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An '''Egyptian fraction''' is a finite sum of distinct [[unit fraction]]s, such as
<math display=block>\frac{1}{2}+\frac{1}{3}+\frac{1}{16}.</math>
That is, each [[
== 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.<ref>{{harvtxt|Dick|Ogle|2018}}; {{harvtxt
<math display=block>\frac{5}{8}=\frac{1}{2}+\frac{1}{8}</math>
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. Exercises in performing this sort of [[fair division]] of food are a standard classroom example in teaching students to work with unit fractions.{{sfnp|Wilson|Edgington|Nguyen|Pescosolido|2011}}
Egyptian fractions can provide a solution to [[rope-burning puzzle]]s, 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 <math>1/x</math>, burning a rope so that it always has <math>x</math> 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.{{sfnp|Winkler|2004}}
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=== Notation ===
To write the unit fractions used in their Egyptian fraction notation, in hieroglyph script, the Egyptians placed the [[Egyptian hieroglyphs|hieroglyph]]:
{| border=0 style="margin-left: 1.6em;
|}
(''er'', "<nowiki>[one]</nowiki> among" or possibly ''re'', mouth) above a number to represent the [[Multiplicative inverse|reciprocal]] of that number. Similarly in hieratic script they drew a line over the letter representing the number. For example:
{|
|<hiero>D21:Z1*Z1*Z1</hiero>
| style="padding-right:1em;" |<math>= \frac{1}{3}</math>
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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.
{| cellpadding="1em" style="margin-left: 1.6em;
|<hiero>Aa13</hiero>
| style="padding-right:1em;" |<math>= \frac{1}{2}</math>
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* For small odd prime denominators <math>p</math>, the expansion <math display=block>\frac{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}{p} = \frac{1}{A} + \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 <math>(2A-p)/Ap</math> was expanded by representing the number <math>2A-p</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{1}{24}+\tfrac{1}{111}+\
* For some composite denominators, factored as <math>p\cdot q</math>, the expansion for <math>\tfrac{2}{pq}</math> has the form of an expansion for <math>\tfrac{2}{p}</math> with each denominator multiplied by <math>q</math>. 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}=\frac{1}{p(p+q)/2}+\frac{1}{q(p+q)/2}.</math> For instance, Ahmes expands <math>\tfrac{2}{35}=\tfrac{2}{5\cdot 7}=\tfrac{1}{30}+\tfrac{1}{42}</math>. Later scribes used a more general form of this expansion, <math display=block>\frac{n}{pq}=\frac{1}{p(p+q)/n}+\frac{1}{q(p+q)/n},</math> which works when <math>p+q</math> is a multiple of <math>n</math>.{{sfnp|Eves|1953}}
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== See also ==
*[[List of sums of reciprocals]]
*[[17-animal inheritance puzzle]]
== Notes ==
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| volume = 43
| year = 1993
| issue = 2
}}
*{{citation
| doi = 10.2307/2688508
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| volume = 14
| year = 1964
| doi=10.2140/pjm.1964.14.85| s2cid = 2629869 }}
*{{citation
| last = Graham | first = Ronald L. | authorlink = Ronald Graham
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| publisher = János Bolyai Math. Soc., Budapest
| series = Bolyai Soc. Math. Stud.
| title =
| contribution-url = http://www.math.ucsd.edu/~ronspubs/13_03_Egyptian.pdf
| volume = 25
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| title = Mathematica in Action
| year = 1999}}
*{{citation
| last1 = Wilson | first1 = P. Holt
| last2 = Edgington | first2 = Cynthia P.
| last3 = Nguyen | first3 = Kenny H.
| last4 = Pescosolido | first4 = Ryan C.
| last5 = Confrey | first5 = Jere
| date = November 2011
| doi = 10.5951/mathteacmiddscho.17.4.0230
| issue = 4
| journal = Mathematics Teaching in the Middle School
| jstor = 10.5951/mathteacmiddscho.17.4.0230
| pages = 230–236
| title = Fractions: how to fair share
| volume = 17}}
*{{citation
| last = Winkler | first = Peter | author-link = Peter Winkler
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| publisher = A K Peters
| title = Mathematical Puzzles: A Connoisseur's Collection
| year = 2004}}
*{{citation
| last = Yokota | first = Hisashi
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| title = On a problem of Bleicher and Erdős
| volume = 30
| year = 1988
}}
{{refend}}
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