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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 /> Chemin faisant - linenn


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vendredi 15 janvier 2010 à
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15:50 |
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Arithmétique | Page principale Jeux et Mathématiques

Route
Coucher de soleil

Jeudi soir j'ai filmé d'un bus, non pas le coucher de soleil, mais les alignements d'arbres à l'horizon. J'espérais trouver une ligne d'arbres plantés régulièrement pour illustrer une propriété d'arithmétique.
L'image du film est imparfaite mais suffira peut-être.
Vous apercevez, très flou, un arbre près du centre de l'image et tout à fait à droite de l'image un second arbre qui commence à apparaître.
On pourrait aussi exploiter a ligne d'abres que l'on voit au loin au bord d'un champ.

Division

La propriété est la suivante : Si en roulant (en bus, en train, en voiture...), vous filmez une ligne régulièrement plantée d'arbres et parallèle à la route, vous verrez sur certaines images n arbres et sur d'autres images n+1 arbres.

(Le nombre n est un entier positif ou nul qui dépend uniquement de l'espacement régulier des arbres, de leur éloignement et des caractéristiques optiques de l'appareil de prises de vues).

Fenêtre
Fenêtres

La distance entre les arbres est d et la largeur de la fenêtre est D sur le second schéma. Pour trouver le nombre d'arbres dans la fenêtre l'idée automatique est de calculer D/d qui donne le nombre nd'arbres dans la fenêtre de couleur gris clair, alors que le nombre d'arbres de l'autre fenêtre est n+1.

En effet D/d est le nombre n d'intervalles d que l'on peut placer entièrement en écrivant D = n d + a avec a < d. Dans ce cas nous avons n+1 arbres.
On peut aussi écrire D = u + (n-1)d + v avec u+v = d+a et u et v inférieurs à d. Dans ce cas nous avons n arbres.

Il l'agit de résoudre D = u + k d + v ou kd = D-(u+v) avec de 0<= u < d et 0<=v < d. Le nombre entier d'arbres est k+1. D et d sont deux réels positifs non nuls.
1) Les encadrements suivants sont équivalents (en utilisant D = u + k d + v )
0 <= u+v < 2d
D-2d < D-(u+v) ≤ D
D-2d < kd ≤ D/d
D/d -2 < k ≤ D/d
D/d -1 ≤ k ≤ D/d
D/d ≤ k+1 ≤ 1 +D/d
2) Ce qu'on peut interpréter en disant que le nombre d'arbres est entier et est donc la partie entière de D/d plus 0 ou plus 1.

arbres
Deux ou trois

Autre propriété

La propriété à montrer est : si A>B≥0 , alors [A]-[B] = [A-B] +0 ou +1

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