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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 develop into either a reference and a version. Their impact used to be decisive at the improvement of algebra in Arabic at the least in Latin and Hebrew. there'll be present in the current ebook 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 via an exhaustive observation, immediately mathematical and historical."

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

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X46 •x 4 s + IOa:^ + 20 = jc\ Changement de variable y = x^. ^|^ x^ +100 = 40x x =2o± ^^m . Abü Kâmil ne donne que la racine 20 - V300 qui est bien la solution du problème (en effet l’autre racine de l’équation quadratique ne vérifie pas la condition 10 - 2 a: > 0). Ce problème n’est pas anodin, car à partir du problème suivant, Abü Kâmil considère des équations à coefficients irrationnels. •^]5ÿ + lOy + 20 = y^ (lO + V3Ô)y + 20 = y^ y = 5 + V ^ + ^(5 + V ^ y +20 = 5 + V ^ + ■^/52,5 + ^/750• - 41 - On considère l’équation ^Ü£zf) = V6.

7 V 7 7-7 7 7 V 7. 1 -- 1 -2 ) =JC^+24. L’unique racine positive est 4. Première méthode : On pose y = tique en y, par l’algorithme. il faut résoudre l’équation quadra­ Deuxième méthode : On pose ^ - y ' ^ - 24. Alors = /• Autre méthode : On pose y^ = x^ - 3x, puis on élimine jc. En remplaçant le radical par y dans l’équation initiale, on obtient une équation linéaire en x et y, qui donne immédiatement 3x = 20 - Ay. On déduit de cette dernière équation la valeur de x^ en fonction de y : Ces valeurs de 3x et de x'^, reportées dans l’équation y'^ = x'^ - 3x, donnent finalement : Abu Kâmil dit que r - 2 4 + i + /(^l + | j - f l 3 + ^ V équation dont la seule racine positive est 7 = 6 .

Fjcy = 60 |(A: + 3)(y-26) = 20 d’où, si on multiplie tout par — : 26 On élimine y, et on obtient : ^24 X + 1H----UC—6 H----. I 1 26j Dans tous ces problèmes depuis le 10, on peut penser que l’équation formulée par Abû Kâmil, immédiatement justifiée par une démonstration géométrique, était aussi l’aboutissement d’un raisonnement qui devait être familier à celui qui pratiquait alors les mathématiques, à savoir : j ’enlève 26 dirhams à chacun des premiers, je rends (60 - 20) dirhams de ce que j ’obtiens ainsi, et je répartis le reste entre les trois hommes supplémentaires.

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

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