![]() | ||
An Egyptian fraction is a finite sum of distinct unit fractions, such as
Contents
- Motivating applications
- Early history
- Hieroglyphic notation
- Modern notation
- Calculation methods
- Methods for odd denominators
- Methods for odd prime denominators
- Methods for semiprime denominators 1
- Methods for semiprime denominators 2
- Methods for some other composite denominators
- Methods for 101
- Expression table of Rhind Papyrus
- Later usage
- Modern number theory
- Open problems
- References
Motivating applications
Beyond their historical use, Egyptian fractions have some practical advantages over other representations of fractional numbers. For instance, Egyptian fractions can help in dividing a number of objects into equal shares (Knott). For example, if one wants to divide 5 pizzas equally among 8 diners, the Egyptian fraction
means that each diner gets half a pizza plus another eighth of a pizza, e.g. by splitting 4 pizzas into 8 halves, and the remaining pizza into 8 eighths.
Similarly, although one could divide 13 pizzas among 12 diners by giving each diner one pizza and splitting the remaining pizza into 12 parts (perhaps destroying it), one could note that
and split 6 pizzas into halves, 4 into thirds and the remaining 3 into quarters, and then give each diner one half, one third and one quarter.
Early history
For more information on this subject, see Egyptian numerals, Eye of Horus, and Egyptian mathematics.Egyptian fraction notation was developed in the Middle Kingdom of Egypt, altering the Old Kingdom's Eye of Horus numeration system. Five early texts in which Egyptian fractions appear were the Egyptian Mathematical Leather Roll, the Moscow Mathematical Papyrus, the Reisner Papyrus, the Kahun Papyrus and the Akhmim Wooden Tablet. A later text, the Rhind Mathematical Papyrus, introduced improved ways of writing Egyptian fractions. The Rhind papyrus was written by Ahmes and dates from the Second Intermediate Period; it includes a table of Egyptian fraction expansions for rational numbers 2/n, as well as 84 word problems. Solutions to each problem were written out in scribal shorthand, with the final answers of all 84 problems being expressed in Egyptian fraction notation. 2/n tables similar to the one on the Rhind papyrus also appear on some of the other texts. However, as the Kahun Papyrus shows, vulgar fractions were also used by scribes within their calculations.
Hieroglyphic notation
To write the unit fractions used in their Egyptian fraction notation, in hieroglyph script, the Egyptians placed the hieroglyph
(er, "[one] among" or possibly re, mouth) above a number to represent the reciprocal of that number. Similarly in hieratic script they drew a line over the letter representing the number. For example:
The Egyptians had special symbols for 1/2, 2/3, and 3/4 that were used to reduce the size of numbers greater than 1/2 when such numbers were converted to an Egyptian fraction series. The remaining number after subtracting one of these special fractions was written using as a sum of distinct unit fractions according to the usual Egyptian fraction notation.
The Egyptians also used an alternative notation modified from the Old Kingdom to denote a special set of fractions of the form 1/2k (for k = 1, 2, ..., 6) and sums of these numbers, which are necessarily dyadic rational numbers. These have been called "Horus-Eye fractions" after a theory (now discredited) that they were based on the parts of the Eye of Horus symbol. They were used in the Middle Kingdom in conjunction with the later notation for Egyptian fractions to subdivide a hekat, the primary ancient Egyptian volume measure for grain, bread, and other small quantities of volume, as described in the Akhmim Wooden Tablet. If any remainder was left after expressing a quantity in Eye of Horus fractions of a hekat, the remainder was written using the usual Egyptian fraction notation as multiples of a ro, a unit equal to 1/320 of a hekat.
Modern notation
The unit fraction 1/n is expressed as n, and the fraction 2/n is expressed as n, and the plus sign “+” is omitted. For example, 2/3 = 1/2 + 1/6 is expressed as 3 = 2 6.
Calculation methods
Modern historians of mathematics have studied the Rhind papyrus and other ancient sources in an attempt to discover the methods the Egyptians used in calculating with Egyptian fractions. In particular, study in this area has concentrated on understanding the tables of expansions for numbers of the form 2/n in the Rhind papyrus. Although these expansions can generally be described as algebraic identities, the methods used by the Egyptians may not correspond directly to these identities. Additionally, the expansions in the table do not match any single identity; rather, different identities match the expansions for prime and for composite denominators, and more than one identity fits the numbers of each type:
Methods for odd denominators
For small odd prime denominators p = 2m + 1, the expansion 2/2m + 1 = 1/m + 1 + 1/(m + 1)(2m + 1) was used. This method is available for not only odd prime denominators but also all odd denominators.
Methods for odd prime denominators
For larger prime denominators, an expansion of the form 2/p = 1/A + 2A − p/Ap was used, where A is a number with many divisors (such as a practical number) between p/2 and p. The remaining term 2A − p/Ap was expanded by representing the number 2A − p/Ap as a sum of divisors of A and forming a fraction d/Ap for each such divisor d in this sum. As an example, Ahmes' expansion 1/24 + 1/111 + 1/296 for 2/37 fits this pattern with A = 24 and 2A − p/Ap = 11 = 3 + 8, as 1/24 + 1/111 + 1/296 = 1/24 + 3/24 × 37 + 8/24 × 37. There may be many different expansions of this type for a given p; however, as K. S. Brown observed, the expansion chosen by the Egyptians was often the one that caused the largest denominator to be as small as possible, among all expansions fitting this pattern.
Methods for semiprime denominators (1)
For semiprime denominators, factored as p×q, one can expand 2/pq using the identity 2/pq = 1/aq + 1/apq, where a = p + 1/2. For instance, applying this method for pq = 21 gives p = 3, q = 7, and a = 3+1/2 = 2, producing the expansion 2/21 = 1/14 + 1/42 from the Rhind papyrus. Some authors have preferred to write this expansion as 2/A × A/pq, where A = p + 1; replacing the second term of this product by p/pq + 1/pq, applying the distributive law to the product, and simplifying leads to an expression equivalent to the first expansion described here. This method appears to have been used for many of the composite numbers in the Rhind papyrus, but there are exceptions, notably 2/35, 2/91, and 2/95.
Methods for semiprime denominators (2)
One can also expand 2/pq as 1/pr + 1/qr, where r = p + q/2. For instance, Ahmes expands 2/35 = 1/30 + 1/42, where p = 5, q = 7, and r = 5+7/2 = 6. Later scribes used a more general form of this expansion, n/pq = 1/pr + 1/qr, where r = p + q/n, which works when p + q is a multiple of n (Eves 1953). If the semiprime denominator is square number, then r = p = q, and 2/pq = 1/pr + 1/pr, so this method is not available for square semiprime denominators.
Methods for some other composite denominators
For some other composite denominators, the expansion for 2/pq has the form of an expansion for 2/q with each denominator multiplied by p. For instance, 95=5×19, and 2/19 = 1/12 + 1/76 + 1/114 (as can be found using the method for primes with A = 12), so 2/95 = 1/5×12 + 1/5×76 + 1/5×114 = 1/60 + 1/380 + 1/570. This expression can be simplified as 1/380 + 1/570 = 1/228 but the Rhind papyrus uses the unsimplified form.
Methods for 101
The final (prime) expansion in the Rhind papyrus, 2/101, does not fit any of these forms, but instead uses an expansion 2/p = 1/p + 1/2p + 1/3p + 1/6p that may be applied regardless of the value of p. That is, 2/101 = 1/101 + 1/202 + 1/303 + 1/606. A related expansion was also used in the Egyptian Mathematical Leather Roll for several cases.
Expression table of Rhind Papyrus
The expression table of 2/n in Rhind Papyrus is described as follows. The fraction 2/3 is denoted by a spacial sign. The expression of sum of distinct unit fractions is not unique, but the only expression is described in Rhind Papyrus, it is not simplest one. For expamle, the simplest expression 2/13 = 1/7 + 1/91, but the expression 2/13 = 1/8 + 1/52 + 1/104 is described in Rhind Papyrus.
The cell whose background color is Aqua expresses the expansion which is described in Rhind Papyrus.
Later usage
For more information on this subject, see Liber Abaci and Greedy algorithm for Egyptian fractions.Egyptian fraction notation continued to be used in Greek times and into the Middle Ages, despite complaints as early as Ptolemy's Almagest about the clumsiness of the notation compared to alternatives such as the Babylonian base-60 notation. An important text of medieval mathematics, the Liber Abaci (1202) of Leonardo of Pisa (more commonly known as Fibonacci), provides some insight into the uses of Egyptian fractions in the Middle Ages, and introduces topics that continue to be important in modern mathematical study of these series.
The primary subject of the Liber Abaci is calculations involving decimal and vulgar fraction notation, which eventually replaced Egyptian fractions. Fibonacci himself used a complex notation for fractions involving a combination of a mixed radix notation with sums of fractions. Many of the calculations throughout Fibonacci's book involve numbers represented as Egyptian fractions, and one section of this book provides a list of methods for conversion of vulgar fractions to Egyptian fractions. If the number is not already a unit fraction, the first method in this list is to attempt to split the numerator into a sum of divisors of the denominator; this is possible whenever the denominator is a practical number, and Liber Abaci includes tables of expansions of this type for the practical numbers 6, 8, 12, 20, 24, 60, and 100.
The next several methods involve algebraic identities such as
In the rare case that these other methods all fail, Fibonacci suggests a greedy algorithm for computing Egyptian fractions, in which one repeatedly chooses the unit fraction with the smallest denominator that is no larger than the remaining fraction to be expanded: that is, in more modern notation, we replace a fraction x/y by the expansion
where
Fibonacci suggests switching to another method after the first such expansion, but he also gives examples in which this greedy expansion was iterated until a complete Egyptian fraction expansion was constructed:
As later mathematicians showed, each greedy expansion reduces the numerator of the remaining fraction to be expanded, so this method always terminates with a finite expansion. However, compared to ancient Egyptian expansions or to more modern methods, this method may produce expansions that are quite long, with large denominators, and Fibonacci himself noted the awkwardness of the expansions produced by this method. For instance, the greedy method expands
while other methods lead to the much better expansion
Sylvester's sequence 2, 3, 7, 43, 1807, ... can be viewed as generated by an infinite greedy expansion of this type for the number one, where at each step we choose the denominator
After his description of the greedy algorithm, Fibonacci suggests yet another method, expanding a fraction
Modern number theory
For more information on this subject, see Erdős–Graham conjecture, Znám's problem, and Engel expansion.Although Egyptian fractions are no longer used in most practical applications of mathematics, modern number theorists have continued to study many different problems related to them. These include problems of bounding the length or maximum denominator in Egyptian fraction representations, finding expansions of certain special forms or in which the denominators are all of some special type, the termination of various methods for Egyptian fraction expansion, and showing that expansions exist for any sufficiently dense set of sufficiently smooth numbers.
Open problems
For more information on this subject, see odd greedy expansion and Erdős–Straus conjecture.Some notable problems remain unsolved with regard to Egyptian fractions, despite considerable effort by mathematicians.
Guy (2004) describes these problems in more detail and lists numerous additional open problems.