From:  (Darrah Chavey)
Date:           Fri, 15 Nov 1996 10:36:09 -0600
Newsgroups:     sci.math
Subject:        Re: Egyptian Fractions

In article <56aark$>, (Donald T.
Davis) wrote:

>Le Compte de Beaudrap <> writes:
>> what is an Egyptian Fraction?
>egyptian scribes did arithmetic calculations in a
>seemingly bizarre way. when they had to handle a
>fractional quantity, they represented it as a sum
>of an integer and several "unit fractions," each of
>the form 1/n. so, for example, they handled 4 5/6
>as 4 + 1/2 + 1/3.  fractions with big denominators
>were very cumbersome in this system, and both addition
>and multiplication of fractional quantities required
>a lot of table-lookup, so as to reduce 2/n terms to
>standardized sums of distinct 1/m terms.

They did have a special number for 2/3-rds.

>no-one knows why the egyptians found this style
>necessary; it may be that they just couldn't conceive
>of a better way, or that they found it more practical
>for the problems that they had to solve. ...

There are a couple of reasons. First, they wrote a number 1/n
as the number n with an oval above it. This makes it difficult
to come up with a notation for m/n without really inventing a
completely new notation, which is always difficult.

   Second, the unary fractions come up naturally in their method
for division. For example, to divide 53 by 8, they would proceed
as follows: Begin with two columns, one headed by "1" and the other
by "8". Double each column until just less than 53:
               1      8
               2     16
               4     32
Now start over from the 1 and 4 and halve each number until the 
second column is "1". I use () to stand for the oval that should
really be above the numbers in the first column:
   1/2 =    () 2      4
   1/4 =    () 4      2
   1/8 =    () 8      1
Now mark the numbers in the second column which add up to 53 (by the notion 
of binary expansion of a number, there will be a unique way to do this)
               1      8
               2     16  *
               4     32  *
   1/2 =    () 2      4  *
   1/4 =    () 4      2
   1/8 =    () 8      1  *
The corresponding marked numbers in column 1 give you your answer:
    53 / 8  =  2 + 4 + ()2 + ()8, or 2 + 4 + 1/2 + 1/8.
Thus the notion of unary fractions, 1/n, comes up naturally from this
algorithm for division. (Division by numbers other than powers of 2
make life more interesting for the Egyptians.)

   Dominic Alivastro, "Ancient Puzzles", suggests a third reason why 
this use of unary fractions is good. Consider the problem Ahmes poses 
of dividing 3 loaves of bread between 5 people. We would answer "each
person gets 3/5-ths of a loaf". If we implemented our solution, we might
then cut 2 loaves into 3/5 | 2/5 pieces, with bread for 3 people; then
cut one of the smaller pieces in half, giving the other two people 
2/5 + 1/5 pieces. Mathematically acceptable, but try this with kids and
they will insist that it is not an even division. Some have larger pieces,
some have smaller. Ahmes would calculate 3/5 as :
   3/5 = ()3 + ()5 + ()15    [ = 1/3 + 1/5 + 1/15 ]
Now cut one loaf into fifths, cut two more into thirds, then take one of
the 1/3-rd pieces and cut it into 5-ths (for the 1/15-th pieces), and you
can now distribute everyone's 3/5-ths share in a way that _looks_ equal,
since they will have exactly the same size pieces. (And no, I don't want
to argue about the crust.)

Darrah Chavey             -- I'm dead, Horatio.  (Hamlet)
Math/CS Dept.             -- He's dead Jim!      (Not quite Hamlet)
Beloit College
700 College St. Beloit, WI 53511