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 until eventually 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 at least in Latin and Hebrew. there'll be present in the current book the 1st conscientiously serious variation of Ab? K?mil s works, in addition to the 1st ever translation right into a sleek language. textual content and translation are preceded by way of an exhaustive statement, instantaneously mathematical and historical."

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

Example text

20 = a: \ Système facile à résoudre si on s’inspire du problème 33. - 37 - Soit jc = Changement de variable y = x^. 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.

Mais la démarche idéale est d’adjoindre, lorsque faire se peut, aux calculs algébriques une démonstration forgée à partir de la géométrie. Inauguré par Abû Kâmil, cet état de choses se prolongera durant six siècles. Telle était la situation des mathématiciens tant qu’ils procédaient par la théorie des proportions. Il n’y rencontraient pas de difficultés particulières, puisque la structure de la géométrie classique est celle que l’on appelle à présent corps commutatif ordonné. Des deux chapitres évoqués, le premier, relativement bref, débute par l’étude du signe du quatrième terme du produit, selon l’habitude des arith­ méticiens contemporains d’Abû Kâmil^ {a ± b ){c± d ) et {a±b){c + d), puis expose la règle des signes en général.

Autre méthode : On échange jc et (10 - x). ^(V2)%(40-hV8ÜÔ) = V 4 2 - h V ^ - V 2 . - 43 - On considère l’équation = VÏÔ(lO-x) x = -4245 + ^]2,5 + 4 m ô . - 44 - On considère l’équation = 5+ x ' = V ^ 5 + ^/2 J+ ^/2 5 0 . - 46 - On considère l’équation (V 2 ? + 2)x = 30, c’est-à-dire (si on divise tout par V2) {x^ + 1 0 ) 4 5 = x \ x^ + 4 ^ = ^450 {y + \ 0 ) S = y' 2 ^2 On pose y = 1 + 1 + -V4M. 4 ^ + y45 = y^ 84 Chapitre II Le calcul algébrique et ses applications - 47 - On considère l’équation (symétrique en ;c et (10 - x)) ^ 10 - +10:11 = 75.

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

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