Download Abu Kamil Algèbre et analyse diophantienne. Édition, by Roshdi Rashed PDF

By Roshdi Rashed

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 study that proved fertile up till the 17th century, and have been quickly to develop into either a reference and a version. Their effect was once decisive at the improvement of algebra in Arabic a minimum of in Latin and Hebrew. there'll be present in the current e-book the 1st carefully severe variation of Ab? K?mil s works, in addition to the 1st ever translation right into a glossy language. textual content and translation are preceded by means of an exhaustive statement, instantly mathematical and historical."

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Abu Kamil Algèbre et analyse diophantienne. Édition, traduction et commentaire

The mathematical works of Ab? ok? mil (floruit circa 880) have been produced generations after the works of Al-Khwarizm? , the founding father of algebra. They spread out 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 impression used to be decisive at the improvement of algebra in Arabic at least in Latin and Hebrew.

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

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10-JC X On résout cette équation comme dans 29. - 32 - On considère l’équation 10- j: Il ajoute cette quantité de part et d’autre dans l’équation initiale et obtient ; JC 10-JC^ JC= -ir O 2 5 - 2 jc, j: ------- + --------+10 U = 73. X 10 - a: Autre méthode : {y-z)x =5 X Alors 10 y = 10-JC -A: = 73-10jr-(10-A :). Z= 10-X Si on additionne membre à membre avec Ici, le « complément » est 2z ; or 2zx = 20 - 2x, donc 10-JC (10-x) = a:, (y + z)x = 25 - 2x on obtient X 10-JC 10 10-x = 6 3 - 8 jc, Z= 10-x d’où On résout cela comme dans 29.

24 - On considère l’équation 10- X _ 1 ------- + X = 5 + —. X 2 Autre méthode, très probablement du même Abu Yûsuf : On pose J = x^. /2 3 V3 =y Alors /5k\ (*) - y =y 13 Id -----= 5c+ - 1- x X 2 et, si on multiplie tout par x : 10-x = ^5 + ^ j x - x ^ , - 22 - On considère l’équation 3x + 2-Vjc^ - 3x = x' 2^x^ - 3 x = x^ - 3x. équation quadratique dont Abu Kâmil ne détaille pas la résolution (on a x = 4). - 25 - On considère l’équation 1 0 -x En divisant tout par le radical puis en mettant chacun des deux membres au carré, on obtient x ^ - 3 x = 4.

6 des Éléments d’Euclide. Dans la démonstration qu’il qualifie d’évidente, il se sert des propriétés du gnomon. Abû Kàmil, après avoir exposé et démontré l’algorithme pour la racine de l’équation, donne aussi un algorithme pour le carré de cette racine. Il est obligé de le démontrer géométriquement. Sa démonstration est analogue à celle de l’algorithme pour la racine. On note qu’Abû Kâmil représente ici le carré de l’inconnue par un segment de droite. Cette surabondance de démonstrations semble exprimer une norme essentielle à l’algèbre selon la conception d’Abû Kâmil, ainsi peut-être qu’un souci didactique.

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