Egyptian fraction
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An Egyptian fraction is a sum of distinct unit fractions, i.e. fractions whose numerators are equal to 1, whose denominators are positive integers, and all of whose denominators differ from each other. It can be shown that every positive rational number can be written in that form. Refer to the harmonic series for the proof that you can express all numbers, even very large, in this way. One algorithm to produce an Egyptian fraction representation for a given rational number r=a/b between 0 and 1 is the following greedy algorithm due to James Joseph Sylvester:
- Find the largest unit fraction just less than r. The denominator can be found by dividing b by a, discarding the remainder, and adding one. (If there's no remainder, we're done right away because r is itself a unit fraction.)
- Subtract the found unit fraction from r. Then continue with step 1, using this new smaller value for r.
Example: convert 19/20 into an Egyptian fraction.
- 20/19 = 1 with some remainder, so our first unit fraction is 1/2.
- 19/20 - 1/2 = 9/20.
- 20/9 = 2 with some remainder, so our second unit fraction is 1/3.
- 9/20 - 1/3 = 7/60
- 60/7 = 8 with some remainder, so our third unit fraction is 1/9.
- 7/60 - 1/9 = 1/180 which is itself a unit fraction.
So our result is
- <math>\frac{19}{20} = \frac{1}{2}+\frac{1}{3}+\frac{1}{9}+\frac{1}{180}<math>
Note that the representation of a given rational number as an Egyptian fraction is not unique, and that the above algorithm does not always yield the shortest such representation:
- <math>\frac{19}{20} = \frac{1}{2}+\frac{1}{4}+\frac{1}{5}<math>
Fractions in Egypt
Mathematical historians sometimes describe algebra as having developed in three primary stages:
- rhetorical algebra, wherein the problem was stated in words of the language of the ancient mathematician;
- syncopated algebra, wherein some words of the problem were abbreviated, for easier comprehension;
- symbolic algebra, where in symbols for operators and operands made comprehension still easier.
Typical of symbolism is denoting "the unknown" by "x". We know from ancient Egyptian texts that Egyptian priests and scribes, in their rhetorical algebra, used the word "aha" meaning "heap" or "set" for the unknown.
This is shown in the Rhind Mathematical Papyrus (Second Intermediate Period) in the British Museum in London in a translation of one of its "aha" problems:
"Problem 24: A quantity and its 1/7 added together become 19. What is the quantity?
"Assume 7. 7 and 1/7 of 7 is 8. As many times as 8 must be multiplied to give 19, so many times 7 must be multiplied to give the required number."
In modern symbolic form, x + x/7 = 8x/7 = 19, or x = 133/8. Proof: 133/8 + 133/(7 · 8) = 133/8 + 19/8 = 152/8 = 19.
Note the fractions in this problem. Ancient Egyptians calculated by unit fractions, such as 1/2, 1/3, 1/4, 1/10, ....
The hieroglyph for an open mouth denoted the fractional solidus, with a number hieroglyph written below this "open mouth" icon to denote the denominator of the fraction.
| <hiero>D21:Z1*Z1*Z1</hiero> | <math>= \frac{1}{3}<math> | <hiero>D21:V20</hiero> | <math>= \frac{1}{10}<math> |
Any fraction we write with a non-unit numerator was written by ancient Egyptians as a sum of unit fractions no two of whose denominators are the same.
These sums of unit fractions have, therefore, become known as "Egyptian fractions".