Egyptian fraction

An Egyptian fraction is a finite series of distinct unit fractions, such as \tfrac{1}{2}+\tfrac{1}{3}+\tfrac{1}{16}. 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 a/b; for instance the Egyptian fraction above sums to 43/48. Every positive rational number can be represented by an Egyptian fraction. Sums of this type, and similar sums also including 2/3 and 3/4 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 In modern mathematical notation, Egyptian fractions have been superseded by vulgar fractions and decimal notation.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.

Motivating applications

Beyond their historical use, Egyptian fractions have some practical advantages over other representations of fractional numbers.

Comparing the size of some fractions

Egyptian fractions sometimes make it easier to compare the sizes of a pair of fractions (Knott). For example, if one wants to find out whether 4/5 is larger than 3/4, one could convert them to Egyptian fractions with shared parts:

Hence, 4/5 is larger by 1/20 .

Similarly, comparing 3/11 and 2/7:

As 1/44 < 1/28, this implies that 3/11 < 2/7 .

Equally distributing objects

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.
Eye of Horus

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.

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
= \frac{1}{3}
D21
V20
= \frac{1}{10}

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.

Aa13
= \frac{1}{2}
D22
= \frac{2}{3}
D23
= \frac{3}{4}

The Egyptians also used an alternative notation modified from the Old Kingdom and based on the parts of the Eye of Horus 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 "Horus-Eye fractions" 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.

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:

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 (Struik 1967), 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 (Sigler 2002, chapter II.7) 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 \tfrac{a}{ab-1}=\tfrac{1}{b}+\tfrac{1}{b(ab-1)}. For instance, Fibonacci represents the fraction \tfrac{8}{11} by splitting the numerator into a sum of two numbers, each of which divides one plus the denominator: \tfrac{8}{11}=\tfrac{6}{11}+\tfrac{2}{11}. Fibonacci applies the algebraic identity above to each these two parts, producing the expansion \tfrac{8}{11}=\tfrac{1}{2}+\tfrac{1}{22}+\tfrac{1}{6}+\tfrac{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

\frac{x}{y}=\frac{1}{\lceil y/x\rceil}+\frac{(-y)\,\bmod\, x}{y\lceil y/x\rceil},

where \lceil \ldots \rceil represents the ceiling function.

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: \tfrac{4}{13}=\tfrac{1}{4}+\tfrac{1}{18}+\tfrac{1}{468} and \tfrac{17}{29}=\tfrac{1}{2}+\tfrac{1}{12}+\tfrac{1}{348}.

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

\frac{5}{121}=\frac{1}{25}+\frac{1}{757}+\frac{1}{763309}+\frac{1}{873960180913}+\frac{1}{1527612795642093418846225},

while other methods lead to the much better expansion

\frac{5}{121}=\frac{1}{33}+\frac{1}{121}+\frac{1}{363}.

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 \lfloor y/x\rfloor+1 instead of \lceil y/x\rceil, and sometimes Fibonacci's greedy algorithm is attributed to 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

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.

\sum_{n\in S}1/n = 1.
The conjecture was proven in 2003 by Ernest S. Croot, III.
\sum\frac1{x_i} + \prod\frac1{x_i}=1.
For instance, the primary pseudoperfect number 1806 is the product of the prime numbers 2, 3, 7, and 43, and gives rise to the Egyptian fraction 1 = 1/2 + 1/3 + 1/7 + 1/43 + 1/1806.
\frac1k+\frac1k=\frac2{k+1}+\frac2{k(k+1)}
if k is odd, or simply by replacing 1/k+1/k by 2/k if k is even. This result was first proven by Takenouchi (1921).
\frac1k+\frac1k=\frac1k+\frac1{k+1}+\frac1{k(k+1)}.
This method can lead to long expansions with large denominators, such as
\frac45=\frac15+\frac16+\frac17+\frac18+\frac1{30}+\frac1{31}+\frac1{32}+\frac1{42}+\frac1{43}+\frac1{56}+
\frac1{930}+\frac1{931}+\frac1{992}+\frac1{1806}+\frac1{865830}.
Botts (1967) had originally used this replacement technique to show that any rational number has Egyptian fraction representations with arbitrarily large minimum denominators.
O\left(y \log y (\log\log y)^4 (\log\log\log y)^2\right)
(Yokota 1988) and a representation with at most
O\left(\sqrt{\log y}\right)
terms (Vose 1985). It is known that \Omega(\log\log y) terms may sometimes be needed (for instance for the fractions in the sequence 1/2, 2/3, 6/7, 42/43, 1806/1807, ... whose denominators form Sylvester's sequence), and conjectured that Vose's bound can be improved to O(\log\log y) (Erdős 1950). It is also possible to find representations in which both the maximum denominator and the number of terms are small (Tenenbaum & Yokota 1990).
\left[0,\frac{\pi^2}{6}-1\right)\cup\left[1,\frac{\pi^2}{6}\right).
x=\frac{1}{a_1}+\frac{1}{a_1a_2}+\frac{1}{a_1a_2a_3}+\cdots.
In addition, the sequence of multipliers ai is required to be nondecreasing. Every rational number has a finite Engel expansion, while irrational numbers have an infinite Engel expansion.
\frac{5}{12}=\frac{1}{4}+\frac{1}{10}+\frac{1}{15}=\frac{1}{5}+\frac{1}{6}+\frac{1}{20}.
Unlike the ancient Egyptians, they allow denominators to be repeated in these expansions. They apply their results for this problem to the characterization of free products of Abelian groups by a small number of numerical parameters: the rank of the commutator subgroup, the number of terms in the free product, and the product of the orders of the factors.

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.

\frac4n=\frac1x+\frac1y+\frac1z
exist for every n? It is known to be true for all n < 1014, and for all but a vanishingly small fraction of possible values of n, but the general truth of the conjecture remains unknown.

Guy (2004) describes these problems in more detail and lists numerous additional open problems.

See also

References

External links

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