Re: Egyptian arithmetic and its remaidners

There is a simple form of remainder arithmetic
found in the Reisner Papyri. It is also reported
several times in the RMP, such as problems
24- 38. It says that quotients and remainders
were found whenever a vulgar fraction appeared.

That is, RMP # 65 asked 100 to be divided by 13,
with a quotient 7 and a remainder 9/13 mentally
cited by Ahmes. The proof is given in the Egyptian
fraction series 2/3 1/19 which equals 9/13.

Additional proof is given by RMP 31 and 33 where
the unknown value x was asked to be found

#31 (97/42)x = 33 and

#33 (97/42)x = 37

#33 is clearly 16 + 2/19 with

2/19 - 1/56 = (112 - 97)/(56*97)

or,

2/97 = 1/56 + (8 + 7)/(56*97)

= 1/56 + 1/674 + 1/776

More interesting is

#31 where x = 14 + 28/97

which Ahmes solved by

x = 14 2/97 + 26/97

with

26/97 - 1/4 = (104 -97)/(4*97)

or,

26/97 = (4 + 2 + 1)/(4*97)

= 1/97 + 1/194 + 1/388

There is more, if anyone is interested.

Look up :

http://egyptianmath.blogspot.com

and several links will appear with the
basic discussion of this subject.

Happy New Year to all,

Milo Gardner
.

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