The Rhind Mathematical Papyrus

An Egyptian fraction is a finite sum of distinct unit fractions, such as

That is, each fraction in the expression has a numerator equal to 1 and a denominator that is a positive integer, and all the denominators differ from each other. The value of an expression of this type is a positive rational number ; for instance the Egyptian fraction above sums to . Every positive rational number can be represented by an Egyptian fraction. Sums of this type, and similar sums also including and as summands, were used as a serious notation for rational numbers by the ancient Egyptians, and continued to be used by other civilizations into medieval times. In modern mathematical notation, Egyptian fractions have been superseded by vulgar fractions and decimal notation. However, Egyptian fractions continue to be an object of study in modern number theory and recreational mathematics, as well as in modern historical studies of ancient mathematics.

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 food or other objects into equal shares.[1] 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, for example 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.

Egyptian fractions can provide a solution to rope-burning puzzles, in which a given duration is to be measured by igniting non-uniform ropes which burn out after a unit time. Any rational fraction of a unit of time can be measured by expanding the fraction into a sum of unit fractions and then, for each unit fraction , burning a rope so that it always has simultaneously lit points where it is burning. For this application, it is not necessary for the unit fractions to be distinct from each other. However, this solution may need an infinite number of re-lighting steps.[2]

Early history

Further information: Egyptian numerals and Egyptian mathematics

Egyptian fraction notation was developed in the Middle Kingdom of Egypt. 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 , 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. Tables of expansions for 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.

Notation

To write the unit fractions used in their Egyptian fraction notation, in hieroglyph script, the Egyptians placed the hieroglyph

D21

(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:

D21
Z1 Z1 Z1
D21
V20

The Egyptians had special symbols for , , and that were used to reduce the size of numbers greater than when such numbers were converted to an Egyptian fraction series. The remaining number after subtracting one of these special fractions was written as a sum of distinct unit fractions according to the usual Egyptian fraction notation.

Aa13
D22
D23

The Egyptians also used an alternative notation modified from the Old Kingdom to denote a special set of fractions of the form (for ) and sums of these numbers, which are necessarily dyadic rational numbers. These have been called "Horus-Eye fractions" after a theory (now discredited)[3] 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 of a hekat.

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 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:

Later usage

Further information: Liber Abaci and Greedy algorithm for Egyptian fractions

Egyptian fraction notation continued to be used in Greek times and into the Middle Ages,[8] despite complaints as early as Ptolemy's Almagest about the clumsiness of the notation compared to alternatives such as the Babylonian base-60 notation. Related problems of decomposition into unit fractions were also studied in 9th-century India by Jain mathematician Mahāvīra.[9] An important text of medieval European 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[10] 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

For instance, Fibonacci represents the fraction 8/11 by splitting the numerator into a sum of two numbers, each of which divides one plus the denominator: 8/11 = 6/11 + 2/11. Fibonacci applies the algebraic identity above to each these two parts, producing the expansion 8/11 = 1/2 + 1/22 + 1/6 + 1/66. Fibonacci describes similar methods for denominators that are two or three less than a number with many factors.

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 ⌈ ⌉ represents the ceiling function; since (−y) mod x < x, this method yields a finite expansion.

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: 4/13 = 1/4 + 1/18 + 1/468 and 17/29 = 1/2 + 1/12 + 1/348.

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 shorter 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 1, where at each step we choose the denominator y/x ⌋ + 1 instead of y/x, and sometimes Fibonacci's greedy algorithm is attributed to James Joseph Sylvester.

After his description of the greedy algorithm, Fibonacci suggests yet another method, expanding a fraction a/b by searching for a number c having many divisors, with b/2 < c < b, replacing a/b by ac/bc, and expanding ac as a sum of divisors of bc, similar to the method proposed by Hultsch and Bruins to explain some of the expansions in the Rhind papyrus.

Modern number theory

Further information: Erdős–Graham problem, 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

Further information: 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.

See also

Notes

References