Egyptian Fractions

Prepared by:

Joseph Malkevitch
Mathematics and Computing Department
York College (CUNY)
Jamaica, NY 11451

email:

malkevitch@york.cuny.edu

web page:

http://york.cuny.edu/~malk/

The ancient Egyptians represented fractions as the sum of fractions with unit numerators (e.g. the numerator was the number 1) and distinct denominators.

Most of our information Egyptian fractions has been gleaned from information in the Egyptian Mathematical Leather Roll (EMLR), which was "unrolled" in 1927. It contains examples of ways of expanding fractions in the form of the sum of unit fractions (usually) with different denominators. An extensive attempt was made recently to determine what methods might have been used to obtain these expressions involving unit fractions by Milo Gardner. Gardner points out that he knows of no method which generates all of the expansions for all the expansions shown in the EMLR. He offers methods by which the fraction expansions shown might have been obtained.

Here are some of the methods he mentions, used to represent unit fractions in terms of the sum of other unit fractions.

METHOD 1

1/(2n) = (1/2)(1/n) = (1/3 + 1/6)(1/n)

In other words this method expands fractions with an even denominator (4 or more) by using the fact that 1/2 = 1/3 + 1/6.

Examples:

1/6 = (1/2)(1/3) = (1/3 + 1/6)(1/3) = 1/9 + 1/18.

1/12 = (1/2)(1/6) = (1/3 + 1/6)(1/6) = 1/18 + 1/36

METHOD 2 (Splitting)

1/n = 1/(2n) +1/(2n)
which can be used in conjucntions with

METHOD 3

1/n = (1/n)(1/2 + 1/3 + 1/6)

(Here we use the fact that 1/2 + 1/3 + 1/6 = 1.)

Example (Using Methods 2 and 3)

1/5 = 1/10 + 1/10 = 1/10 + (1/10)(1/2 + 1/3 + 1/6) = 1/10 + 1/20 + 1/30 +1/60.

METHOD 4

1/n = 1/(n + 1) + 1/(n)(n+1)

Example:

1/5 = 1/6 + 1/30.

Note that from the previous example, we can conclude that:

1/6 = 1/10 + 1/20 + 1/60.

This, in turn, shows: 1/3 = 1/5 + 1/10 + 1/30.

METHOD 5

1/n= (1/A)(A/n)

Example 1:

Expand 1/4.

Choose A to be 7.

We have 1/4 = (1/7 )(7/4) = 1/7( 1 + 1/2 + 1/4) = 1/7 + 1/14 + 1/28.

Note that 1/4 can also be expanded using the choice A = 5.

Thus, 1/4 = (1/5)(5/4) = (1/5)(1 + 1/4) = 1/5 + 1/20. (This coincides with using Method 4.)

Example 2:

We can also expand 1/15 by this method using A = 25.

1/15 = (1/25)(25/15) = (1/25)(10/6) = (1/25)(1 + 1/2 + 1/6) = 1/25 + 1/50 + 1/150.

Using method 4 for 1/15 we get 1/16 + 1/240.

The two different results here are typical of the trade-off between number of summands and the size of the denominators involved. 1/15 has a three term expansion a denominator of 150 and a two term expandsion with a denominator of 240.

Note that 1/15 = 1/20 + 1/60 as well! This seems better than the previous two expansions.

Here is another interesting observation for constructing Egyptian fractions.

Suppose a + b = c.

I claim that 1/(a)(b) = 1/(c) (1/a + 1/b). This follows since 1/a + 1/b = (a + b)/ab.

Here is an an example:

Since 4 + 7 = 11

we can conclude that 1/28 = 1/11((1/4) + 1/7)) = 1/44 + 1/77.

What should be meant by the "best" Egpytian fraction expansion of a given fraction?

References:

Gilling, R., Mathematics in the Time of the Pharoahs, MIT Press, Cambridge, 1972 (reprinted by Dover Press, New York).

Gardner, M., The Egyptian Mathematical Leather Roll, Attested Short Term and Long Term, in The History of the Mathematical Sciences, ed. I. Grattan-Guiness, Hindustan Book Agency (distributor, American Mathematical Society), New Delhi, 2004, p. 119-134,