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### 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

```

```
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/
```
Associated Topics:
High School History/Biography
Middle School Fractions
Middle School History/Biography

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