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Linenn - Jeux et Mathematiques - <br /> <b>Deprecated</b>: Non-static method plxUtils::strCheck() should not be called statically in <b>/homepages/3/d170528928/htdocs/jm/bn/core/lib/class.plx.show.php</b> on line <b>127</b><br /> George Boole fait la différence - linenn


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George Boole fait la différence

Par
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jp, le
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mardi 16 février 2010 à
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00:10 |
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Logique | Page principale Jeux et Mathématiques

$$\sum_{i=0}^n i^3 = \left(\sum_{i=0}^n i\right)^2$$

Démonstration : l'égalité est vraie pour $n = 0, 1, 2, 3, 4$ donc est vraie pour tout $n$.

Algébrisation

J'ai pris la liberté de modifier une borne de l'égalité de la page 8 de l'article [1], on comprendra pourquoi.
Ce qui m'intéresse dans la démonstration est qu'elle repose – sans le dire – sur une étude par George Boole des différences finies. Ces différences sont les termes i3 et i des sommes, elles sont des polynômes de degrés 3 et 1 de la variable i. Les deux membres sont donc des polynômes de mêmes degrés 3+1=4 et (1+1)×2=4. Il n'est même pas nécessaire de déterminer les polynômes, il suffit montrer que l'égalité est vraie en 5 points distincts (des valeurs des indices), par exemple en 0, 1, 2, 3, 4. Si c'est le cas les polynômes sont aussi identiques, (identiques dans N ou Z et aussi dans Q, R ou C).

Polynôme

Aujourd'hui encore on peut se procurer le livre [3] "Calculus of Finite Differences" de Georges Boole (une dizaine d'euros sur le web).

Je n'ai pas trouvé dans ce livre de raisonnement semblable à celui de [1]. Si on y trouve bien que la différence première d'un polynôme de degré n est un polynôme de degré n-1, une réciproque n'apparaît qu'au moment de l'expression du polynôme à partir de ses différences finies. Notations ci-dessous x(2)=x(x-1), x(3)=x(x-1)(x-2) etc.

Polynôme

Documents, prolongements

[1] An Enquiry Concerning Human (and Computer!) [Mathematical] Understanding par Doron Zeilberger.

Je recommande la vidéo [2] dans laquelle Pierre Cartier explique cette "algébrisation" qui remonte donc, pour nos polynômes, au moins à George Boole. [2] L’algebrisation du continu : de Boole à Sato par Pierre Cartier.

[3] "Calculus of Finite Differences" de George Boole, 368 pages, "Forgotten Books".

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