An Egyptian fraction is the sum of distinct unit fractions, such as . That is, each fraction in the series has a numerator equal to 1 and a denominator that is a positive integer, and all the denominators differ from each other. The sum of an expression of this type is a positive rational number a/b; for instance the Egyptian fraction above sums to 43/48. It can be shown that every positive rational number can be represented in this way by an Egyptian fraction. This type of sum was used as a serious notation for fractions by the ancient Egyptians, continuing 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 In modern mathematical notation, Egyptian fractions have been superseded by vulgar fractions and decimal notation.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. A unit fraction is a rational number written as a fraction where the numerator is one and the denominator is a positive integer. ... A cake divided into four equal quarters. ... In algebra, a vulgar fraction consists of one integer divided by a non-zero integer. ... In algebra, a vulgar fraction consists of one integer divided by a non-zero integer. ... The integers are commonly denoted by the above symbol. ... A negative number is a number that is less than zero, such as −3. ... In mathematics, a rational number (commonly called a fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... In arithmetic, a vulgar fraction (or common fraction) consists of one integer divided by a non-zero integer. ... The decimal (base ten or occasionally denary) numeral system has ten as its base. ... Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ... Recreational mathematics includes many mathematical games, and can be extended to cover such areas as logic and other puzzles of deductive reasoning. ...

Ancient Egypt

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. Four early texts in which Egyptian fractions appear were the Egyptian Mathematical Leather Roll, the Moscow Mathematical Papyrus, the Reisner 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. The system of Egyptian numerals was a numeral system used in ancient Egypt. ... Hieroglyphic version of the Eye of Horus The Eye of Horus (originally, The Eye of Ra) is an ancient Egyptian symbol of protection and power, from the deity Horus or Ra. ... This article or section is in need of attention from an expert on the subject. ... The Middle Kingdom is a period in the history of ancient Egypt stretching from the establishment of the Eleventh Dynasty to the end of the Fourteenth Dynasty, roughly between 2030 BC and 1640 BC. The period comprises of 2 phases, the 11th Dynasty, which ruled from Thebes and the 12th... Hieroglyphic version of the Eye of Horus The Eye of Horus (originally, The Eye of Ra) is an ancient Egyptian symbol of protection and power, from the deity Horus or Ra. ... The Egyptian Mathematical Leather Roll (also referred to as EMLR) was a 10 x 17 leather roll purchased by Alexander Henry Rhind in 1858. ... The Moscow and Rhind Mathematical Papyri are two of the oldest mathematical texts discovered. ... THE REISNER PAPYRUS is one of the most basic of the hieratic mathematical texts. ... The Akhmim Wooden Tablet, is an ancient Egyptian artifact that has been dated to 2000 BC, near to the beginning of the Egyptian Middle Kingdom. ... The Rhind Mathematical Papyrus ( papyrus British Museum 10057 and pBM 10058), is named after Alexander Henry Rhind, a Scottish antiquarian, who purchased the papyrus in 1858 in Luxor, Egypt; it was apparently found during illegal excavations in or near the Ramesseum. ... Ahmes (more accurately Ahmose) was an Egyptian scribe who lived during the Second Intermediate Period. ... The Second Intermediate Period marks a period when Ancient Egypt once again fell into disarray between the end of the Middle Kingdom, and the start of the New Kingdom. ... In mathematics education, the term word problem is often used to refer to any mathematical exercise for students stated in a way that does not let them avoid awareness of the verbal way in which the problem is posed. ...

To write the unit fractions used in their Egyptian fraction notation, in hieroglyph script, the Egyptians placed the hieroglyph Hieroglyphs at the Memphis museum with a statue of Ramesses II in the background. ...

(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 reciprocal function: y = 1/x. ...

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 the usual Egyptian fraction representations.

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 1/2^k (for k = 1, 2, ..., 6), that is, 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, the subject of 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 regular Egyptian fraction notation as multiples of a ro, a unit equal to 1/320 of a hekat. Hieroglyphic version of the Eye of Horus The Eye of Horus (originally, The Eye of Ra) is an ancient Egyptian symbol of protection and power, from the deity Horus or Ra. ... In mathematics, a dyadic fraction or dyadic rational is a rational number that when written as a vulgar fraction has a denominator that is a power of two, i. ... A hekat is an ancient Egyptian weights and measures volume unit, used to measure grain, bread, and beer, and further sub-divided into other units for medical prescriptions, beginning with the oipe, hin, jar, dja and ro. ... The Akhmim Wooden Tablet, is an ancient Egyptian artifact that has been dated to 2000 BC, near to the beginning of the Egyptian Middle Kingdom. ...

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, they do not match any single identity; rather, different methods were used for prime and for composite denominators, and more than one method was used for numbers of each type: In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors, which are 1 and the prime number itself. ... A composite number is a positive integer which has a positive divisor other than one or itself. ...

• For small odd prime denominators p, the expansion 2/p = 2/(p + 1) + 2/p(p + 1) was used.

• 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) in the range p/2 < A < p. The remaining term (2A-p)/Ap was expanded by representing the number 2A-p as a sum of divisors of A and forming a fraction d/Ap for each such divisor d in this sum (Hultsch 1895, Bruins 1957). As an example, Ahmes' expansion 2/37 = 1/24 + 1/111 + 1/296 fits this pattern, with A = 24 and 2A-p = 11 = 3+8, since 3 and 8 are divisors of 24. 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.

• For composite 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 (Gardner, 2002); 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 (Gillings 1982, Gardner 2002), but there are exceptions, notably 2/35, 2/91, and 2/95 (Knorr 1982).

• 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 (mod n) (Eves, 1953).

• 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 (Eves, 1953). This expression can be simplified as 1/380 + 1/570 = 1/228 but the Rhind papyrus uses the unsimplified form.

• 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.

A practical number or panarithmic number is a positive integer n such that all preceding positive integers are a sum of distinct divisors of n. ...

Medieval mathematics

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), gives us 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. Liber Abaci (1202) is an historic book on arithmetic by Leonardo of Pisa, known later by his nickname Fibonacci. ... In mathematics, an Egyptian fraction is a representation of a natural number as a sum of unit fractions, as e. ... A medieval artists rendition of Claudius Ptolemaeus Claudius Ptolemaeus (Greek: ; c. ... Almagest is the Latin form of the Arabic name (al-kitabu-l-mijisti, i. ... Babylonian mathematics refers to any mathematics of the peoples of Mesopotamia, from the days of the early Sumerians to the fall of Babylon in 539 BC. In contrast to the sparsity of sources in Egyptian mathematics, our knowledge of Babylonian mathematics is derived from some 400 clay tablets unearthed since... Liber Abaci (1202) is an historic book on arithmetic by Leonardo of Pisa, known later by his nickname Fibonacci. ... Drawing of Leonardo Pisano Leonardo of Pisa or Leonardo Pisano (Pisa, c. ...

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 method 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. Mixed radix numeral systems are more general than the usual ones in that the numerical base may vary from position to position. ... A practical number or panarithmic number is a positive integer n such that all preceding positive integers are a sum of distinct divisors of n. ...

The next several methods involve algebraic identities such as For instance, Fibonacci represents the fraction by splitting the numerator into a sum of two numbers, each of which divides one plus the denominator: Fibonacci applies the algebraic identity above to each these two parts, producing the expansion 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 In mathematics, an Egyptian fraction is a representation of a natural number as a sum of unit fractions, as e. ...

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: and

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 instead of , and sometimes Fibonacci's greedy algorithm is attributed to Sylvester. Sylvesters sequence consists of the coprime denominators of an Egyptian fraction that adds up to 1. ... James Joseph Sylvester James Joseph Sylvester (September 3, 1814 London - March 15, 1897 Oxford) was an English mathematician. ...

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 the ancient RMP 2/p expansions.

Modern number theory

For more information on this subject, see Erdős–Graham conjecture, Znám's problem, and Engel expansion.

Modern number theorists have studied many different problems related to Egyptian fractions, including 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. The Erdős–Graham conjecture in combinatorial number theory states that, if {2,3,...} are partitioned into finitely many subsets, then one of the subsets can be used to form an Egyptian fraction representation of unity. ... Známs problem is the question of what sets of integers, of a given length, can be put together such that each integer in the set is a proper divisor of the product of the other integers in the set, plus 1. ... // Definition The Engel expansion of a positive real number x is the unique non-decreasing sequence of positive integers such that: Rational numbers have a finite Engel expansion, while irrational numbers have an infinite Engel expansion. ... In number theory, a positive integer m is called B-smooth if all prime factors of m are such that . For example, 22335654 is 5-smooth since none of its prime factors are greater than 5. ...

• The Erdős–Graham conjecture in combinatorial number theory states that, if the unit fractions are partitioned into finitely many subsets, then one of the subsets can be used to form an Egyptian fraction representation of one. That is, for every r > 0, and every r-coloring of the integers greater than one, there is a finite monochromatic subset S of these integers such that

The conjecture was proven in 2003 by Ernest S. Croot, III.

• Znám's problem and primary pseudoperfect numbers are closely related to the existence of Egyptian fractions of the form

• Egyptian fractions are normally defined as requiring all denominators to be distinct, but this requirement can be relaxed to allow repeated denominators. However, this relaxed form of Egyptian fractions does not allow for any number to be represented using fewer fractions, as any expansion with repeated fractions can be converted to an Egyptian fraction of equal or smaller length by repeated application of the replacement

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).

• Graham and Jewett (Wagon 1991) and Beeckmans (1993) proved that it is similarly possible to convert expansions with repeated denominators to (longer) Egyptian fractions, via the replacement

This method can lead to long expansions with large denominators, such as

Botts (1967) had originally used this replacement technique to show that any rational number has Egyptian fraction representations with arbitrarily large minimum denominators.

• Any fraction x/y has an Egyptian fraction representation in which the maximum denominator is bounded by

(Tenenbaum and Yokota 1990) and a representation with at most

terms (Vose 1985).

• Graham (1964) characterized the numbers that can be represented by Egyptian fractions in which all denominators are nth powers. In particular, a rational number q can be represented as an Egyptian fraction with square denominators if and only if q lies in one of the two half-open intervals

• Martin (1999) showed that any rational number has very dense expansions, using a constant fraction of the denominators up to N for any sufficiently large N.

• Engel expansion, sometimes called an Egyptian product, is a form of Egyptian fraction expansion in which each denominator is a multiple of the previous one:

In addition, the sequence of multipliers a_i is required to be nondecreasing. Every rational number has a finite Engel expansion, while irrational numbers have an infinite Engel expansion.

The Erdős–Graham conjecture in combinatorial number theory states that, if {2,3,...} are partitioned into finitely many subsets, then one of the subsets can be used to form an Egyptian fraction representation of unity. ... Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ... Ernie Croot (Ernest S. Croot III) is a mathematician principally known for his solution of the Erdős–Graham conjecture. ... Známs problem is the question of what sets of integers, of a given length, can be put together such that each integer in the set is a proper divisor of the product of the other integers in the set, plus 1. ... In mathematics, and in particular number theory, a primary pseudoperfect number is a number N that satisfies the Egyptian fraction equation where the sum is over only the prime divisors of N. Equivalently (as can be seen by multiplying this equation by N), Except for the exceptional primary pseudoperfect number... // Definition The Engel expansion of a positive real number x is the unique non-decreasing sequence of positive integers such that: Rational numbers have a finite Engel expansion, while irrational numbers have an infinite Engel expansion. ... In mathematics, an irrational number is any real number that is not a rational number, i. ...

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. In number theory, the odd greedy expansion problem concerns a method for forming Egyptian fractions in which all denominators are odd. ... The Erdős–Straus conjecture states that for all integers n ≥ 2, the rational number 4/n can be expressed as the sum of three unit fractions. ...

• The Erdős–Straus conjecture concerns the length of the shortest expansion for a fraction of the form 4/n. Does an expansion

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.

• It is unknown whether an odd greedy expansion exists for every fraction with an odd denominator. If we modify Fibonacci's greedy method so that it always chooses the smallest possible odd denominator, under what conditions does this modified algorithm produce a finite expansion? An obvious necessary condition is that the starting fraction x/y have an odd denominator y, and it is conjectured but not known that this is also a sufficient condition. It is known (Breusch 1954; Stewart 1954) that every x/y with odd y has an expansion into distinct odd unit fractions, constructed using a different method than the greedy algorithm.

• It is possible to use brute-force search algorithms to find the Egyptian fraction representation of a given number with the fewest possible terms (Stewart 1992) or minimizing the largest denominator; however, such algorithms can be quite inefficient. The existence of polynomial time algorithms for these problems, or more generally the computational complexity of such problems, remains unknown.

The Erdős–Straus conjecture states that for all integers n ≥ 2, the rational number 4/n can be expressed as the sum of three unit fractions. ... In number theory, the odd greedy expansion problem concerns a method for forming Egyptian fractions in which all denominators are odd. ... In computer science, a brute-force search consists of systematically enumerating every possible solution of a problem until a solution is found, or all possible solutions have been exhausted. ... In computational complexity theory, polynomial time refers to the computation time of a problem where the time, m(n), is no greater than a polynomial function of the problem size, n. ... Complexity theory is part of the theory of computation dealing with the resources required during computation to solve a given problem. ...

Bibliography

The Journal of Number Theory (ISSN 0022-314X), often abbreviated J. Number Theory in bibliographies, is a mathematics journal that publishes a broad spectrum of original research in number theory. ... A digital object identifier (or DOI) is a standard for persistently identifying a piece of intellectual property on a digital network and associating it with related data, the metadata, in a structured extensible way. ... Mathematical Reviews is a scientific journal edited by the American Mathematical Society offering reviews of recent mathematical papers. ... Mathematics Magazine is a a bimonthly publication of the Mathematical Association of America. ... Mathematical Reviews is a scientific journal edited by the American Mathematical Society offering reviews of recent mathematical papers. ... The American Mathematical Monthly is a mathematical journal published 10 times each year by the Mathematical Association of America since 1894. ... Howard Whitley Eves (10 January 1911 New Jersey - 6 June 2004) was an American mathematician, known for his work in geometry and the history of mathematics. ... Ronald L. Graham (born October 31, 1935) is a mathematician credited by the American Mathematical Society with being one of the principle architects of the rapid development worldwide of discrete mathematics in recent years[1]. He has done important work in scheduling theory, computational geometry, Ramsey theory, and quasi-randomness. ... link title ... Mathematical Reviews is a scientific journal edited by the American Mathematical Society offering reviews of recent mathematical papers. ... A digital object identifier (or DOI) is a standard for persistently identifying a piece of intellectual property on a digital network and associating it with related data, the metadata, in a structured extensible way. ... Mathematical Reviews is a scientific journal edited by the American Mathematical Society offering reviews of recent mathematical papers. ... Transactions of the American Mathematical Society is a monthly mathematics journal published by the American Mathematical Society. ... A digital object identifier (or DOI) is a standard for persistently identifying a piece of intellectual property on a digital network and associating it with related data, the metadata, in a structured extensible way. ... Mathematical Reviews is a scientific journal edited by the American Mathematical Society offering reviews of recent mathematical papers. ... American Journal of Mathematics, April 2006 issue. ... Mathematical Reviews is a scientific journal edited by the American Mathematical Society offering reviews of recent mathematical papers. ... Ian Stewart, FRS (b. ... Scientific American is a popular-science magazine, published (first weekly and later monthly) since August 28, 1845, making it the oldest continuously published magazine in the United States. ... Dirk Struik (September 30, 1894-October 21, 2000) was a mathematician and Marxian theoretician in the United States. ... The Journal of Number Theory (ISSN 0022-314X), often abbreviated J. Number Theory in bibliographies, is a mathematics journal that publishes a broad spectrum of original research in number theory. ... A digital object identifier (or DOI) is a standard for persistently identifying a piece of intellectual property on a digital network and associating it with related data, the metadata, in a structured extensible way. ... Mathematical Reviews is a scientific journal edited by the American Mathematical Society offering reviews of recent mathematical papers. ... The London Mathematical Society (LMS) is the leading mathematical society in England. ... Mathematical Reviews is a scientific journal edited by the American Mathematical Society offering reviews of recent mathematical papers. ...

External links

• Eppstein, David. Egyptian Fractions.

• Knott, R.. Egyptian fractions.

• O'Connor, J. J.; Robertson, E. F. (2000). Mathematics in Egyptian Papyri.

• Weisstein, Eric W. Egyptian Fraction. MathWorld–A Wolfram Web Resource.

• Brown, Kevin. RMP 2/nth table.

• Gardner, Milo. History of Egyptian fractions.