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

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 spread out fields of study that proved fertile up till the 17th century, and have been quickly to turn into either a reference and a version. Their effect was once decisive at the improvement of algebra in Arabic not less than in Latin and Hebrew. there'll be present in the current e-book the 1st carefully serious 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 way of an exhaustive remark, straight away mathematical and historical."

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

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Dernière méthode ; L’équivalence énoncée dans la remarque ci-dessus conduit à une dernière méthode pour résoudre le problème 67 : | m+v = 10 [u + 24Û = V- 2Vv On met chaque membre au carré X^(10-X^) = X ^ + 4 X ' + 4 X ' On pose X = Vm, alors Vv = X + 2, d’après la remarque. On remplace Met Vpar leurs valeurs dans la première équation : X^ + (X + 2Ÿ = 10. X2+2X-3 = 0 X = 1 ou -3. Cette équation a pour racines 1 et (-3). Abü Kàmil donne à présent une démonstration géométrique de l’équivalence énoncée dans la dernière remarque, au moyen de la figure suivante : - 68 - On considère le système d’équations 10 = x + y 10 10 ^ 1 B X y 4 L F Lemme : E 0D A9 H 10 10 y X ^ y — + — = - + - + 2.

I —X2 +. -1X 2 =x 2, alors ls~t =2 Alors r = V2-1, d’où ( 2x 1], etc. x^ =4 +- +- +^^8+ [5 +- + l ^ . 2 3 - 51 - On considère l’équation - 53 - On considère l’équation [x^ - ( 2 x + 10)f = 8 x \ (*) 2x + J - x ^ + J - x ^ =20. Abu Kâmil en déduit x^ - 2 x - W = xVS, Voir plus loin, p. 438. V 3 V3 90 Le calcul algébrique et ses applications Chapitre II Abü Kâmil isole les deux radicaux puis met chaque membre au carré : ^ '1 2 '3 y 91 - 57 - On considère l’équation =4x\ = (20-2x)% Première méthode : On pose y - yr.

21 - On considère l’équation __ 2 /2x ' =x -7x '-2 3-3 j~ 3 On pose yr - 3x. L’équation devient alors V3 4}; - 4 = y'^. Elle a pour solution y^ = 4, et x^ - 3x = 4 a pour solution = 16 (Abù Kâmil ne détaille pas la résolution de ces deux équations quadratiques). - 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.

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


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