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Multiplication et division \[EAcute]gyptiennes\
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 "La m\[EAcute]thode utilis\[EAcute]e par les Egyptiens est attest\[EAcute]e \
par le papyrus de ",
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documents du ",
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Pour parodier ",
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\[EAcute]s en langage ",
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Soit, par exemple, le produit suivant \[AGrave] effectuer\
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Une m\[EAcute]thode consiste \[AGrave] exprimer le premier nombre en base 2\
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Pour terminer le calcul, deux op\[EAcute]rations arithm\[EAcute]tiques \
suffisent: la multiplication de nombres par 2 et l'addition de nombres.
En particulier, elle ne fait pas appel \[AGrave] des tables de \
multiplication.\
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Pour convertir l'entier 181 en base 2, on peut proc\[EAcute]der comme suit:
1-\[EGrave]re \[EAcute]tape:
     dresser la liste des puissances de 2, tant qu'elles ne sont pas sup\
\[EAcute]rieures \[AGrave] 181;
2-\[EGrave]me \[EAcute]tape: on passe en revue chaque ligne, pour \
d\[EAcute]terminer celles qui doivent \[EHat]tre biff\[EAcute]es;
    dans la colonne 2, on trouve le chiffre en base 2; un 0 indique que la \
ligne est biff\[EAcute]e;
    dans la colonne 3, on trouve le reste qui n'est pas encore converti; le \
dernier reste doit \[EHat]tre nul.
     puisque 181 \[GreaterEqual] 128, il faut prendre 128 et former la diff\
\[EAcute]rence 181-128=53;
     puisque  53 < 64, il faut biffer 64;
     puisque 53 \[GreaterEqual] 32, il faut prendre 32 et former la diff\
\[EAcute]rence 53-32=21;
     puisque 21 \[GreaterEqual] 16, il faut prendre 16 et former la diff\
\[EAcute]rence   21-16=5;
     puisque  5 < 8, il faut biffer 8;
     puisque 5 \[GreaterEqual] 4, il faut prendre 4 et former la \
diff\[EAcute]rence  5-4=1;
     puisque  1 < 2, il faut biffer 2;
     puisque 1 \[GreaterEqual] 1, prendre 1 et former la diff\[EAcute]rence \
1-1=0.\
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Cell["\<\
Pour multiplier 181 par 273,  on peut proc\[EAcute]der comme suit:
1-\[EGrave]re \[EAcute]tape:
     dans la premi\[EGrave]re colonne: dresser la liste des puissances de 2, \
tant qu'elles ne sont pas sup\[EAcute]rieures \[AGrave] 181;
     dans la deuxi\[EGrave]me colonne:en regard de 2, 4, 8, ..., par \
doublements successifs, on calcule 2 \[Times] 273, 4 \[Times] 273, 8 \[Times] \
273, ... 
2-\[EGrave]me \[EAcute]tape:
    colonnes 1 \[AGrave] 3: on exprime 181 en base 2 (voir les commentaires \
plus haut);
    colonne 4: on ne retient que les multiples de 273 situ\[EAcute]s sur des \
lignes non biff\[EAcute]es.
3-\[EGrave]me \[EAcute]tape:
    colonne 1: les multiples de 273 qui ont \[EAcute]t\[EAcute] retenus \
doivent \[EHat]tre additionn\[EAcute]s;
    colonne 2: sommes partielles; la derni\[EGrave]re somme est le \
r\[EAcute]sultat final.\
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 "Il s' agit d' une division euclidienne : le quotient est un entier et le \
reste est entier. Alors que les Egyptiens savaient manipuler certaines \
fractions, dans le papyrus de ",
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 ", le reste est g\[EAcute]n\[EAcute]ralement ignor\[EAcute]."
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A titre d'exemple, consid\[EAcute]rons le quotient de 95 432 par 285.\
\>", "Text",
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Pour diviser 95432 par 285,  on peut proc\[EAcute]der comme suit:
1-\[EGrave]re \[EAcute]tape:
     dans la premi\[EGrave]re colonne: multiplier 285 \
r\[EAcute]p\[EAcute]titivement par 2, tant que le r\[EAcute]sultat n'est pas \
sup\[EAcute]rieur \[AGrave] 95432;
     dans la deuxi\[EGrave]me colonne:en regard, \[EAcute]crire les multiples \
de 2.
2-\[EGrave]me \[EAcute]tape:
    colonnes 1 \[AGrave] 3:
    dans la colonne 2, un 0 signifie que la ligne doit \[EHat]tre biff\
\[EAcute]e;
    dans la colonne 3 se trouve le reste qui sera, si possible, \
trait\[EAcute] plus bas; le dernier reste est le reste de la division.
           puisque 95432 \[GreaterEqual] 72960, il faut prendre 256 et former \
la diff\[EAcute]rence 95432-72960=22472;
           puisque  22472 < 36480, il faut biffer 128;
           puisque 22472 \[GreaterEqual] 18240, il faut prendre 64 et former \
la diff\[EAcute]rence 22472-18240=4232;
           puisque 4232 < 9120, il faut biffer 32;
           puisque  4232 < 4560, il faut biffer 16;
           puisque 4232 \[GreaterEqual] 2280, il faut prendre 8 et former la \
diff\[EAcute]rence  4232-2280=1952;
           puisque 1952 \[GreaterEqual] 1140, il faut prendre 4 et former la \
diff\[EAcute]rence  1952-1140=812;
           puisque 812 \[GreaterEqual] 570, il faut prendre 2 et former la \
diff\[EAcute]rence  812-570=242;
           puisque  242 < 285, il faut biffer 1;
          le reste de la division est 242.
     colonne 4: on ne retient que les multiples de 2 situ\[EAcute]s sur des \
lignes non biff\[EAcute]es.
3-\[EGrave]me \[EAcute]tape:
    colonne 1: les multiples de 2 qui ont \[EAcute]t\[EAcute] retenus doivent \
\[EHat]tre additionn\[EAcute]s;
    colonne 2: sommes partielles; la derni\[EGrave]re somme est le quotient.
R\[EAcute]ponse: le quotient est de 334 et le reste est de 242.\
\>", "Text",
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Justification de la division \[EAcute]gyptienne\
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