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Download e-book for iPad: Abu Kamil Algèbre et analyse diophantienne. Édition, by Roshdi Rashed

By Roshdi Rashed

ISBN-10: 3110295679

ISBN-13: 9783110295672

The mathematical works of Ab? K?mil (floruit circa 880) have been produced generations after the works of Al-Khwarizm?, the founding father of algebra. They unfolded fields of analysis that proved fertile up till the 17th century, and have been quickly to turn into either a reference and a version. Their impression used to be decisive at the improvement of algebra in Arabic a minimum of in Latin and Hebrew. there'll be present in the current book the 1st conscientiously severe version of Ab? K?mil s works, in addition to the 1st ever translation right into a glossy language. textual content and translation are preceded through an exhaustive remark, without delay mathematical and historical."

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Additional info for Abu Kamil Algèbre et analyse diophantienne. Édition, traduction et commentaire

Example text

Alors V5ÔÔÔÔ - 200-f-x' = lOx jc = 5±V 225-V 5Ô æO . Abü Kâmil pouvait sûrement vérifier cette égalité au moyen de la théorie du livre X des Éléments d’Euclide. Autre méthode ; On pose >' = 10-X Si X est l’une des deux racines, (10 - x) est l’autre. yx = 10 - X Autre méthode : On pose u = —- et v = —- — X 10-X + V= V5 ^ 10 - X = V5- En multipliant la seconde équation par (10 - x), puis en y remplaçant yx par sa valeur (10 - x), on obtient [mv = 1 fyx = 1 0 - X Alors V5m- [x + 10y = (1 0 -x ) + V M - x 4 5 =1 La deuxième équation donne alors une valeur de y linéaire en x.

Mais AH = EH = OM. D’autre part, on a MI = NI, d’où MI • OM = KD ■ND, d’où Algorithme pour déterminer le carré de l ’inconnue : aire {MS) + aire {EK) = aire (7D) + aire {EK) = aire {ED) - c. b -> b ^ -^b ^c , Il est aussi évident que HB - EB = EH = lE - MI - ME, d’où b ^ y ^ 7 , b ^ Ÿ ,2 ,2 — + b^c-^ Jb c + \ ^J —^ 2 J + b^c. + C+ 2 \ ^J Dans le cas général, pour a ^ 1, on divise par a pour normaliser l’équation. aire {HM) = HE^ ~ I 2 ' On a alors aire {HK) = aire {ED) + aire {HM) = c + 1- 1 , Démonstration de l ’algorithme : Démonstration 1 : Soit l’aire carrée (ABCD) = et l’aire rectangulaire {AN) = bx {et Fig.

PROBLÈMES DIVERS Alors 10 = 5x et X = 2 ; - 7 - On considère le système (type ax = b). flO = X + y | y ^ - x ^ = 8 0 - 4 - On considère le système JlO = X + y l9y = x^ Alors 100 - 20x = 80, d’où x = 1. Abu Kâmil calcule aussi l’équation en y : 2 0 y - 100 = 80. 58 Chapitre II Le calcul algébrique et ses applications 59 Il indique enfin une autre méthode X X-5-U Il ajoute que cette méthode n’est pas connue des algébristes et qu’elle est particulièrement utile pour résoudre les équations quadratiques trinômes (elle permet de les réduire à des équations binômes par élimination du terme de degré 1 en x).

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Abu Kamil Algèbre et analyse diophantienne. Édition, traduction et commentaire by Roshdi Rashed

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