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 turn into either a reference and a version. Their impact used to be decisive at the improvement of algebra in Arabic at least in Latin and Hebrew. there'll be present in the current ebook the 1st carefully severe variation of Ab? K?mil s works, in addition to the 1st ever translation right into a smooth language. textual content and translation are preceded by means of an exhaustive statement, right now mathematical and historical."

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

Sample text

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. CHAPITRE II LE CALCUL ALGÉBRIQUE ET SES APPLICATIONS Tout comme al-Khwàrizmï, Abû Kàmil fait suivre le chapitre consacré à la théorie des équations des deux premiers degrés de deux autres, qui se succèdent dans cet ordre ; le premier porte sur les calculs algébriques sur les polynômes associés à ces équations ; le second sur les transformations et les opérations algébriques pour ramener un problème donné à l’une ou l’autre des six équations canoniques.

10 2 + V3 . Abü Kamil veut éliminer le radical du y V ^ + 20 = y \ dénominateur. 2x + ^l3x^ = 10 C’est une équation trinôme avec un coefficient irrationnel : V 3 ? = 1 0 -2 x . Si on met au carré chaque membre, on a yVô + V26. - 40 - On considère l’équation 3x^ = 100 + 4x^ - 40x. Mais cette mise au carré introduit des solutions superflues. Pour les reconnaître ultérieurement, retenons que l’on doit aussi avoir 10 - 2;r > 0. x46 •x 4 s + IOa:^ + 20 = jc\ Changement de variable y = x^. ^|^ x^ +100 = 40x x =2o± ^^m .

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. 7). On a donc AE = ^ et aire {ED) = aire (AD) - aire {AN), soit x^ - b x = c.

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