki matematik yutuqlarni ko’rmagan, chunki arablar, yangi fath qilingan imperiyasi bilan, hali hech qanday intellektual g’ayratga ega bo’lmagan va dunyoning boshqa qismlarida izlanishlar susaygan edi. 8-asrning ikkinchi yarmida Islomda madaniy uyg’onish yuz berdi va matematikada va fanlarda izlanishlar kuchaydi. [49] Musulmon Abbosiy xalifa al-Mamun (809–833) Aristotel paydo bo’lganida tush ko’rgan va shu sababli al-Mamun arabcha tarjimani iloji boricha ko’proq yunoncha asarlardan, shu jumladan Ptolomeyning asarlaridan nusxa ko’chirishni buyurgan. Almagest va Evklidnikidir Elementlar. Yunon asarlari musulmonlarga Vizantiya imperiyasi shartnomalar evaziga, chunki ikki imperiya tinch bo’lmagan tinchlik. [49] Ushbu yunoncha asarlarning aksariyati tarjima qilingan Sobit ibn Qurra (826-901), u Evklid, Arximed, Apollonius, Ptolomey va Evtosius tomonidan yozilgan kitoblarni tarjima qilgan. [50]

Arab matematiklari algebra mustaqil fan sifatida asos solgan va unga “algebra” nomini bergan (al-jabr). Ular birinchilardan bo’lib algebrani an elementar shakl va o’zi uchun. [51] Arab algebrasining kelib chiqishi to’g’risida uchta nazariya mavjud. Birinchisi hindlarning ta’siriga, ikkinchisi Mesopotamiya yoki fors-suriyalik ta’siriga, uchinchisi yunon ta’siriga urg’u beradi. Ko’pgina olimlar, bu uchta manbaning birlashishi natijasi deb hisoblashadi. [52]

Hukmronlik davrida arablar to’liq ritorik algebradan foydalanganlar, bu erda ko’pincha hatto raqamlar so’zlar bilan yozilgan. Arablar oxir-oqibat yozilgan raqamlarni (masalan, yigirma ikkita) raqam bilan almashtiradilar Arab raqamlari (masalan, 22), ammo arablar sinxronlashtirilgan yoki ramziy algebrani qabul qilmagan yoki rivojlantirmagan [50] ishiga qadar Ibn al-Banna, 13-asrda ramziy algebra ishlab chiqqan va undan keyin Abu al-Hasan ibn Al-al-Kalasodiy XV asrda.

Al-jabr val muqobala

Shuningdek qarang: Tugatish va muvozanatlash bo’yicha hisoblash bo’yicha ixcham kitob

Chapda: Algebra kitobining asl arabcha qo’lyozmasi Al-Xorazmiy. O’ngda: Algebra ning sahifasi Al-Xorazmiy Fredrik Rozen tomonidan, yilda Ingliz tili.

Musulmon [53] Fors tili matematik Muhammad ibn Muso al-Xuvrizmi fakulteti a’zosi bo’lganDonolik uyi ” (Baytul-Hikma) Al-Ma’mun tomonidan tashkil etilgan Bag’dodda. Milodiy 850 yilda vafot etgan Al-Xorazmiy yarim o’ndan ortiq matematik va astronomik asarlar yozgan, ularning ba’zilari hindistonliklarga asoslangan. Sindxind. [49] Al-Xorazmiyning eng taniqli kitoblaridan biri shunday nomlangan Al-jabr val muqobala yoki Tugatish va muvozanatlash bo’yicha hisoblash bo’yicha ixcham kitob va bu polinomlarni ikkinchi darajagacha echish haqida to’liq ma’lumot beradi. [54] Kitobda “ning asosiy tushunchasi ham berilgankamaytirish Chiqarilgan atamalarni tenglamaning boshqa tomoniga ko’chirishni, ya’ni tenglamaning qarama-qarshi tomonlarida o’xshash atamalarni bekor qilishni nazarda tutgan holda “va” muvozanatlash “. Bu Al-Xorazmiy dastlab shunday ta’riflagan operatsiya al-jabr. [55] “Algebra” nomi “al-jabr“kitobining sarlavhasida.

R. Rashed va Anjela Armstrong yozadilar:

“Al-Xorazmiy matni nafaqat matndan ajralib turishini ko’rish mumkin Bobil tabletkalari, shuningdek, dan Diofant ‘ Arifmetika. Bu endi bir qatorga tegishli emas muammolar hal qilinishi kerak, ammo ekspozitsiya bu ibtidoiy atamalardan boshlanadi, unda kombinatsiyalar tenglamalarning barcha mumkin bo’lgan prototiplarini berishi kerak, bu esa aniq o’rganishning haqiqiy ob’ektini tashkil etadi. Boshqa tomondan, o’zi uchun tenglama g’oyasi boshidanoq paydo bo’ladi va, masalan, muammoni echish jarayonida paydo bo’lmagani kabi, umumiy tarzda aytish mumkin, lekin maxsus chaqirilgan muammolarning cheksiz sinfini aniqlang. ” [56]

Al-Jabr oltita bobga bo’lingan bo’lib, ularning har biri turli xil formulalar bilan bog’liq. Ning birinchi bobi Al-Jabr kvadratlari uning ildizlariga (bolta) teng keladigan tenglamalar bilan shug’ullanadi 2 = bx), ikkinchi bob songa (axta) teng kvadratchalar haqida 2 = c), uchinchi bobda songa teng ildizlar (bx = c), to’rtinchi bobda kvadratlar va ildizlar songa teng (ax) 2 + bx = c), beshinchi bobda kvadratchalar va sonlarning teng ildizlari (bolta) haqida so’z boradi 2 + c = bx), va oltinchi va oxirgi bob kvadratlarga teng ildizlar va sonlar (bx + c = ax 2 ). [57]

Kitobning XIV asrdagi arabcha nusxasidan, ikkita kvadrat tenglamaning geometrik echimlarini ko’rsatadigan sahifalar

Yilda Al-Jabr, al-Xorazmiy geometrik isbotlardan foydalanadi, [16] u x = 0 ildizini tan olmaydi, [57] va u faqat ijobiy ildizlar bilan shug’ullanadi. [58] Shuningdek, u buni tan oladi diskriminant ijobiy bo’lishi va usulini tavsiflashi kerak kvadratni to’ldirish, garchi u protsedurani oqlamasa ham. [59] Yunonlarning ta’siri Al-Jabr ‘geometrik asoslar [52] [60] va Herondan olingan bitta muammo bo’yicha. [61] U harfli diagrammalardan foydalanadi, ammo uning barcha tenglamalarida barcha koeffitsientlar aniq sonlardir, chunki u geometrik ravishda nimani ifoda etishini parametrlar bilan ifodalashga imkoni yo’q edi; uslubning umumiyligi nazarda tutilgan bo’lsa-da. [16]

Al-Xorazmiy, ehtimol Diofantnikini bilmagan Arifmetika, [62] arablarga X asrdan biroz oldin ma’lum bo’lgan. [63] Garchi al-Xorazmiy, ehtimol, Braxmagupta asarini bilgan bo’lsa ham, Al-Jabr raqamlar hatto so’zlar bilan yozilgan holda to’liq ritorik. [62] Masalan, biz nima yozar edik

x 2 + 10 x = 39

Diophantus shunday yozgan bo’lar edi [64]

Δ Υ a̅ ςi̅ ‘ίσ Μ λ̅θ̅

Va al-Xorazmiy shunday deb yozgan bo’lar edi [64]

Bir kvadrat va o’nta ildiz bir xil miqdordagi o’ttiz to’qqizga teng dirhemlar; ya’ni o’nta ildizga ko’paytirilganda o’ttiz to’qqizga teng bo’lgan kvadrat nima bo’lishi kerak?

Aralashtirilgan tenglamalarda mantiqiy zaruriyatlar

Abd al-Hamud ibn Turk nomli qo’lyozma muallifi Aralashtirilgan tenglamalarda mantiqiy zaruriyatlar, bu al-Xvarzimiynikiga juda o’xshash Al-Jabr va u bilan bir vaqtning o’zida yoki hatto undan ham oldinroq nashr etilgan Al-Jabr. [63] Qo’lyozma xuddi shu geometrik namoyishda topilgan Al-Jabrva bitta holatda xuddi shu misolda topilgan Al-Jabr, va hatto undan tashqariga chiqadi Al-Jabr agar diskriminant manfiy bo’lsa, kvadrat tenglamaning echimi yo’qligiga geometrik dalil berish orqali. [63] Ushbu ikki asarning o’xshashligi ba’zi tarixchilarning arab algebrasi al-Xorazmiy va Abdul al-Hamid davrida yaxshi rivojlangan bo’lishi mumkin degan xulosaga keldi. [63]

Abu Komil va al-Karxi

Arab matematiklari davolanishdi mantiqsiz raqamlar kabi algebraik ob’ektlar. [65] The Misrlik matematik Abu Komil Shuja ibn Aslam (taxminan 850–930) irratsional sonlarni birinchi bo’lib qabul qilgan (ko’pincha a shaklida) kvadrat ildiz, kub ildizi yoki to’rtinchi ildiz ) echimlari sifatida kvadrat tenglamalar yoki kabi koeffitsientlar ichida tenglama. [66] U uchta chiziqli bo’lmagan narsani birinchi bo’lib hal qildi bir vaqtning o’zida tenglamalar uchta noma’lum o’zgaruvchilar. [67]

Al-Karxi (953–1029), shuningdek Al-Karaji nomi bilan ham tanilgan, voris bo’lgan Abul al-Vafo al-Bozjoniy (940–998) va u ax shaklidagi tenglamalarning birinchi raqamli echimini topdi 2n + bx n = c. [68] Al-Karxi faqat ijobiy ildizlarni ko’rib chiqdi. [68] Al-Karxi, shuningdek, algebradan ozod bo’lgan birinchi shaxs sifatida qaraladi geometrik operatsiyalari va ularni turi bilan almashtiring arifmetik bugungi kunda algebra asosidagi operatsiyalar. Uning algebra va polinomlar, polinomlarni boshqarish uchun arifmetik amallarni bajarish qoidalarini berdi. The matematika tarixchisi F. Vupke, yilda Du Faxri ekstremali, Abou Bekr Muhammad Ben Alhacan Alkarkhi nomidagi al’èbre traité (Parij, 1853), Al-Karaji “algebraik nazariyani birinchi bo’lib kiritganligi uchun maqtagan hisob-kitob “Bundan kelib chiqqan holda, Al-Karaji tergov o’tkazdi binomial koeffitsientlar va Paskal uchburchagi. [69]

Omar Xayyom, Sharaf al-Din va al-Kashi

Omar Xayyom

Uchinchi darajali tenglamani echish uchun x 3 + a 2 x = b Xayyom qurdi parabola x 2 = ay, a doira diametri bilan b/a 2 va kesishish nuqtasi orqali vertikal chiziq. Yechim gorizontal chiziq segmentining boshlanishidan vertikal chiziqning kesishganigacha va uzunligi bilan berilgan x-aksis.

Omar Xayyom (taxminan 1050 – 1123) algebra bo’yicha kitob yozgan Al-Jabr uchinchi darajadagi tenglamalarni kiritish. [70] Omar Xayyom kvadrat tenglamalar uchun ham arifmetik, ham geometrik echimlarni taqdim etdi, ammo u faqat umumiy uchun geometrik echimlarni berdi kub tenglamalar chunki u arifmetik echimlarni mumkin emas deb noto’g’ri ishongan. [70] Uning kesishgan konikalar yordamida kubik tenglamalarni echish usuli ishlatilgan Menaechmus, Arximed va Ibn al-Xaysam (Alhazen), ammo Umar Xayyom barcha kub tenglamalarni ijobiy ildizlar bilan qoplash usulini umumlashtirdi. [70] U faqat ijobiy ildizlarni ko’rib chiqdi va u uchinchi darajadan o’tmadi. [70] Shuningdek, u Geometriya va Algebra o’rtasidagi kuchli munosabatlarni ko’rdi. [70]

12-asrda, Sharaf al-Din at-Tsī (1135-1213) yozgan Al-Muadalat (Tenglamalar to’g’risida risola), ijobiy echimlarga ega bo’lgan sakkiz turdagi kubik tenglamalari va ijobiy echimlarga ega bo’lmaydigan kubik tenglamalarning besh turi ko’rib chiqildi. U keyinchalik “deb nomlanadigan narsadan foydalangan”Ruffini -Horner usuli “ga raqamli ravishda taxminan ildiz kub tenglamaning Shuningdek, u tushunchalarini ishlab chiqdi maksimal va minima ijobiy echimlarga ega bo’lmagan kubik tenglamalarni echish uchun egri chiziqlar. [71] U muhimligini tushundi diskriminant ning kubik tenglamasi va ning dastlabki versiyasidan foydalanilgan Kardano formulasi [72] kubik tenglamalarning ayrim turlariga algebraik echimlarni topish. Roshdi Rashed kabi ba’zi olimlar, Sharafuddin kashf etgan deb ta’kidlaydilar lotin kubik polinomlardan tashkil topgan va uning ahamiyatini anglagan, boshqa olimlar uning echimini Evklid va Arximed g’oyalari bilan bog’lashgan. [73]

Sharafiddin ham a tushunchasini ishlab chiqqan funktsiya. [ iqtibos kerak ] Tenglamani tahlil qilishda x 3 + d = b x 2 masalan, u tenglama shaklini o’zgartirgandan boshlanadi x 2 ( b − x ) = d . Keyin u tenglamaning echimi bor-yo’qligi masalasi chap tomonda joylashgan “funktsiya” ning qiymatga etish-etmasligiga bog’liqligini aytadi. d . Buni aniqlash uchun u funktsiya uchun maksimal qiymatni topadi. U maksimal qiymat qachon paydo bo’lishini isbotlaydi x = 2 b 3 > , bu funktsional qiymatni beradi 4 b 3 27 >> . Keyin Sharaf ad-Din, agar bu qiymat kamroq bo’lsa, deb ta’kidlaydi d , ijobiy echimlar mavjud emas; agar u teng bo’lsa d , keyin bitta echim bor x = 2 b 3 > ; va agar u kattaroq bo’lsa d , keyin ikkita echim bor, biri o’rtasida 0 va 2 b 3 > va bittasi 2 b 3 > va b . [74]

XV asr boshlarida, Jamshid al-Koshiy ning dastlabki shaklini ishlab chiqdi Nyuton usuli tenglamani raqamli echish uchun x P − N = 0 ning ildizlarini topish N . [75] Al-Koshiy ham rivojlandi kasr kasrlari and claimed to have discovered it himself. However, J. Lennart Berggrenn notes that he was mistaken, as decimal fractions were first used five centuries before him by the Bag’dodiy matematik Abu’l-Hasan al-Uqlidisiy X asrdayoq. [67]

Al-Hassār, Ibn al-Banna, and al-Qalasadi

Al-Hassar, matematik Marokash ixtisoslashgan Islom meros huquqshunosligi during the 12th century, developed the modern symbolic matematik yozuv uchun kasrlar, qaerda raqamlovchi va maxraj gorizontal chiziq bilan ajratilgan. This same fractional notation appeared soon after in the work of Fibonachchi XIII asrda. [ iqtibos kerak ]

Abu al-Hasan ibn Al-al-Kalasodiy (1412–1486) was the last major medieval Arab algebraist, who made the first attempt at creating an algebraik yozuv beri Ibn al-Banna two centuries earlier, who was himself the first to make such an attempt since Diofant va Braxmagupta qadimgi davrlarda. [76] The syncopated notations of his predecessors, however, lacked symbols for matematik operatsiyalar. [38] Al-Qalasadi “took the first steps toward the introduction of algebraic symbolism by using letters in place of numbers” [76] and by “using short Arabic words, or just their initial letters, as mathematical symbols.” [76]

Europe and the Mediterranean region

Just as the death of Gipatiya signals the close of the Iskandariya kutubxonasi as a mathematical center, so does the death of Boetsiy signal the end of mathematics in the G’arbiy Rim imperiyasi. Although there was some work being done at Afina, it came to a close when in 529 the Vizantiya imperator Yustinian yopildi butparast philosophical schools. The year 529 is now taken to be the beginning of the medieval period. Scholars fled the West towards the more hospitable East, particularly towards Fors, where they found haven under King Xsrolar and established what might be termed an “Athenian Academy in Exile”. [77] Under a treaty with Justinian, Chosroes would eventually return the scholars to the Sharqiy imperiya. During the Dark Ages, European mathematics was at its nadir with mathematical research consisting mainly of commentaries on ancient treatises; and most of this research was centered in the Vizantiya imperiyasi. The end of the medieval period is set as the fall of Konstantinopol uchun Turklar 1453 yilda.

So’nggi o’rta asrlar

The 12th century saw a flood of translations dan Arabcha ichiga Lotin and by the 13th century, European mathematics was beginning to rival the mathematics of other lands. In the 13th century, the solution of a cubic equation by Fibonachchi is representative of the beginning of a revival in European algebra.

As the Islamic world was declining after the 15th century, the European world was ascending. And it is here that Algebra was further developed.

Symbolic algebra

Ushbu bo’lim is missing information about most important results in algebra that are more recent than 15th century, and are completely ignored. Iltimos, ushbu ma’lumotni kiritish uchun bo’limni kengaytiring. Qo’shimcha tafsilotlar munozara sahifasi. ( 2017 yil yanvar )

Modern notation for arithmetic operations was introduced between the end of the 15th century and the beginning of the 16th century by Yoxannes Vidmann va Maykl Stifel. At the end of 16th century, François Viette introduced symbols, presently called o’zgaruvchilar, for representing indeterminate or unknown numbers. This created a new algebra consisting of computing with symbolic expressions as if they were numbers.

Another key event in the further development of algebra was the general algebraic solution of the cubic and quartic equations, developed in the mid-16th century. A g’oyasi aniqlovchi tomonidan ishlab chiqilgan Yapon matematikasi Kowa Seki in the 17th century, followed by Gotfrid Leybnits ten years later, for the purpose of solving systems of simultaneous linear equations using matritsalar. Gabriel Kramer also did some work on matrices and determinants in the 18th century.

Belgisi x

By tradition, the first unknown o’zgaruvchan in an algebraic problem is nowadays represented by the belgi x > ; if there is a second or a third unknown, these are labeled y > va z >> navbati bilan. Algebraik x is conventionally printed in kursiv turi to distinguish it from the sign of multiplication.

Mathematical historians [78] generally agree that the use of x in algebra was introduced by Rene Dekart and was first published in his treatise La Géémetrie (1637). [79] [80] In that work, he used letters from the beginning of the alphabet (a, b, v. ) for known quantities, and letters from the end of the alphabet (z, y, x. ) for unknowns. [81] It has been suggested that he later settled on x (o’rniga z) for the first unknown because of its relatively greater abundance in the French and Latin typographical fonts of the time. [82]

Three alternative theories of the origin of algebraic x were suggested in the 19th century: (1) a symbol used by German algebraists and thought to be derived from a cursive letter r, mistaken for x; [83] (2) the numeral 1 with oblique chizilgan; [84] and (3) an Arabic/Spanish source (see below). But the Swiss-American historian of mathematics Florian Kajori examined these and found all three lacking in concrete evidence; Cajori credited Descartes as the originator, and described his x, yva z as “free from tradition[,] and their choice purely arbitrary.” [85]

Nevertheless, the Hispano-Arabic hypothesis continues to have a presence in ommaviy madaniyat Bugun. [86] It is the claim that algebraic x is the abbreviation of a supposed qarz from Arabic in Old Spanish. The theory originated in 1884 with the German sharqshunos Pol de Lagard, shortly after he published his edition of a 1505 Spanish/Arabic bilingual glossary [87] in which Spanish kosa (“thing”) was paired with its Arabic equivalent, شىء (shay ʔ ), transcribed as xei. (The “sh” sound in Qadimgi ispan was routinely spelled x.) Evidently Lagarde was aware that Arab mathematicians, in the “rhetorical” stage of algebra’s development, often used that word to represent the unknown quantity. He surmised that “nothing could be more natural” (Nichts war also natürlicher. ) than for the initial of the Arabic word—romanlashtirilgan as the Old Spanish x—to be adopted for use in algebra. [88] A later reader reinterpreted Lagarde’s conjecture as having “proven” the point. [89] Lagarde was unaware that early Spanish mathematicians used, not a transkripsiya of the Arabic word, but rather its tarjima in their own language, “cosa”. [90] There is no instance of xei or similar forms in several compiled historical vocabularies of Spanish. [91] [92]

Gotfrid Leybnits

Although the mathematical notion of funktsiya was implicit in trigonometric and logarithmic tables, which existed in his day, Gotfrid Leybnits was the first, in 1692 and 1694, to employ it explicitly, to denote any of several geometric concepts derived from a curve, such as abstsissa, ordinat, teginish, akkord, va perpendikulyar. [93] In the 18th century, “function” lost these geometrical associations.

Leibniz realized that the coefficients of a system of chiziqli tenglamalar could be arranged into an array, now called a matritsa, which can be manipulated to find the solution of the system, if any. This method was later called Gaussni yo’q qilish. Leibniz also discovered Mantiqiy algebra va ramziy mantiq, also relevant to algebra.

Mavhum algebra

The ability to do algebra is a skill cultivated in matematik ta’lim. As explained by Andrew Warwick, Kembrij universiteti students in the early 19th century practiced “mixed mathematics”, [94] qilish mashqlar based on physical variables such as space, time, and weight. Over time the association of o’zgaruvchilar with physical quantities faded away as mathematical technique grew. Eventually mathematics was concerned completely with abstract polinomlar, murakkab sonlar, giperkompleks raqamlar and other concepts. Application to physical situations was then called amaliy matematika yoki matematik fizika, and the field of mathematics expanded to include mavhum algebra. For instance, the issue of constructible numbers showed some mathematical limitations, and the field of Galua nazariyasi ishlab chiqilgan.

The father of algebra

The title of “the father of algebra” is frequently credited to the Persian mathematician Al-Xorazmiy, [95] [96] [97] tomonidan qo’llab-quvvatlanadi historians of mathematics, kabi Karl Benjamin Boyer, [95] Solomon Gandz va Bartel Leendert van der Vaerden. [98] However, the point is debatable and the title is sometimes credited to the Ellistik matematik Diofant. [95] [99] Those who support Diophantus point to the algebra found in Al-Jabr being more boshlang’ich than the algebra found in Arifmetika va Arifmetika being syncopated while Al-Jabr is fully rhetorical. [95] However, the mathematics historian Kurt Vogel argues against Diophantus holding this title, [100] as his mathematics was not much more algebraic than that of the ancient Bobilliklar. [101]

Those who support Al-Khwarizmi point to the fact that he gave an exhaustive explanation for the algebraic solution of quadratic equations with positive roots, [102] and was the first to teach algebra in an elementar shakl and for its own sake, whereas Diophantus was primarily concerned with the theory of numbers. [51] Al-Khwarizmi also introduced the fundamental concept of “reduction” and “balancing” (which he originally used the term al-jabr to refer to), referring to the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation. [55] Other supporters of Al-Khwarizmi point to his algebra no longer being concerned “with a series of muammolar hal qilinishi kerak, ammo ekspozitsiya which starts with primitive terms in which the combinations must give all possible prototypes for equations, which henceforward explicitly constitute the true object of study.” They also point to his treatment of an equation for its own sake and “in a generic manner, insofar as it does not simply emerge in the course of solving a problem, but is specifically called on to define an infinite class of problems.” [56] Viktor J. Kats hurmat bilan Al-Jabr hali ham mavjud bo’lgan birinchi haqiqiy algebra matni sifatida. [103]

Shuningdek qarang

Izohlar va iqtiboslar

^Boyer (1991 yil):229)
^ Jeffrey A. Oaks, Haitham M. Alkhateeb, Simplifying equations in Arabic algebra, Historia Mathematica, 34 (2007), 45-61, ISSN0315-0860, [1]
^ (Boyer 1991 yil, “Revival and Decline of Greek Mathematics” p.180) “It has been said that three stages of in the historical development of algebra can be recognized: (1) the rhetorical or early stage, in which everything is written out fully in words; (2) a syncopated or intermediate state, in which some abbreviations are adopted; and (3) a symbolic or final stage. Such an arbitrary division of the development of algebra into three stages is, of course, a facile oversimplification; but it can serve effectively as a first approximation to what has happened””
^ (Boyer 1991 yil, “Mesopotamia” p. 32) “Until modern times there was no thought of solving a quadratic equation of the form x 2 + p x + q = 0 +px+q=0> , where p and q are positive, for the equation has no positive root. Consequently, quadratic equations in ancient and Medieval times—and even in the early modern period—were classified under three types: (1) x 2 + p x = q +px=q> (2) x 2 = p x + q =px+q> (3) x 2 + q = p x +q=px> “
^ Kats, Viktor J.; Barton, Bill (October 2007), “Stages in the History of Algebra with Implications for Teaching”, Matematikadan o’quv ishlari, 66 (2): 185–201, doi:10.1007/s10649-006-9023-7, S2CID120363574
^ abvde (Boyer 1991 yil, “Mesopotamia” p. 30) “Babylonian mathematicians did not hesitate to interpolate by proportional parts to approximate intermediate values. Linear interpolation seems to have been a commonplace procedure in ancient Mesopotamia, and the positional notation lent itself conveniently to the rile of three. [. ] a table essential in Babylonian algebra; this subject reached a considerably higher level in Mesopotamia than in Egypt. Many problem texts from the Old Babylonian period show that the solution of the complete three-term quadratic equation afforded the Babylonians no serious difficulty, for flexible algebraic operations had been developed. They could transpose terms in an equations by adding equals to equals, and they could multiply both sides by like quantities to remove fractions or to eliminate factors. By adding 4ab to (a − b) 2 they could obtain (a + b) 2 for they were familiar with many simple forms of factoring. [. ]Egyptian algebra had been much concerned with linear equations, but the Babylonians evidently found these too elementary for much attention. [. ] In another problem in an Old Babylonian text we find two simultaneous linear equations in two unknown quantities, called respectively the “first silver ring” and the “second silver ring.””
^ Joyce, David E. (1995). “Plimpton 322”. The clay tablet with the catalog number 322 in the G. A. Plimpton Collection at Columbia University may be the most well known mathematical tablet, certainly the most photographed one, but it deserves even greater renown. It was scribed in the Old Babylonian period between -1900 and -1600 and shows the most advanced mathematics before the development of Greek mathematics. Iqtibos jurnali talab qiladi | jurnal = (Yordam bering)
^ (Boyer 1991 yil, “Mesopotamia” p. 31) “The solution of a three-term quadratic equation seems to have exceeded by far the algebraic capabilities of the Egyptians, but Neugebauer in 1930 disclosed that such equations had been handled effectively by the Babylonians in some of the oldest problem texts.”
^ ab (Boyer 1991 yil, “Mesopotamia” p. 33) “There is no record in Egypt of the solution of a cubic equations, but among the Babylonians there are many instances of this. [. ] Whether or not the Babylonians were able to reduce the general four-term cubic, ax 3 + bx 2 + cx = d, to their normal form is not known.”
^ (Boyer 1991 yil, “Egypt” p. 11) “It had been bought in 1959 in a Nile resort town by a Scottish antiquary, Henry Rhind; hence, it often is known as the Rhind Papyrus or, less frequently, as the Ahmes Papyrus in honor of the scribe by whose hand it had been copied in about 1650 BC. The scribe tells us that the material is derived from a prototype from the Middle Kingdom of about 2000 to 1800 BCE.”
^ (Boyer 1991 yil, “Egypt” p. 19) “Much of our information about Egyptian mathematics has been derived from the Rhind or Ahmes Papyrus, the most extensive mathematical document from ancient Egypt; but there are other sources as well.”
^ ab (Boyer 1991 yil, “Egypt” pp. 15–16) “The Egyptian problems so far described are best classified as arithmetic, but there are others that fall into a class to which the term algebraic is appropriately applied. These do not concern specific concrete objects such as bread and beer, nor do they call for operations on known numbers. Instead they require the equivalent of solutions of linear equations of the form x + a x = b yoki x + a x + b x = v , where a and b and c are known and x is unknown. The unknown is referred to as “aha,” or heap. [. ] The solution given by Ahmes is not that of modern textbooks, but one proposed characteristic of a procedure now known as the “method of false position,” or the “rule of false.” A specific false value has been proposed by 1920s scholars and the operations indicated on the left hand side of the equality sign are performed on this assumed number. Recent scholarship shows that scribes had not guessed in these situations. Exact rational number answers written in Egyptian fraction series had confused the 1920s scholars. The attested result shows that Ahmes “checked” result by showing that 16 + 1/2 + 1/8 exactly added to a seventh of this (which is 2 + 1/4 + 1/8), does obtain 19. Here we see another significant step in the development of mathematics, for the check is a simple instance of a proof.”
^Bill Kasselman. “One of the Oldest Extant Diagrams from Euclid”. Britaniya Kolumbiyasi universiteti . Olingan 2008-09-26 .
^ abvde (Boyer 1991 yil, “Euclid of Alexandria” p.109) “Book II of the Elementlar is a short one, containing only fourteen propositions, not one of which plays any role in modern textbooks; yet in Euclid’s day this book was of great significance. This sharp discrepancy between ancient and modern views is easily explained—today we have symbolic algebra and trigonometry that have replaced the geometric equivalents from Greece. For instance, Proposition 1 of Book II states that “If there be two straight lines, and one of them be cut into any number of segments whatever, the rectangle contained by the two straight lines is equal to the rectangles contained by the uncut straight line and each of the segments.” This theorem, which asserts (Fig. 7.5) that AD (AP + PR + RB) = AD·AP + AD·PR + AD·RB, is nothing more than a geometric statement of one of the fundamental laws of arithmetic known today as the distributive law: a (b + c + d) = ab + ac + ad. In later books of the Elementlar (V and VII) we find demonstrations of the commutative and associative laws for multiplication. Whereas in our time magnitudes are represented by letters that are understood to be numbers (either known or unknown) on which we operate with algorithmic rules of algebra, in Euclid’s day magnitudes were pictured as line segments satisfying the axions and theorems of geometry. It is sometimes asserted that the Greeks had no algebra, but this is patently false. They had Book II of the Elementlar, which is geometric algebra and served much the same purpose as does our symbolic algebra. There can be little doubt that modern algebra greatly facilitates the manipulation of relationships among magnitudes. But it is undoubtedly also true that a Greek geometer versed in the fourteen theorems of Euclid’s “algebra” was far more adept in applying these theorems to practical mensuration than is an experienced geometer of today. Ancient geometric “algebra” was not an ideal tool, but it was far from ineffective. Euclid’s statement (Proposition 4), “If a straight line be cut at random, the square on the whole is equal to the squares on the segments and twice the rectangle contained by the segments, is a verbose way of saying that ( a + b ) 2 = a 2 + 2 a b + b 2 =a^+2ab+b^> ,”
^ abv (Boyer 1991 yil, “The Heroic Age” pp. 77–78) “Whether deduction came into mathematics in the sixth century BCE or the fourth and whether incommensurability was discovered before or after 400 BCE, there can be no doubt that Greek mathematics had undergone drastic changes by the time of Plato. [. ] A “geometric algebra” had to take the place of the older “arithmetic algebra,” and in this new algebra there could be no adding of lines to areas or of areas to volumes. From now on there had to be strict homogeneity of terms in equations, and the Mesopotamian normal form, xy = A, x ± y = b, were to be interpreted geometrically. [. ] In this way the Greeks built up the solution of quadratic equations by their process known as “the application of areas,” a portion of geometric algebra that is fully covered by Euclid’s Elementlar. [. ] The linear equation ax = bc, for example, was looked upon as an equality of the areas ax and bc, rather than as a proportion—an equality between the two ratios a:b and c:x. Consequently, in constructing the fourth proportion x in this case, it was usual to construct a rectangle OCDB with the sides b = OB and c = OC (Fig 5.9) and then along OC to lay off OA = a. One completes the rectangle OCDB and draws the diagonal OE cutting CD in P. It is now clear that CP is the desired line x, for rectangle OARS is equal in area to rectangle OCDB”
^ abv (Boyer 1991 yil, “Europe in the Middle Ages” p. 258) “In the arithmetical theorems in Euclid’s Elementlar VII–IX, numbers had been represented by line segments to which letters had been attached, and the geometric proofs in al-Khwarizmi’s Algebra made use of lettered diagrams; but all coefficients in the equations used in the Algebra are specific numbers, whether represented by numerals or written out in words. The idea of generality is implied in al-Khwarizmi’s exposition, but he had no scheme for expressing algebraically the general propositions that are so readily available in geometry.”
^ abv (Heath 1981a, “The (‘Bloom’) of Thymaridas” pp. 94–96) Thymaridas of Paros, an ancient Pythagorean already mentioned (p. 69), was the author of a rule for solving a certain set of n simultaneous simple equations connecting n unknown quantities. The rule was evidently well known, for it was called by the special name [. ] the ‘flower’ or ‘bloom’ of Thymaridas. [. ] The rule is very obscurely worded, but it states in effect that, if we have the following n equations connecting n noma’lum miqdorlar x, x₁, x₂ . x_n-1, namely [. ] Iamblichus, our informant on this subject, goes on to show that other types of equations can be reduced to this, so that they rule does not ‘leave us in the lurch’ in those cases either.”
^ (Flegg 1983, “Unknown Numbers” p. 205) “Thymaridas (fourth century) is said to have had this rule for solving a particular set of n linear equations in n unknowns:
If the sum of n quantities be given, and also the sum of every pair containing a particular quantity, then this particular quantity is equal to 1/ (n – 2) of the difference between the sums of these pairs and the first given sum.”
^ abv (Boyer 1991 yil, “Euclid of Alexandria” p. 100) “but by 306 BCE control of the Egyptian portion of the empire was firmly in the hands of Ptolemy I, and this enlightened ruler was able to turn his attention to constructive efforts. Among his early acts was the establishment at Alexandria of a school or institute, known as the Museum, second to none in its day. As teachers at the school he called a band of leading scholars, among whom was the author of the most fabulously successful mathematics textbook ever written—the Elementlar (Stoichia) of Euclid. Considering the fame of the author and of his best seller, remarkably little is known of Euclid’s life. So obscure was his life that no birthplace is associated with his name.”
^ (Boyer 1991 yil, “Euclid of Alexandria” p. 101) “The tale related above in connection with a request of Alexander the Great for an easy introduction to geometry is repeated in the case of Ptolemy, who Euclid is reported to have assured that “there is no royal road to geometry.””
^ (Boyer 1991 yil, “Euclid of Alexandria” p. 104) “Some of the faculty probably excelled in research, others were better fitted to be administrators, and still some others were noted for teaching ability. It would appear, from the reports we have, that Euclid very definitely fitted into the last category. There is no new discovery attributed to him, but he was noted for expository skills.”
^ (Boyer 1991 yil, “Euclid of Alexandria” p. 104) “The Elementlar was not, as is sometimes thought, a compendium of all geometric knowledge; it was instead an introductory textbook covering all boshlang’ich mathematics.”
^ (Boyer 1991 yil, “Euclid of Alexandria” p. 110) “The same holds true for Elementlar II.5, which contains what we should regard as an impractical circumlocution for a 2 − b 2 = ( a + b ) ( a − b ) “
^ (Boyer 1991 yil, “Euclid of Alexandria” p. 111) “In an exactly analogous manner the quadratic equation bolta + x 2 = b 2 is solved through the use of II.6: If a straight line be bisected and a straight line be added to it in a straight line, the rectangle contained by the whole (with the added straight line) and the added straight line together with the square on the half is equal to the square on the straight line made up of the half and the added straight line. [. ] with II.11 being an important special case of II.6. Here Euclid solves the equation bolta + x 2 = a 2 “
^ abv (Boyer 1991 yil, “Euclid of Alexandria” p. 103) “Euclid’s Ma’lumotlar, a work that has come down to us through both Greek and the Arabic. It seems to have been composed for use at the schools of Alexandria, serving as a companion volume to the first six books of the Elementlar in much the same way that a manual of tables supplements a textbook. [. ] It opens with fifteen definitions concerning magnitudes and loci. The body of the text comprises ninety-five statements concerning the implications of conditions and magnitudes that may be given in a problem. [. ] There are about two dozen similar statements serving as algebraic rules or formulas. [. ] Some of the statements are geometric equivalents of the solution of quadratic equations. For example[. ] Eliminating y bizda . bor (a – x)dx = b 2 v yoki dx 2 – adx + b 2 c = 0 , undan x = a / 2 ± √ ( a / 2 ) 2 – b 2 ( v / d ) . The geometric solution given by Euclid is equivalent to this, except that the negative sign before the radical is used. Statements 84 and 85 in the Data are geometric replacements of the familiar Babylonian algebraic solutions of the systems xy = a 2 , x ± y = b , which again are the equivalents of solutions of simultaneous equations.”
^ (Boyer 1991 yil, “The Euclidean Synthesis” p. 103) “Eutocius and Proclus both attribute the discovery of the conic sections to Menaechmus, who lived in Athens in the late fourth century BC. Proclus, quoting Eratosthenes, refers to “the conic section triads of Menaechmus.” Since this quotation comes just after a discussion of “the section of a right-angled cone” and “the section of an acute-angled cone,” it is inferred that the conic sections were produced by cutting a cone with a plane perpendicular to one of its elements. Then if the vertex angle of the cone is acute, the resulting section (calledoxytome) is an ellipse. If the angle is right, the section (orthotome) is a parabola, and if the angle is obtuse, the section (amblytome) is a hyperbola (see Fig. 5.7).”
^ ab (Boyer 1991 yil, “The age of Plato and Aristotle” p. 94–95) “If OP=y and OD = x are coordinates of point P, we have y 2 = R).OV, or, on substituting equals,
y 2 =R’D.OV=AR’.BC/AB.DO.BC/AB=AR’.BC 2 /AB 2 .x
Inasmuch as segments AR’, BC, and AB are the same for all points P on the curve EQDPG, we can write the equation of the curve, a “section of a right-angled cone,” as y 2 =lx, where l is a constant, later to be known as the latus rectum of the curve. [. ] Menaechmus apparently derived these properties of the conic sections and others as well. Since this material has a string resemblance to the use of coordinates, as illustrated above, it has sometimes been maintains that Menaechmus had analytic geometry. Such a judgment is warranted only in part, for certainly Menaechmus was unaware that any equation in two unknown quantities determines a curve. In fact, the general concept of an equation in unknown quantities was alien to Greek thought. [. ] He had hit upon the conics in a successful search for curves with the properties appropriate to the duplication of the cube. In terms of modern notation the solution is easily achieved. By shifting the curring plane (Gig. 6.2), we can find a parabola with any latus rectum. If, then, we wish to duplicate a cube of edge a, we locate on a right-angled cone two parabolas, one with latus rectum a and another with latus rectum 2a. [. ] It is probable that Menaechmus knew that the duplication could be achieved also by the use of a rectangular hyperbola and a parabola.”
^ ab (Boyer 1991 yil, “China and India” pp. 195–197) “estimates concerning the Chou Pei Suan Ching, generally considered to be the oldest of the mathematical classics, differ by almost a thousand years. [. ] A date of about 300 B.C. would appear reasonable, thus placing it in close competition with another treatise, the Chiu-chang suan-shu, composed about 250 B.C., that is, shortly before the Han dynasty (202 B.C.). [. ] Almost as old at the Chou Pei, and perhaps the most influential of all Chinese mathematical books, was the Chui-chang suan-shu, yoki Matematik san’at bo’yicha to’qqiz bob. This book includes 246 problems on surveying, agriculture, partnerships, engineering, taxation, calculation, the solution of equations, and the properties of right triangles. [. ] Chapter eight of the To’qqiz bob is significant for its solution of problems of simultaneous linear equations, using both positive and negative numbers. The last problem in the chapter involves four equations in five unknowns, and the topic of indeterminate equations was to remain a favorite among Oriental peoples.”
^ ab (Boyer 1991 yil, “China and India” p. 204) “Li Chih (yoki Li Yeh, 1192–1279), Pekin matematikasi, Xublayxon tomonidan 1206 yilda hukumat lavozimiga taklif qilingan, ammo muloyimlik bilan uni rad etish uchun bahona topgan. Ts’e-yuan xay-ching (Doira o’lchovlarining dengiz oynasi) to’rtinchi darajali tenglamalarga olib keladigan ba’zi muammolar [. ] bilan shug’ullanadigan 170 ta masalani o’z ichiga oladi. Garchi u o’zining tenglamalarni echish uslubini, shu jumladan oltinchi darajadagi ba’zi usullarini ta’riflamagan bo’lsa-da, Chu Shih-chie va Horner tomonidan qo’llanilgan bu juda boshqacha shakl emas edi. Horner usulini qo’llagan boshqalar Chin Chiu-shao (taxminan 1202 – 1261 yillarda) va Yang Xuy (taxminan 1261 – 1275 yillarda). Birinchisi printsipial bo’lmagan gubernator va vazir bo’lib, lavozimiga kirishgandan keyin yuz kun ichida ulkan boyliklarga ega bo’ldi. Uning Shu-shu chiu-chang (To’qqiz qismda matematik risola) xitoylik noaniq tahlilning eng yuqori nuqtasini belgilaydi, shu bilan bir vaqtda mosliklarni echish tartiblarini ixtiro qildi. “
^ ab (Boyer 1991 yil, “Xitoy va Hindiston” p. 197) “Xitoyliklar patterlarni juda yaxshi ko’rar edilar; shuning uchun sehrli maydonning birinchi yozuvlari (qadimiy, ammo noma’lum) u erda paydo bo’lishi ajablanarli emas. [. ] Bunday naqshlarga bo’lgan tashvish muallifni tark etdi To’qqiz bob bir vaqtning o’zida chiziqli tenglamalar tizimini [. ] matritsada ustunli operatsiyalarni bajarish orqali hal qilish [. ] ga kamaytirish uchun [. ] Ikkinchi shakl 36z = 99, 5y + z = 24 tenglamalarni ifodalaydi, va 3x + 2y + z = 39 dan z, y va x qiymatlari ketma-ketlik bilan osonlik bilan topiladi. “
^ (Boyer 1991 yil, “Xitoy va Hindiston” 204–205 betlar) “Xuddi shu” Horner “moslamasini hayoti haqida deyarli hech narsa ma’lum bo’lmagan va faqat qisman omon qolgan Yang Xui ishlatgan. Uning hissalari orasida eng qadimgi mavjud. Uchdan kattaroq xitoylik sehrli kvadratchalar, shu jumladan to’rtdan sakkizgacha buyruqlarning ikkitasi va har biri to’qqiz va o’ninchi buyruqlar.
^ (Boyer 1991 yil, “Xitoy va Hindiston” p. 203) “Sung matematiklarining oxirgisi va buyuklari Chu Chih-chie edi (fl. 1280-1303), ammo biz u haqida kam ma’lumotga egamiz-, [. ] ko’proq tarixiy va matematik qiziqish uyg’otadi Ssy-yüan yü-chien (To’rt elementning qimmatli ko’zgusi) 1303 yil. XVIII asrda bu ham Xitoyda g’oyib bo’ldi, faqat keyingi asrda qayta kashf etildi. Osmon, er, odam va materiya deb nomlangan to’rtta element to’rtta noma’lum kattaliklarning bir xil tenglamadagi tasviridir. Kitob Xitoy algebra rivojlanishining eng yuqori cho’qqisini belgilaydi, chunki u bir vaqtning o’zida tenglamalar va o’n to’rtdan yuqori darajadagi tenglamalar bilan shug’ullanadi. Unda muallif o’zi chaqiradigan transformatsiya usulini tasvirlaydi fan fa, elementlari Xitoyda ancha oldin paydo bo’lgan, ammo umuman yarim ming yil o’tib yashagan Horner nomini olgan “.
^ ab (Boyer 1991 yil, “Xitoy va Hindiston” p. 205) “ichida topilgan qator yig’indilardan bir nechtasi Qimmat oyna quyidagilar: [. ] Biroq, hech qanday dalillar keltirilmagan va bu mavzu taxminan XIX asrga qadar Xitoyda yana davom ettirilmaganga o’xshaydi. [. ] The Qimmat oyna G’arbda noo’rin ravishda “paskal uchburchagi” nomi bilan tanilgan arifmetik uchburchakning diagrammasi bilan ochiladi. (Rasmga qarang.) [. ] Chu uchburchak uchun kreditni rad etib, uni “sakkizinchi va quyi kuchlarni topish uchun eski uslubning diagrammasi” deb ataydi. Oltinchi kuch orqali shunga o’xshash koeffitsientlar tartibi Yang Xuyning ishlarida paydo bo’lgan, ammo dumaloq nol belgisiz. “
^ (Boyer 1991 yil, “Yunon matematikasining tiklanishi va pasayishi” p. 178) Diofantning hayotiga nisbatan noaniqlik shu qadar kattaki, biz uning qaysi asrda yashaganligini aniq bilmaymiz. Umuman olganda u taxminan milodiy 250 yilda gullab-yashnagan deb taxmin qilinadi, ammo ba’zida bir asr yoki undan ko’proq oldinroq yoki keyinroq bo’lgan sanalar ba’zida [. ] Agar bu jumboq tarixiy jihatdan to’g’ri bo’lsa, Diofant sakson to’rt yoshda yashagan. [. ] Bizga ma’lum bo’lgan asosiy Diophantine asari bu Arifmetika, traktat dastlab o’n uchta kitobda mavjud bo’lib, ulardan faqat dastlabki oltitasi saqlanib qolgan. “
^ abvd (Boyer 1991 yil, “Yunon matematikasining tiklanishi va tanazzuli” 180–182-betlar) “” Bu jihatdan uni avvalgi buyuk klassikalar bilan taqqoslash mumkin. Iskandariya yoshi; ammo bu bilan yoki aslida an’anaviy yunon matematikasi bilan deyarli hech qanday o’xshashligi yo’q. Bu asosan yangi filialni ifodalaydi va boshqa yondashuvdan foydalanadi. Geometrik usullardan ajralgan holda, u Bobil algebrasiga juda o’xshaydi. Ammo Bobil matematiklari birinchi navbatda tashvishga tushishgan taxminiy ning echimlari aniqlang tenglamalar uchinchi darajaga qadar Arifmetika Diophantus (bizda bo’lgani kabi) deyarli butunlay bag’ishlangan aniq ikkala tenglamaning echimi aniqlang va noaniq. [. ] Omon qolgan oltita kitob davomida Arifmetika raqamlarning kuchlari va munosabatlar va operatsiyalar uchun qisqartirishlardan muntazam foydalanish mavjud. Noma’lum raqam yunoncha letter harfiga o’xshash belgi bilan ifodalanadi (ehtimol arifmosning oxirgi harfi uchun). [. ] Buning o’rniga 150 ga yaqin muammolarning to’plami mavjud bo’lib, ularning barchasi aniq raqamli misollar asosida ishlab chiqilgan, ammo ehtimol usulning umumiyligi nazarda tutilgan. Postulyatsiyani rivojlantirish mavjud emas va barcha mumkin bo’lgan echimlarni topishga harakat qilinmaydi. Ikki musbat ildizga ega bo’lgan kvadrat tenglamalarda faqat kattaroq kattaroq bo’ladi va salbiy ildizlar tan olinmaydi. Belgilangan va noaniq muammolar o’rtasida aniq farq yo’q, hatto echimlar soni odatda cheksiz bo’lgan ikkinchisi uchun ham faqat bitta javob beriladi. Diofant bir nechta noma’lum sonlar bilan bog’liq muammolarni barcha noma’lum miqdorlarni mohirona ifoda etib, iloji boricha ulardan bittasi nuqtai nazaridan ifodaladi.
^“Diophantus tarjimai holi”. www-history.mcs.st-and.ac.uk . Olingan 2017-12-18 .
^Hermann Hankel “Muallifimiz [Diophantos] bilan umumiy, keng qamrovli uslubning zarracha izi sezilmaydi; har bir muammo ba’zi bir maxsus usullarni talab qiladi, hatto eng yaqin bo’lgan muammolar uchun ham ishlashdan bosh tortadi. Shu sababli zamonaviy olim uchun bu qiyin Diophantosning 100 ta echimini o’rganib chiqqandan keyin ham 101-masalani hal qilish. (Hankel H.,Geschichte der matematik im altertum und mittelalter, Leypsig, 1874, Ulrix Lirechtda ingliz tiliga tarjima qilingan va keltirilgan. XIII asrda Xitoy matematikasi, Dover nashrlari, Nyu-York, 1973.)
^ ab (Boyer 1991 yil, “Yunon matematikasining tiklanishi va pasayishi” p. 178) “Diofantin sinxopatsiyasi va zamonaviy algebraik yozuvlar o’rtasidagi asosiy farq operatsiyalar va munosabatlar uchun maxsus belgilarning, shuningdek eksponensial yozuvlarning etishmasligidadir.”
^ abv (Derbishir 2006 yil, “Algebra otasi” 35-36 betlar)
^ (Kuk 1997 yil, “Rim imperiyasida matematika” 167–168 betlar).
^ (Boyer 1991 yil, “O’rta asrlarda Evropa” p. 257) “Kitobda Diofantda paydo bo’lgan va arablar tomonidan keng qo’llanilgan shaxsiyatlardan [. ] tez-tez foydalaniladi.”
^ (Boyer 1991 yil, “Hindlar matematikasi” p. 197) “Hind matematikasi bo’yicha saqlanib qolgan eng qadimgi hujjatlar miloddan avvalgi I ming yillikning o’rtalarida, taxminan Fales va Pifagoralar yashagan davrda yozilgan asarlarning nusxalari. [. ] Miloddan avvalgi VI asrdan boshlab.”
^ ab (Boyer 1991 yil, “Xitoy va Hindiston” p. 222) “The Livavanti, kabi Vija-Ganita, hindlarning sevimli mavzulari bilan bog’liq ko’plab muammolarni o’z ichiga oladi; aniq va noaniq chiziqli va kvadrat tenglamalar, oddiy mensuratsiya, arifmetik va geometrik progressiyalar, surdlar, Pifagor uchliklari va boshqalar. “
^ (Boyer 1991 yil, “Hindular matematikasi” p. 207) “U musbat tamsayılarning boshlang’ich segmentining kvadratlari va kublari yig’indisi uchun yanada oqlangan qoidalar berdi. Atama sonidan, hadlar sonidan ortiqcha bitta va ikki martadan iborat uchta miqdor hosilasining oltinchi qismi. hadlar soni plyus bitta – kvadratlar yig’indisi. ketma-ket yig’indining kvadrati kublar yig’indisidir. “
^ (Boyer 1991 yil, “Xitoy va Hindiston” p. 219) “Aryabhatadan bir asrdan ko’proq vaqt o’tgach, Markaziy Hindistonda yashagan Braxmagupta (628-rasm) o’zining eng taniqli asari trigonometriyasida [. ] Brahmasphuta Siddhanta, [. ] bu erda biz kvadrat tenglamalarning umumiy echimlarini topamiz, shu jumladan ikkitasi ildizlari ulardan biri salbiy bo’lgan holatlarda ham. “
^ (Boyer 1991 yil, “Xitoy va Hindiston” p. 220) “Hind algebra, noaniq tahlilni ishlab chiqishda alohida e’tiborga loyiqdir, bunda Braxmagupta bir necha hissa qo’shgan. Birinchidan, uning ishida m, 1/2 (m) shaklida ifodalangan Pifagor uchliklarini shakllantirish qoidasini topamiz. 2 / n – n), 1/2 (m 2 / n + n); ammo bu faqat eski Bobil hukmronligining o’zgartirilgan shakli bo’lib, u u bilan tanish bo’lishi mumkin edi. “
^ abvd (Boyer 1991 yil, “Xitoy va Hindiston” p. 221) “u a bergan birinchi kishi edi umumiy a + b = c chiziqli Diofant tenglamasining yechimi, bu erda a, b va c butun sonlardir. [. ] Bu uning bergan Braxmagupaning obro’si uchun juda muhimdir barchasi chiziqli Diofant tenglamasining ajralmas echimlari, Diofantning o’zi esa noaniq tenglamaning bitta echimini berishdan mamnun edi. Braxmagupta Diofant singari ba’zi bir misollardan foydalanganligi sababli, biz yana yunonlarning Hindistonga ta’sir qilish ehtimoli yoki ularning ikkalasi ham umumiy manbadan, ehtimol Bobildan foydalanganlik ehtimolini yana bir bor ko’ramiz. Diophantus singari Braxmagupta algebrasi hamohang bo’lganligi ham qiziq. Qo’shish, subtrahend ustiga nuqta qo’yish orqali ajratish, ajratish va bo’linishni dividend ostiga qo’yish orqali bo’linish, bizning kasrli yozuvimizda bo’lgani kabi, ammo barsiz. Ko’paytirish va evolyutsiya (ildizlarni olish) operatsiyalari, shuningdek noma’lum miqdorlar tegishli so’zlarning qisqartirishlari bilan ifodalangan. [. ] XII asrning etakchi matematikasi Bxaskara (1114 – taxminan 1185). Aynan u Pell tenglamasining umumiy echimini berish va nolga bo’linish masalasini ko’rib chiqish kabi Brahmagupta asarlaridagi ayrim bo’shliqlarni to’ldirgan “.
^ ab (Boyer 1991 yil, “Xitoy va Hindiston” 222–223 betlar.) “Doira va sharni davolashda Lilavati aniq va taxminiy gaplarni farqlay olmaydi. [. ] Bhaskaraning ko’plab muammolari Livavati va Vija-Ganita shubhasiz, avvalgi hindu manbalaridan olingan; shuning uchun muallif noaniq tahlil bilan eng yaxshi darajada shug’ullanishini ta’kidlash ajablanarli emas “.
^ abv (Boyer 1991 yil, “Arabcha gegemonlik” p. 227) “Musulmon imperiyasining birinchi asrida ilmiy yutuqlardan mahrum bo’lgan edi. Bu davr (taxminan 650 yildan 750 yilgacha), aslida, ehtimol matematikaning rivojlanishida nodir bo’lgan, chunki arablar hali intellektual intilishlariga erishmagan edilar. va dunyoning boshqa qismlarida o’rganish haqida qayg’urish susaygan edi. Agar sakkizinchi asrning ikkinchi yarmida Islomda kutilmaganda madaniy uyg’onish bo’lmaganida edi, qadimiy ilm-fan va matematikaning aksariyati yo’qolgan bo’lar edi. [. ] Aynan al-Mamun xalifaligi davrida (809–833) arablar o’zlarining tarjimaga bo’lgan ishtiyoqlarini to’liq qondirishgan edi.Xalifaning tushida Aristotel paydo bo’lganligi va natijada al-Mamun qaror qilgani aytilgan. u qo’l qo’yishi mumkin bo’lgan barcha yunon asarlaridan, shu jumladan Ptolomeydan arabcha nusxalarini yaratish Almagest va Evklidning to’liq versiyasi Elementlar. Arablar bezovta tinchlikni saqlagan Vizantiya imperiyasidan yunon qo’lyozmalari tinchlik shartnomalari orqali olingan. Al-Mamun Bag’dodda qadimgi Iskandariyadagi muzey bilan taqqoslanadigan “Donolik uyi” ni (Baytul-hikma) tashkil etdi. Fakultet a’zolari orasida matematik va astronom Muhammad ibn Muso al-Xorazmiy ham bor edi, uning nomi Evklid singari keyinchalik G’arbiy Evropada xalq so’ziga aylanishi kerak edi. 850 yilga qadar vafot etgan olim, o’nlab astronomik va matematik asarlarni yozgan, ularning eng qadimgi asarlari, ehtimol Sindxod Hindistondan olingan “.
^ ab (Boyer 1991 yil, “Arabcha gegemonlik” p. 234) “ammo al-Xorazmiyning asari jiddiy nuqsonga ega edi, uni zamonaviy dunyoda o’z maqsadiga samarali xizmat qilishidan oldin uni olib tashlash kerak edi: ritorik shakl o’rnini bosadigan ramziy belgi ishlab chiqish kerak edi. Arablar bu qadamni hech qachon boshlamadilar, faqat raqamli so’zlarni raqam belgilariga almashtirish uchun. [. ] Sobit tarjimonlar maktabining asoschisi edi, ayniqsa yunon va suriy tillaridan va biz unga Evklid, Arximed, asarlarining arab tiliga tarjimalari uchun juda katta qarzdormiz. Apollonius, Ptolomey va Evtosius ».
^ ab Gandz va Saloman (1936), Al-Xorazmiy algebra manbalari, Osiris i, p. 263–277: “Xorazmiy ma’lum ma’noda Diofantga qaraganda” algebra otasi “deb nomlanishga ko’proq haqlidir, chunki Xorazmiy birinchi bo’lib algebrani elementar shaklda o’rgatgan va o’zi uchun Diofant birinchi navbatda nazariya bilan shug’ullanadi. raqamlar “.
^ ab (Boyer 1991 yil, “Arabcha gegemonlik” p. 230) “Al-Xorazmiy so’zlarini davom ettirdi:” Biz oltita turdagi tenglamalar haqida raqamlarga kelsak, etarli darajada aytdik. Ammo, endi biz raqamlar bilan izohlagan bir xil muammolarning haqiqatini geometrik tarzda namoyish etishimiz zarur. “Ushbu parchaning halqasi Bobil yoki Hindistonga qaraganda yunoncha ekanligi aniq. Shuning uchun uchta asosiy fikr maktabi mavjud arab algebrasining kelib chiqishi to’g’risida: biri hindlarning ta’siriga urg’u beradi, boshqasi Mesopotamiya yoki suriya-fors an’analariga urg’u beradi, uchinchisi yunon ilhomiga ishora qiladi. Haqiqat, agar uchta nazariyani birlashtirsak, yaqinlashadi. “
^ (Boyer 1991 yil, “Arabcha gegemonlik” 228–229 betlar) “muallifning arab tilidagi muqaddimasi Muhammad payg’ambarga va al-Ma’munga” Mo’minlar qo’mondoni “ni behuda maqtashdi.”
^ (Boyer 1991 yil, “Arabcha gegemonlik” p. 228) “Arablar umuman olganda dastlabki xulosadan yaxshi aniq dalillarni va sistematik tashkilotni yaxshi ko’rar edilar – bu erda na Diofant va na hindular ustunlik qildilar”.
^ ab (Boyer 1991 yil, “Arabcha gegemonlik” p. 229) “Qanday shartlarda ekanligi aniq emas al-jabr va muqobala degan ma’noni anglatadi, ammo odatdagi talqin yuqoridagi tarjimada nazarda tutilganga o’xshashdir. So’z al-jabr “tiklanish” yoki “tugatish” kabi bir narsani anglatishi mumkin va bu olib tashlangan atamalarni tenglamaning boshqa tomoniga ko’chirishni nazarda tutadi, bu traktatda aniq ko’rinadi; so’z muqobala “qisqartirish” yoki “muvozanatlash” degan ma’noni anglatadi, ya’ni tenglamaning qarama-qarshi tomonlarida o’xshash atamalarni bekor qilish. “
^ ab Rashed, R .; Armstrong, Anjela (1994), Arab matematikasining rivojlanishi, Springer, 11-2 betlar, ISBN978-0-7923-2565-9 , OCLC29181926
^ ab (Boyer 1991 yil, “Arabcha gegemonlik” p. 229) “oltita qisqa bobda, uchta turdagi miqdorlardan tashkil topgan oltita tenglamadan: ildizlar, kvadratlar va sonlar (ya’ni x, x 2 va raqamlar). I bob, uchta qisqa xatboshida, zamonaviy yozuvlarda x shaklida ifodalangan, ildizlarga teng kvadratlarning holatini o’z ichiga oladi 2 = 5x, x 2 / 3 = 4x va 5x 2 = 10x, mos ravishda x = 5, x = 12 va x = 2 javoblarni beradi. (X = 0 ildizi tan olinmadi.) II bob sonlarga teng kvadratlarni, III bob esa ildizlarning sonlarga teng holatlarini echadi, yana bobda uchta rasm bilan koeffitsient koeffitsienti qoplanadi. o’zgaruvchan atama birga teng, ko’p yoki kichik. IV, V va VI boblar qiziqroq, chunki ular o’z navbatida uch davrli kvadrat tenglamalarning uchta klassik holatini o’z ichiga oladi: (1) kvadratlarga va ildizlarga sonlarga teng, (2) kvadratlarga va ildizlarga teng sonlarga va (3) ) kvadratlarga teng ildizlar va sonlar. “
^ (Boyer 1991 yil, “Arabcha gegemonlik” 229–230-betlar. “” Bu echimlar “kvadratni to’ldirish” uchun “oshpazlar kitobi” qoidalari bo’lib, muayyan holatlarga nisbatan qo’llaniladi. [. ] Har holda, faqat ijobiy javob beriladi. [. ] ] Yana bittasi uchun faqat bitta ildiz berilgan, ikkinchisi uchun manfiy. [. ] Yuqorida keltirilgan oltita tenglama, ijobiy ildizlarga ega bo’lgan chiziqli va kvadrat tenglamalarning barcha imkoniyatlarini tugatdi. “
^ (Boyer 1991 yil, “Arabcha gegemonlik” p. 230) “Al-Xorazmiy bu erda biz diskriminant deb belgilagan narsamiz ijobiy bo’lishi kerakligiga e’tibor qaratadi:” Siz shuni ham tushunishingiz kerakki, ildizlarning yarmini tenglamaning bu shaklida olib, so’ngra yarmini o’z-o’zidan ko’paytiring. ; agar ko’paytma natijasida hosil bo’ladigan yoki kvadratga hamroh bo’ladigan yuqorida ko’rsatilgan birliklardan kam bo’lsa, sizda tenglama mavjud. “[. ] Yana bir marta kvadratni to’ldirish bosqichlari puxta ko’rsatiladi, asossiz”
^ (Boyer 1991 yil, “Arabcha gegemonlik” p. 231) “The Algebra al-Xorazmiyning shubhasiz ellin elementlariga xiyonat qilishi “
^ (Boyer 1991 yil, “Arabcha gegemonlik” p. 233) “Al-Xorazmiyning bir nechta muammolari arablarning Bobil-Heron matematik oqimiga bog’liqligini aniq ko’rsatib beradi. Ulardan biri, ehtimol, to’g’ridan-to’g’ri Herondan olingan, chunki shakl va o’lchovlar bir xil”.
^ ab (Boyer 1991 yil, “Arabcha gegemonlik” p. 228) “al-Xorazmiy algebrasi puxta ritorikdir, biron bir sinxronizatsiya yunon tilida topilmagan Arifmetika yoki Brahmagupta asarida. Hatto raqamlar ramzlar o’rniga so’zlar bilan yozilgan! Diofantning asarini al-Xorazmiy bilishi ehtimoldan yiroq emas, lekin u hech bo’lmaganda Braxmaguptaning astronomik va hisoblash qismlarini yaxshi bilgan bo’lishi kerak; hali na al-Xorazmiy va na boshqa arabshunoslar sinxronizatsiya yoki manfiy sonlardan foydalanmadilar. “
^ abvd (Boyer 1991 yil, “Arabcha gegemonlik” p. 234) “The Algebra al-Xorazmiyning asari odatda ushbu mavzu bo’yicha birinchi asar sifatida qaraladi, ammo yaqinda Turkiyada nashr etilgan nashr bu borada ba’zi savollarni tug’dirdi. Abd-al-Hamid ibn-Turkning “Aralashtirilgan tenglamalardagi mantiqiy zaruriyatlar” deb nomlangan asarining qo’lyozmasi kitobning bir qismi edi. Al-jabr val muqobala shubhasiz al-Xorazmiy bilan deyarli bir xil bo’lgan va bir vaqtning o’zida nashr etilgan, ehtimol undan ham oldinroq. “Mantiqiy zaruriyatlar” ning saqlanib qolgan boblari aynan al-Xorazmiy kabi geometrik namoyishlarning bir xil turini beradi. Algebra va bitta holatda xuddi shu misolli misol x 2 + 21 = 10x. Abd-Hamadning ekspozitsiyasi bir jihatdan al-Xorazmiynikiga qaraganda ancha puxta, chunki agar u diskriminant manfiy bo’lsa, kvadrat tenglamada echim yo’qligini isbotlash uchun geometrik raqamlarni keltirib chiqaradi. Ikki kishining asarlaridagi o’xshashliklar va ulardagi tizimli tashkilot algebra ularning davridagi odatdagidek rivojlanish emasligini ko’rsatmoqda. An’anaviy va yaxshi buyurtma qilingan ekspozitsiyaga ega darsliklar bir vaqtning o’zida paydo bo’lganda, mavzu shakllanish bosqichidan ancha kattaroq bo’lishi mumkin. [. ] Diofant va Pappuslarning e’tiborsizligiga e’tibor bering, ular dastlab Arabistonda taniqli bo’lmagan mualliflar, garchi Diofantin Arifmetika X asr oxiridan oldin tanish bo’lgan “.
^ ab (Derbishir 2006 yil, “Algebra otasi” p. 49)
^O’Konnor, Jon J.; Robertson, Edmund F., “Arab matematikasi: unutilgan yorqinlikmi?”, MacTutor Matematika tarixi arxivi, Sent-Endryus universiteti
. “Algebra birlashtiruvchi nazariya bo’lib, u ratsional sonlar, irratsional sonlar, geometrik kattaliklar va boshqalarga” algebraik ob’ektlar “sifatida qarashga imkon berdi.”
^ Jak Sesiano, “Islom matematikasi”, p. 148, yilda Selin, Xeleyn; D’Ambrosio, Ubiratan, tahrir. (2000), Madaniyatlar bo’ylab matematika: g’arbiy matematikaning tarixi, Springer, ISBN978-1-4020-0260-1
^ ab Berggren, J. Lennart (2007). “O’rta asr islomida matematika”. Misr, Mesopotamiya, Xitoy, Hindiston va Islom matematikasi: Manba kitobi. Prinston universiteti matbuoti. p. 518. ISBN978-0-691-11485-9 .
^ ab (Boyer 1991 yil, “Arabcha gegemonlik” p. 239) “Abu’l Vefa trionometr bilan bir qatorda qobiliyatli algebraist edi. [. ] Uning vorisi al-Karxi, shubhasiz, ushbu tarjimadan Diofantusning arab shogirdi bo’lish uchun foydalangan, ammo Diofantin tahlilisiz! [. ] In Xususan, al-Karxiga ax shaklidagi tenglamalarning birinchi sonli echimi berilgan 2n + bx n = c (faqat ijobiy ildizlarga ega tenglamalar ko’rib chiqildi), “
^O’Konnor, Jon J.; Robertson, Edmund F., “Abu Bekr ibn Muhammad ibn al-Husayn Al-Karaji”, MacTutor Matematika tarixi arxivi, Sent-Endryus universiteti
.
^ abvde (Boyer 1991 yil, “Arabcha gegemonlik” 241–242-betlar) “Omar Xayyom (taxminan 1050 – 1123),” chodir quruvchi “, Algebra uchinchi darajali tenglamalarni o’z ichiga olgan al-Xorazmiydan tashqarida. Arab Xayyom ham o’zidan oldingi arablar kabi kvadratik tenglamalarni ham arifmetik, ham geometrik echimlarni taqdim etgan; umumiy kubik tenglamalar uchun u ishondi (yanglishib, XVI asr keyinroq ko’rsatganidek), arifmetik echimlar mumkin emas; shuning uchun u faqat geometrik echimlarni berdi. Kublarni echish uchun kesishgan koniklardan foydalanish sxemasi ilgari Menaxmus, Arximed va Alhazan tomonidan qo’llanilgan, ammo Omar Xayyom barcha uchinchi darajali tenglamalarni (ijobiy ildizlarga ega) qamrab olish usulini umumlashtirishning maqtovli qadamini qo’ydi. .. Uch darajadan yuqori darajadagi tenglamalar uchun, Umar Xayyom, shubhasiz, o’xshash geometrik usullarni o’ylamagan edi, chunki kosmik uch o’lchovdan ko’proq narsani o’z ichiga olmaydi, [. ] Arab eklektizmining eng samarali hissalaridan biri bu yopilish tendentsiyasi edi. raqamli va geometrik algebra orasidagi bo’shliq. Bu yo’nalishdagi qat’iyatli qadam Dekart bilan ancha o’tib ketdi, ammo Umar Xayyom shunday deb yozgan edi: “Kimki algebrani noma’lum narsalarni olishda hiyla deb o’ylasa, uni behuda deb o’ylagan. Algebra ekanligiga e’tibor bermaslik kerak va geometriya tashqi ko’rinishiga ko’ra har xil. Algebralar bu isbotlangan geometrik faktlar. “
^O’Konnor, Jon J.; Robertson, Edmund F., “Sharafuddin al-Muzaffar at-Tusi”, MacTutor Matematika tarixi arxivi, Sent-Endryus universiteti
.
^ Rashed, Roshdi; Armstrong, Anjela (1994), Arab matematikasining rivojlanishi, Springer, 342-3-betlar, ISBN978-0-7923-2565-9
^ Berggren, J. L. (1990), “Sharafiddin at-Tusiyning” Muadalat “dagi yangilik va an’ana”, Amerika Sharq Jamiyati jurnali, 110 (2): 304–9, doi:10.2307/604533, JSTOR604533, Rashedning ta’kidlashicha, Sharafuddin kubik polinomlarning hosilasini kashf etgan va uning kubik tenglamalari echilishi mumkin bo’lgan sharoitlarni o’rganish uchun uning ahamiyatini anglagan; ammo, boshqa olimlar Sharfiddinning fikrini Evklid yoki Arximedda topilgan matematikaga bog’laydigan juda farqli tushuntirishlarni taklif qilishgan.
^ Viktor J. Kats, Bill Barton (2007 yil oktyabr), “Algebra tarixining bosqichlari o’qitishga taalluqli”, Matematikadan o’quv ishlari, 66 (2): 185–201 [192], doi:10.1007 / s10649-006-9023-7, S2CID120363574
^ Tjalling J. Ypma (1995), “Nyuton-Rafson uslubining tarixiy rivojlanishi”, SIAM sharhi37 (4): 531–51, doi:10.1137/1037125
^ abvO’Konnor, Jon J.; Robertson, Edmund F., “Abu’l Hasan ibn Ali al-Qalasadiy”, MacTutor Matematika tarixi arxivi, Sent-Endryus universiteti
.
^ (Boyer 1991 yil, “Iskandariya Evklidi. 192-193-betlar)” Boetsiyning o’limi G’arbiy Rim imperiyasida qadimiy matematikaning oxiri bo’lganligi uchun qabul qilinishi mumkin, chunki Gipatiyaning o’limi Iskandariyaning matematik markazi sifatida yopilishini belgilab qo’ygan edi; Ammo Afinada ish bir necha yil davom etdi. [. ] 527 yilda Yustinian Sharqda imperator bo’lganida, u Afinadagi Akademiya va boshqa falsafiy maktablarni butparastlik bilan o’rganish pravoslav nasroniylik uchun tahdid deb o’ylagan edi; shuning uchun 529 yilda falsafiy maktablar yopilib, olimlar tarqalib ketishdi. O’sha paytda Rim olimlar uchun juda mehmondo’st uy bo’lgan va Simplicius va boshqa ba’zi faylasuflar Sharqdan boshpana qidirganlar. Buni ular Forsda topdilar, u erda shoh Xsrous boshchiligida “Suriyadagi Afina akademiyasi” deb nomlanishi mumkin edi (Sarton 1952; 400-bet). “
^ Masalan, Bashmakova va Smirnova (2000 yil:78) harvcoltxt xatosi: maqsad yo’q: CITEREFBashmakovaSmirnova2000 (Yordam bering) , Boyer (1991 yil):180), Berton (1995):319), Derbishir (2006):93), Katz va Parshall (2014):238), Sesiano (1999 yil): 125) va Shvets (2013 yil.):110)
^Dekart (1637:301–303)
^Dekart (1925):9–14)
^Kajori (1919:698); Kajori (1928:381–382)
^Enestrom (1905):317)
^ Masalan, Tropfke (1902): 150). Ammo Gustaf Enestrom (1905: 316-317) Dekart 1619 yilda yozgan maktubida nemis belgisini o’ziga xos farqli o’laroq ishlatganligini ko’rsatdi. x.
^ Kesilgan raqam 1 tomonidan ishlatilgan Pietro Cataldi noma’lum birinchi kuch uchun. Ushbu konventsiya va x Cajori tomonidan berilgan Gustav Vertxaym, ammo Kajori (1919: 699; 1928: 382) buni tasdiqlovchi dalil topmadi.
^Kajori (1919:699)
^ Masalan, ga qarang TED nutqi Terri Mur tomonidan yozilgan “Nima uchun” x “noma’lum?”, 2012 yilda chiqarilgan.
^Alkala (1505)
^Lagard (1884).
^Yoqub (1903):519).
^Chavandoz (1982) XVI asrda algebra bo’yicha ispan tilida nashr etilgan beshta risolaning ro’yxati keltirilgan bo’lib, ularning barchasida “cosa” ishlatilgan: Aurel (1552), Ortega (1552), Diyez (1556), Peres de Moya (1562) va Nunes (1567). Oxirgi ikkita asar ham qisqartiradi kosa kabi “ko.“- shunday qiladi Puig (1672).
^ Shakllar mavjud emas Alonso (1986), Kasten va Kodi (2001), Oelschläger (1940), Ispaniya Qirollik akademiyasi Ispaniyaning onlayn diaxronik korpusi (KORDE ) va Devis “s Corpus del Español.
^“Nega x?” . Olingan 2019-05-30 .
^ Struik (1969), 367
^ Endryu Uorvik (2003) Nazariya magistrlari: Kembrij va matematik fizikaning yuksalishi, Chikago: Chikago universiteti matbuoti
ISBN 0-226-87374-9
^ abvd (Boyer 1991 yil, “Arabcha gegemonlik” p. 228) “Diophantus ba’zan” algebra otasi “deb nomlanadi, ammo bu unvon ko’proq mos ravishda Abu Abdulloh bin mirsmi al-Xorazmiyga tegishli. To’g’ri, al-Xorazmiyning ishi ikki jihatdan Diofantnikidan orqaga chekinishni anglatadi. Birinchidan, bu Diofantin muammolarida topilganidan ancha oddiy darajada, ikkinchidan, al-Xorazmiy algebrasi puxta ritorikdir, yunon tilida sinxoplashning birortasi ham mavjud emas Arifmetika yoki Brahmagupta asarida. Hatto raqamlar ramzlar o’rniga so’zlar bilan yozilgan! Diofantning asarini al-Xorazmiy bilishi ehtimoldan yiroq emas, lekin u hech bo’lmaganda Braxmaguptaning astronomik va hisoblash qismlarini yaxshi bilgan bo’lishi kerak; hali na al-Xorazmiy va na boshqa arabshunoslar sinxronizatsiya yoki manfiy sonlardan foydalanmadilar. “
^ Xerskovik, Nikolas; Linchevski, Liora (1994 yil 1-iyul). “Arifmetik va algebra o’rtasidagi kognitiv bo’shliq”. Matematikadan o’quv ishlari. 27 (1): 59–78. doi:10.1007 / BF01284528. ISSN1573-0816. S2CID119624121. Bu algebraning otasi deb hisoblangan al-Xorazmiy uchun ajablanib bo’lishi mumkin edi (Boyer / Merzbax, 1991), uni IX asr atrofida O’rta er dengizi bilan tanishtirdi.
^ Dodge, Yadolah (2008). Statistikaning qisqacha ensiklopediyasi . Springer Science & Business Media. p.1. ISBN9780387317427 . Algoritm atamasi Bag’dodda yashagan va algebra otasi bo’lgan IX asr matematiki al-Xorazmiy ismining lotincha talaffuzidan kelib chiqadi.
^ (Derbishir 2006 yil, “Algebra otasi” p. 31) “Van der Vaerden matematik al-Xorazmiydan boshlab algebra ota-onasini keyinchalik bir nuqtaga suradi”
^ (Derbishir 2006 yil, “Algebra otasi” p. 31) “Diofant, algebra otasi, uning sharafiga men ushbu bobni nomlaganman. Milodning 1, 2 yoki 3 asrlarida Iskandariyada, Rim Misrida yashagan”.
^ J. Sesiano, K. Vogel, “Diophantus”, Ilmiy biografiya lug’ati (Nyu-York, 1970-1990), “Diofantus, ko’pincha uni algebra otasi deb atashmagan”.
^ (Derbishir 2006 yil, “Algebra otasi” p. 31) “Kurt Vogel, masalan, Ilmiy biografiya lug’ati, Diophantausning ishini eski bobilliklarnikidan ancha algebraik deb biladi “
^ (Boyer 1991 yil, “Arabcha gegemonlik” p. 230) “Yuqorida keltirilgan oltita tenglama musbat ildizga ega bo’lgan chiziqli va kvadrat tenglamalar uchun barcha imkoniyatlarni tugatadi. Al-Xorazmiyning ekspozitsiyasi shu qadar tizimli va to’liq bo’lganki, uning o’quvchilari echimlarni o’zlashtirishda ozgina qiyinchilikka duch kelishgan.”
^Kats, Viktor J. (2006). “ALGEBRA TARIXINING O’QITISH UChUN KO’RSATILGAN BOShQALARI” (PDF) . VICTOR J.KATZ, Kolumbiya okrugi universiteti, Vashington, AQSh, AQSh: 190. Arxivlangan asl nusxasi (PDF) 2019-03-27 da . Olingan 2019-08-06 – Vashington shtatidagi Kolumbiya okrugi universiteti orqali. Hozirgacha mavjud bo’lgan birinchi haqiqiy algebra matni – bu Muhammad ibn Muso al-Xorazmiyning 825 yil atrofida Bag’dodda yozgan al-jabr va al-muqobala asaridir.

Adabiyotlar

Alkala, Pedro de (1505), De lingua arabica, Granada (nashr Pol de Lagarde, Göttingen: Arnold Xoyer, 1883) CS1 tarmog’i: joylashuvi (havola)
Alonso, Martin (1986), Diccionario del español o’rta asrlar, Salamanka: Universidad Pontificia de Salamanca
Aurel, Marko (1552), Libro primero de arithmetica algebratica, Valensiya: Joan de Mey
Bashmakova, men va Smirnova, G. (2000) Algebraning boshlanishi va evolyutsiyasi, Dolciani matematik ekspozitsiyalari 23. Abe Shenitsits tarjima qilgan. Amerika matematik assotsiatsiyasi.
Boyer, Karl B. (1991), Matematika tarixi (Ikkinchi tahr.), John Wiley & Sons, Inc., ISBN978-0-471-54397-8
Berton, Devid M. (1995), Bertonning Matematika tarixi: Kirish (3-nashr), Dubuque: Wm. C. Jigarrang
Berton, Devid M. (1997), Matematika tarixi: kirish (Uchinchi nashr), McGraw-Hill Companies, Inc., ISBN978-0-07-009465-9
Kajori, Florian (1919), “Qanday qilib x noma’lum miqdorni anglatadi”, Maktab fanlari va matematika, 19 (8): 698–699, doi:10.1111 / j.1949-8594.1919.tb07713.x
Kajori, Florian (1928), Matematik yozuvlar tarixi, Chikago: Ochiq sud nashriyoti, ISBN9780486161167
Kuk, Rojer (1997), Matematika tarixi: qisqacha dars, Wiley-Interscience, ISBN978-0-471-18082-1
Derbishir, Jon (2006), Noma’lum miqdor: haqiqiy va xayoliy algebra tarixi, Vashington, DC: Jozef Genri Press, ISBN978-0-309-09657-7
Dekart, Rene (1637), La Géémetrie, Leyde: Yan Maire. Onlayn 2008 yil ed. L. Xermann tomonidan, Gutenberg loyihasi.
Dekart, Rene (1925), Rene Dekart geometriyasi, Chikago: Ochiq sud, ISBN9781602066922
Díez, Xuan (1556), Sumario compendioso de las quentas de plata y oro que en los reynos del Piru o’g’li zarur bo’lgan los mercaderes: y todo genero de tratantes, con algunas reglas tocantes al arithmetica, Mexiko
Enestrom, Gustaf (1905), “Kleine Mitteilungen”, Matematikaning bibliotekasi, Ser. 3, 6 (faqat AQShda onlayn kirish)
Flegg, Grem (1983), Raqamlar: ularning tarixi va ma’nosi, Dover nashrlari, ISBN978-0-486-42165-0
Xit, Tomas Little (1981a), Yunon matematikasi tarixi, I tom, Dover nashrlari, ISBN978-0-486-24073-2
Xit, Tomas Little (1981b), Yunon matematikasi tarixi, II jild, Dover nashrlari, ISBN978-0-486-24074-9
Jeykob, Georg (1903), “Voqidadagi Sharq madaniyatining elementlari”, Smitson institutining Regentslar Kengashining yillik hisoboti [. ] 1902 yil 30-iyunda tugagan yil uchun: 509–529
Kasten, Lloyd A.; Cody, Florian J. (2001), O’rta asr ispan tilining taxminiy lug’ati (2-nashr), Nyu-York: O’rta asrlarni o’rganish Ispan seminariyasi
Kats, Viktor J.; Parshall, Karen ochlik (2014), Noma’lumni tamirlash: algebra tarixi antik davrdan yigirmanchi asrning boshlariga qadar, Princeton, NJ: Princeton University Press, ISBN978-1-400-85052-5
Lagard, Pol de (1884), “Woher stammt das x der Mathematiker?”, Mittheilungen, 1, Gyettingen: Dieterichsche Sortimentsbuchhandlung, 134-137 betlar.
Nunes, Pedro (1567), Libro de algebra en arithmetica y geometria, Antverpen: Arnoldo Birkman
Oelschläger, Viktor R. B. (1940), O’rta asr ispancha so’zlar ro’yxati, Madison: Viskonsin universiteti matbuoti
Ortega, Xuan de (1552), Tractado subtilissimo de arismetica y geometriya, Granada: Rene Rabut
Peres de Moya, Xuan (1562), Aritmética práctica y especulativa, Salamanka: Mathias Gast
Puig, Andres (1672), Arithmetica especulativa y Practica; y arte de algebra, Barcelona: Antonio Lacavalleria
Rider, Robin E. (1982), Dastlabki zamonaviy algebra bibliografiyasi, 1500-1800, Berkeley: Berkli tadqiqotlari tarixidagi hujjatlar
Sesiano, Jak (1999), Algebra tarixiga kirish: Mesopotamiya Times dan Uyg’onish davriga tenglamalarni echish, Providence, RI: Amerika Matematik Jamiyati, ISBN9780821844731
Stilluell, Jon (2004), Matematika va uning tarixi (Ikkinchi nashr), Springer Science + Business Media Inc., ISBN978-0-387-95336-6
Svets, Frank J. (2013), Evropaning matematik uyg’onishi: Matematikaning tarixi orqali sayohat, 1000-1800 (2-nashr), Mineola, NY: Dover Publications, ISBN9780486498058
Tropfke, Yoxannes (1902), Geschichte der Elementar-Mathematik in systematischer Darstellung, 1, Leypsig: Von Veit va Comp.

Tashqi havolalar

“Islom shayxi Zakariyo al-Ansoriyning Ibn al-Haimning” Algebra va muvozanat ilmi haqidagi she’ri “ga sharh” Kojentni tushuntirishda Yaratuvchining epifani deb nomlangan ” dan boshlab XV asrda paydo bo’lgan algebra haqidagi asosiy tushunchalarni o’z ichiga olgan Jahon raqamli kutubxonasi.

Mavhum algebra
Kategoriya nazariyasi
Boshlang’ich algebra
K-nazariyasi
Kommutativ algebra
Kommutativ bo’lmagan algebra
Buyurtmalar nazariyasi
Umumjahon algebra

Guruh (nazariya )
Qo’ng’iroq (nazariya )
Modul (nazariya )
Maydon
Polinom halqasi (Polinom )
Tarkibi algebra

Matritsa (nazariya)
Vektor maydoni (Vektor )
Modul
Ichki mahsulot maydoni (nuqta mahsuloti )
Hilbert maydoni

Tensor algebra
Tashqi algebra
Nosimmetrik algebra
Geometrik algebra (Multivektor )

Mavhum algebra
Algebraik tuzilmalar
Guruh nazariyasi
Lineer algebra

Lineer algebra
Maydon nazariyasi
Ring nazariyasi
Buyurtmalar nazariyasi

Qiziqarli malumotlar
Algebra tarixi – History of algebra