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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 ==
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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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| 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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| 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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