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Egyptian Fractions Explained

Egyptians expressed fractions using only unit fractions, which have 1 as the numerator. Common fractions like 1/2 and 2/3 had unique symbols. Other fractions were expressed as the sum of progressively smaller unit fractions. For example, 2/5 was expressed as 1/3 + 1/15. Egyptians would estimate the largest unit fraction less than the given fraction, subtract it, and continue decomposing any remainder into unit fractions through trial and error to find the simplest representation.

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627 views2 pages

Egyptian Fractions Explained

Egyptians expressed fractions using only unit fractions, which have 1 as the numerator. Common fractions like 1/2 and 2/3 had unique symbols. Other fractions were expressed as the sum of progressively smaller unit fractions. For example, 2/5 was expressed as 1/3 + 1/15. Egyptians would estimate the largest unit fraction less than the given fraction, subtract it, and continue decomposing any remainder into unit fractions through trial and error to find the simplest representation.

Uploaded by

oneluckyclover2
Copyright
© Attribution Non-Commercial (BY-NC)
We take content rights seriously. If you suspect this is your content, claim it here.
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SME

 430:  History  of  Mathematics     1/26/10  

Name __________________________
Egyptian Fractions
Egyptians used almost exclusively fractions which have 1 for a numerator. These are called unit
fractions. The symbol for a fraction in hieroglyphic script contains sign/symbol for

mouth - but this is at the same time symbol for the smallest unit of grain which is called
ro.
So Egyptians would write fractions like this:

=2 = 1/2

 =  13        =  1/13    

In order to make writing and computations more easier we’ll use Hindu-Arabic numerals and
we’ll put a line over a number to represent the unit fraction corresponding to that number.

For example 3 means 1/3 or 17 means 1/17 in terms of Egyptian fractions.

Besides unit fractions 2/3 had a special symbol. used to represent 2/3. In order to make
our job easier we will put two lines over 3 to represent 2/3.

2/3 or any unit fraction (fraction with one as the numerator, like 1/7) were expressed in a simple,
straightforward way. 1/2 had a sign of its own, in our way of writing (2), as did 2/3 (3). And the

other unit fractions were just the symbol (meaning "part"), with the denominator
expressed as an integer, under this symbol.

Other fractions are not as simple. They are expressed as the sum of progressively smaller unit
fractions. For this purpose, 2/3 is considered to be a unit fraction. For example, 2/5=1/3 + 1/15.
Notice that 2/5 could also be expressed as 1/5 + 1/5. This is not allowed in Egyptian fractions.
No two unit fractions can be the same.
SME  430:  History  of  Mathematics     1/26/10  

Here are a few examples:


first calculation improvement
2/9=1/5 + 1/45 =1/6+1/18
2/7=1/4 + 1/7 +1/28

Exercises:
1. 3/8= 1/8 + 2/8 = 1/8 + 1/4
2. 2/5= 1/3 + 1/15
3. 3/4= 2/3 + 1/12
4. 3/5= ½ + 1/10
5. 5/8= ½ + 1/8
6. 4/5= ½ + 1/5 + 1/10
7. 5/6= ½ + 1/3
8. 7/8= ½ + ¼ + 1/8
9. 5/7= 2/3 + 1/21
10. 5/9= ½ + 1/18
11. 2/7= ¼ + 1/28
12. 6/7= 2/3 + 1/7 + 1/21
13. 3/6= ½
14. 4/9= 1/3 + 1/9
15. 7/9= 2/3 + 1/9

As you can see, some of these are more complicated than others.

How did we come up with these values? Well, we estimated the fraction with the largest unit
fraction that was just smaller than the given fraction. we subtracted this unit fraction from the
given fraction. If this remainder was still not a unit fraction, we repeated the process, choosing
the largest unit fraction that is smaller than this remainder. And the process could be repeated
over and over.

Let's use 7/8 as an example. We estimate 7/8 with 2/3 (the largest Egyptian unit fraction less than
7/8). We subtract 7/8 - 2/3, which is 5/24, which cannot be simplified into a unit fraction. So we
estimate 5/24 with 1/5 (the largest unit fraction less than 5/24). We subtract 5/24-1/5, and we get
1/120, which is a unit fraction. So, 7/8=2/3 + 1/5 + 1/120.

This process always converges. In other words, it never goes on forever. We have proved this,
but it seems that Fibonacci proved it before we did, about 1200 A.D. But this process does not
guarantee the simplest Egyptian fraction. 7/8 may be expressible in two terms, instead of three.
The only general method for finding the simplest Egyptian fractions, is trial and error.

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