Egyptian fraction: Difference between revisions

{{shortShort description|Finite sum of distinct unit fractions}}

[[File:Rhind_Mathematical_PapyrusRhind Mathematical Papyrus.jpg|thumb|300px|The [[Rhind Mathematical Papyrus]]]]

:<math display=block>\frac{1}{2}+\frac{1}{3}+\frac{1}{16}.</math>

That is, each [[Fraction (mathematics)|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 number|positive]] [[rational number]] {{<math|''>\tfrac{a''/''}{b''}}</math>; for instance the Egyptian fraction above sums to <math>\tfrac{43/}{48}</math>. Every positive rational number can be represented by an Egyptian fraction. Sums of this type, and similar sums also including <math>\tfrac{2/}{3}</math> and <math>\tfrac{3/}{4}</math> as [[summand]]s, 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 In modern mathematical notation, Egyptian fractions have been superseded by [[vulgar fraction]]s and [[decimal]] notation.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 [[History of mathematics|ancient mathematics]].

For instance, Egyptian fractions can help in dividing afood numberor ofother objects into equal shares (Knott).<ref>{{harvtxt|Dick|Ogle|2018}}; {{harvtxt|Koshaleva|Kreinovich|2021}}</ref> For example, if one wants to divide 5 pizzas equally among 8 diners, the Egyptian fraction

{| align="center" border=0 cellpadding=0.5em style="margin-left: 1.6em;

The Egyptians had special symbols for <math>\tfrac{1/}{2}</math>, <math>\tfrac{2/}{3}</math>, and <math>\tfrac{3/}{4}</math> that were used to reduce the size of numbers greater than <math>\tfrac{1/}{2}</math> 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.

The Egyptians also used an alternative notation modified from the Old Kingdom to denote a special set of fractions of the form <math>1/2<sup>''^k''</supmath> (for ''<math>k'' = 1, 2, ...\dots, 6</math>) and sums of these numbers, which are necessarily [[dyadic rational|dyadic rational]] numbers]]. These have been called "Horus-Eye fractions" after a theory (now discredited)<ref>{{harvtxt|Ritter|2002}}. See also {{harvtxt|Katz|2007}} and {{harvtxt|Robson|Stedall|2009}}.</ref> 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 (volume unit)|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 <math>\tfrac{1/}{320}</math> 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 <math>\tfrac{2/''}{n''}</math> 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 number|prime]] and for [[composite number|composite]] denominators, and more than one identity fits the numbers of each type:

* For small odd prime denominators ''<math>p''</math>, the expansion <math display=block>\frac{2}{nowrap|1=2/''p''} = \frac{1/(}{(''p'' + 1) / 2)} + \frac{1/''}{p''((''p'' + 1) / 2)}}</math> was used.

* For larger prime denominators, an expansion of the form <math display=block>\frac{2}{nowrap|1=2/''p''} = \frac{1/''}{A''} + (2''A'' &minus; ''\frac{2A-p'')/''}{Ap''}}</math> was used, where ''<math>A''</math> is a number with many divisors (such as a [[practical number]]) between ''<math>\tfrac{p''/}{2}</math> and ''<math>p''</math>. The remaining term {{nowrap|<math>(2''A'' &minus; ''2A-p'')/''Ap''}}</math> was expanded by representing the number {{nowrap|(2''A'' &minus; ''<math>2A-p'')</''Ap''}}math> as a sum of divisors of ''<math>A''</math> and forming a fraction ''<math>\tfrac{d''/''}{Ap''}</math> for each such divisor ''<math>d''</math> in this sum.<ref>{{harvtxt|Hultsch|1895}}; {{harvtxt|Bruins|1957}}</ref> As an example, Ahmes' expansion <math>\tfrac{2}{37}=\tfrac{nowrap|1/}{24 }+ \tfrac{1/}{111 }+ \tfrac{1/}{296}} for 2</37math> fits this pattern with {{nowrap|1=''<math>A'' = 24}}</math> and {{nowrap|1=(2''A'' &minus; ''<math>2A-p'')/''Ap'' = 11 = 8+3 + 8}}</math>, as <math>\tfrac{1}{nowrap|1111}=1/\tfrac{8}{24\cdot + 137}</111math> +and <math>\tfrac{1/}{296 }= 1/24 + \tfrac{3/(}{24 &times;\cdot 37) + 8}</(24 &times; 37)}}math>. There may be many different expansions of this type for a given ''<math>p''</math>; 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 some composite denominators, factored as ''<math>p''&times;''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,\cdot ''q'' = 7, and ''a'' = (3+1)</2 = 2math>, producing the expansion for <math>\tfrac{2}{pq}</21math> = 1/14 + 1/42 fromhas the Rhindform papyrus.of Some authors have preferred to write thisan expansion asfor <math>\tfrac{2/''A'' &times; ''A''/''pq'', where ''A'' = ''}{p''+1;{{sfnp|Gardner|2002}}</math> replacingwith theeach seconddenominator term of this productmultiplied by ''p''<math>q</''pq'' + 1/''pq'', applying the distributive law to the product, and simplifying leads to an expression equivalent to the first expansion described heremath>. This method appears to have been used for many of the composite numbers in the Rhind papyrus,<ref>{{harvtxt|Gillings|1982}}; {{harvtxt|Gardner|2002}}</ref> but there are exceptions, notably <math>\tfrac{2/}{35}</math>, <math>\tfrac{2/}{91}</math>, and <math>\tfrac{2/}{95}</math>.{{sfnp|Knorr|1982}}

* One can also expand <math display=block>\frac{2/''}{pq'' as }=\frac{1}{p(p+q)/''pr'' 2}+ \frac{1/''qr'', where ''r'' = }{q(''p''+''q'')/2}.</math> For instance, Ahmes expands <math>\tfrac{2/}{35 }=\tfrac{2}{5\cdot 7}=\tfrac{1/}{30 }+ \tfrac{1/}{42, where ''p'' = 5, ''q'' = 7, and ''r'' = (5+7)}</2 = 6math>. Later scribes used a more general form of this expansion, ''<math display=block>\frac{n''/''}{pq'' }= \frac{1}{p(p+q)/''pr'' n}+ \frac{1/''qr'', where ''r'' =}{q(''p'' + ''q'')/''n''},</math> which works when ''<math>p'' + ''q''</math> is a multiple of ''<math>n''</math>.{{sfnp|Eves|1953}}

== Later usage ==

:''For more information on this subject, see ''[[{{further|Liber Abaci]]'' and [[|Greedy algorithm for Egyptian fractions]].''}}

Egyptian fraction notation continued to be used in Greek times and into the Middle Ages,{{sfnp|Struik|1967}} despite complaints as early as [[Ptolemy]]'s [[Almagest]] about the clumsiness of the notation compared to alternatives such as the [[Babylonian mathematics|Babylonian]] [[sexagesimal|base-60 notation]]. Related problems of decomposition into unit fractions were also studied in 9th-century India by Jain mathematician [[Mahāvīra (mathematician)|Mahāvīra]].{{sfnp|Kusuba|2004}} 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.

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: <math>\tfrac{4}{nowrap|1={{sfrac|4|13}} =\tfrac {1}{sfrac|1|4}} +\tfrac {1}{sfrac|1|18}} +\tfrac {1}{sfrac|1|468}</math>}}} and <math>\tfrac{17}{nowrap|1={{sfrac|17|29}} =\tfrac {1}{sfrac|1|2}} +\tfrac {1}{sfrac|1|12}} +\tfrac {1}{sfrac|1|348}}}}.</math>

:<math display=block>\frac{5}{121}=\frac{1}{25}+\frac{1}{757}+\frac{1}{763309763\,309}+\frac{1}{873960180913873\,960\,180\,913}+\frac{1}{15276127956420934188462251\,527\,612\,795\,642\,093\,418\,846\,225},</math>

:<math display=block>\frac{5}{121}=\frac{1}{33}+\frac{1}{121}+\frac{1}{363}.</math>

[[Sylvester's sequence]] 2, 3, 7, 43, 1807, ... can be viewed as generated by an infinite greedy expansion of this type for the number one1, where at each step we choose the denominator <math>\lfloor{{nowrap|⌊ {{sfrac|''y/''|''x\rfloor''}} ⌋ + 1</math>}} instead of <math>\lceil{{nowrap|⌈ {{sfrac|''y/''|''x\rceil</math>''}} ⌉}}, and sometimes Fibonacci's greedy algorithm is attributed to [[James Joseph Sylvester|Sylvester]].

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

*The [[Erdős–Graham conjecture]] in [[Number theory|combinatorial number theory]] states that, if the integers greater than 1 are partitioned into finitely many subsets, then one of the subsets has a finite subset of itself whose reciprocals sum to one. That is, for every {{nowrap|''r'' > 0}}, and every ''r''-coloring of the integers greater than one, there is a finite monochromatic subset ''S'' of these integers such that <math display=block>\sum_{n\in S}\frac{1}{n} = 1.</math> The conjecture was proven in 2003 by [[Ernest S. Croot III]].

*[[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: <math display=block>x=\frac{1}{a_1}+\frac{1}{a_1a_2}+\frac{1}{a_1a_2a_3}+\cdots.</math> In addition, the sequence of multipliers ''a_i'' is required to be nondecreasing. Every rational number has a finite Engel expansion, while [[irrational number]]s have an infinite Engel expansion.

== Open problems ==

:''For more information on this subject, see [[odd{{further|Odd greedy expansion]] and [[|Erdős–Straus conjecture]].''}}

* The [[Erdős–Straus conjecture]]{{sfnp|Erdős|1950}} concerns the length of the shortest expansion for a fraction of the form {{sfrac|4/|''n''}}. Does an expansion <math display=block>\frac4n=\frac1x+\frac1y+\frac1z</math> exist for every ''n''? It is known to be true for all {{nowrap|''n'' < 10¹⁷}}, and for all but a vanishingly small fraction of possible values of ''n'', but the general truth of the conjecture remains unknown.

==Other applicationSee also ==

== Notes ==

{{reflistReflist|30em}}

== References ==

}}.

| issue = 2}}.| doi-access = free

| year = 1967}}.

| year = 1954| jstor = 2307234 }}.

| year = 1957}}.

| year = 2015}}.

| title = Az <math>\textstyle\frac{1}{x_1}+\frac{1}{x_2}+\cdots+\frac{1}{x_n}=\frac{a}{b}</math> egyenlet egész számú megoldásairól

| year = 1950}}.

| year = 1953}}.

| editor-last = Gratton-Guinness| editor-first = Ivor | editor-link=Ivor Grattan-Guinness

| year = 1982}}.

| doi=10.2140/pjm.1964.14.85| s2cid = 2629869 }}.

| title = ErdösErdős centennial

| year = 2013}}.

| year = 2004}}.

| year = 1895}}.

*{{citation|editor-first=VVictor J.|editor-last=Katz|editor-link=Victor J. Katz|title=The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook|location=Princeton|publisher=Princeton University Press|year=2007}}.

| issue = 2}}.

| issue = 1-21–2

| year = 2014| s2cid = 121871250 }}.

*{{citation|last=Ritter|first=Jim|contribution=Closing the Eye of Horus: the Rise and Fall of 'Horus-Eye Fractions'|title=Under One Sky: Astronomy and Mathematics in the ancient Near East|editor1-first=EJ.|editor1-last=Robson|editor1-link=Eleanor RobsonSteele|editor2-first=JA.|editor2-last=StedallImhausen|titleeditor2-link=TheAnnette Oxford Handbook of the History of MathematicsImhausen|location=OxfordMünster|publisher=Oxford University PressUgarit-Verlag|year=20092002|pages=297–323}}.

| year = 1954}}.

| year = 1967}}.

| year = 1921}}.

}}.

| year = 1985}}.

| year = 1999}}.

| year = 1988}}.| doi-access = free

== External links ==

*{{mathworld|urlname=EgyptianFraction|title=Egyptian Fraction|mode=cs2}}