Egyptian fraction
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.
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.
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
Exercises in performing this sort of fair division of food are a standard classroom example in teaching students to work with unit fractions.
[2] 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.
[3] Egyptian fraction notation was developed in the Middle Kingdom of Egypt.
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
However, as the Kahun Papyrus shows, vulgar fractions were also used by scribes within their calculations.
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.
The Egyptians also used an alternative notation modified from the Old Kingdom to denote a special set of fractions of the form
These have been called "Horus-Eye fractions" after a theory (now discredited)[4] 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
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
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: Egyptian fraction notation continued to be used in Greek times and into the Middle Ages,[9] 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.
[10] 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.
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.
where ⌈ ⌉ represents the ceiling function; since (−y) mod x < x, this method yields a finite 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.
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.
Some notable problems remain unsolved with regard to Egyptian fractions, despite considerable effort by mathematicians.
