Egyptian Fractions

Date: 03/21/99
From: Anonymous
Subject: Egyptian Fractions

Consider the fraction 6/7. The book I was reading gave this 
explanation:
     
      1           7 (I assume they use 7 here because it is the 
                     denominator)
      1/2         3+1/2 (halving)

      1/4         1+1/2+1/4 (halving)

But why do they do the next step? Why not halve again?

      1/7         1
      1/14        1/2
      1/28        1/4

Now they note: 1/2+1/4+1/14+1/28 = 6
therefore 6/7 = 1/2+1/4+1/14+1/28

I understand that Egyptian Fractions are unit fractions, that there 
are never more than 4, and that no two are the same size, but I cannot 
follow line 4. 

Thanks for your help.

Date: 03/21/99
From: Doctor Rick
Subject: Re: Egyptian Fractions

I am wondering about the context of this problem. Is it purporting to 
explain exactly how the Egyptians worked with fractions, or is it a
modern set of rules for working with unit fractions in a more-or-less
Egyptian manner?

A book I have (Boyer and Merzbach, _A History of Mathematics_, 2nd 
edition, 1988) gives some examples from a table of Egyptian fractions 
in the Rhind Papyrus, written by the scribe Ahmes in 1650 BC. The 
authors are not able to reach a conclusion about why one decomposition 
into unit fractions (for instance, 2/15 = 1/10 + 1/30) was preferred 
out of the many possible decompositions (such as 1/15 + 1/15, or 1/8 + 
1/120, or 1/12 + 1/20). Apparently there are no known examples in the 
Egyptian papyri of the method they used to decompose a fraction into 
unit fractions. The papyri do contain examples of how to multiply 
using a similar technique (successive doubling, rather than halving); 

See our archives:

  http://mathforum.org/dr.math/problems/wihart.6.4.96.html   

But wherever the rules and the method in your book came from, I think 
I can see how the method works. Let us go through it.

First, what we want to do is to find a number n (expressed as an 
Egyptian fraction) such that 6 = 7 * n. This number n will be equal to 
6/7. We build a table in which each number on the right is 7 times the 
number on the left, by starting with two numbers that obviously have 
this relation, and successively dividing both sides by 2:

  n       7n
--------------
  1       7
  1/2     3+1/2
  1/4     1+1/2+1/4

For the Egyptians to do this, they would need to understand that half 
of 1/n is 1/(2n). We cannot take this for granted, but my book says 
they did understand this.

Now look at the terms on the right. Our goal is to find numbers in the 
righthand column that add to 6. Then we can add the terms on the left 
to get our fraction. If each term on the right is 7 times the number 
on the left, then any sum of terms on the right is 7 times the sum of 
the corresponding terms on the left (by the distributive property).

We have two terms that add to between 5 and 6, namely, (3 + 1/2) +
(1 + 1/2 + 1/4) = 5 + 1/4. All we need now is 3/4, which is less than 
1. So let us start over, with 1/7 in the lefthand column and 1 in the 
righthand column. The righthand column is still 7 times the lefthand 
column.

  n       7n
--------------
  1/7     1
  1/14    1/2
  1/28    1/4

We can stop here because the smallest fraction we need is 1/4. It will 
add to the 5 + 1/4 we have already, bringing the total to 5 + 1/2; 
then we add 1/2, bringing the total to 6 (that is what we want). So, 
we mark off the rows that add up to 6 in the right column, and add 
only these rows in the left column as well:

 n                         7n
---------------------------------------
 1                         7
 1/2                       3+1/2      X
 1/4                       1+1/2+1/4  X

 1/7                       1
 1/14                      1/2        X
 1/28                      1/4        X
---------------------------------------
 1/2 + 1/4 + 1/14 + 1/28   6

The number on the left is 1/7 of 6, so it is our answer.

By the way, if you try this method on 2/15, you will get 1/8 + 1/120, 
which is NOT the decomposition listed in the Rhind papyrus, so this 
must not be exactly the technique used by the author of that document. 
But it is an interesting method nevertheless. I hope my explanation 
helped you.

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/