View source for Rhind Mathematical Papyrus 2/n table

{{Short description|Mathematical table}}
The [[Rhind Mathematical Papyrus]],<ref>{{citation
 | last = Chace | first = Arnold Buffum
 | location = Oberlin
 | publisher = Mathematical Association of America
 | series = Classics in Mathematics Education
 | title = The Rhind Mathematical Papyrus: Free Translation and Commentary with Selected Photographs, Translations, Transliterations and Literal Translations (2 vols.)
 | volume = 8
 | date = 1927–1929}}. Reprint, Reston: National Council of Teachers of Mathematics, 1979, {{ISBN|0-87353-133-7}}.</ref><ref>{{citation
 | last1 = Robins | first1 = Gay | authorlink1 = Gay Robins
 | last2 = Shute | first2 = Charles | authorlink2 = Charles Shute (academic)
 | location = London
 | publisher = British Museum Press
 | title = The Rhind Mathematical Papyrus: an Ancient Egyptian Text
 | year = 1987}}.</ref> an [[ancient Egypt]]ian [[mathematical]] work, includes a [[mathematical table]] for converting [[rational number]]s of the form 2/''n'' into [[Egyptian fraction]]s (sums of distinct [[unit fraction]]s), the form the Egyptians used to write fractional numbers. The text describes the representation of 50 rational numbers. It was written during the [[Second Intermediate Period of Egypt]] (approximately 1650–1550&nbsp;BCE)<ref>{{citation|title=Mathematics in Ancient Egypt: A Contextual History|title-link= Mathematics in Ancient Egypt: A Contextual History |first=Annette|last=Imhausen|authorlink=Annette Imhausen|publisher=Princeton University Press|year=2016|isbn=9780691209074|at=[https://books.google.com/books?id=xkjWDwAAQBAJ&pg=PA65 p. 65]}}</ref> by [[Ahmes]], the first writer of mathematics whose name is known. Aspects of the document may have been copied from an unknown 1850&nbsp;BCE text.

==Table==
The following table gives the expansions listed in the [[papyrus]].
{| class="wikitable" style="margin:0 auto;"
|+ '''2/''n'' table from the Rhind Mathematical Papyrus'''
|-
| 2/3{{figure space}} = 1/2 + 1/6  || 2/5{{figure space}}{{figure space}} = 1/3 + 1/15         || 2/7{{figure space}} = 1/4 + 1/28
|-
| 2/9{{figure space}} = 1/6 + 1/18 || 2/11{{figure space}} = 1/6 + 1/66               || 2/13 = 1/8 + 1/52 + 1/104
|-
| 2/15 = 1/10 + 1/30  || 2/17{{figure space}} = 1/12 + 1/51 + 1/68           || 2/19 = 1/12 + 1/76 + 1/114
|-
| 2/21 = 1/14 + 1/42  || 2/23{{figure space}} = 1/12 + 1/276                 || 2/25 = 1/15 + 1/75 
|-
| 2/27 = 1/18 + 1/54  || 2/29{{figure space}} = 1/24 + 1/58 + 1/174 + 1/232  || 2/31 = 1/20 + 1/124 + 1/155
|-
| 2/33 = 1/22 + 1/66  || 2/35{{figure space}} = 1/30 + 1/42                  || 2/37 = 1/24 + 1/111 + 1/296
|-
| 2/39 = 1/26 + 1/78  || 2/41{{figure space}} = 1/24 + 1/246 + 1/328         || 2/43 = 1/42 + 1/86 + 1/129 + 1/301 
|-
| 2/45 = 1/30 + 1/90  || 2/47{{figure space}} = 1/30 + 1/141 + 1/470         || 2/49 = 1/28 + 1/196 
|-
| 2/51 = 1/34 + 1/102 || 2/53{{figure space}} = 1/30 + 1/318 + 1/795         || 2/55 = 1/30 + 1/330 
|-
| 2/57 = 1/38 + 1/114 || 2/59{{figure space}} = 1/36 + 1/236 + 1/531         || 2/61 = 1/40 + 1/244 + 1/488 + 1/610  
|-
| 2/63 = 1/42 + 1/126 || 2/65{{figure space}} = 1/39 + 1/195                 || 2/67 = 1/40 + 1/335 + 1/536   
|-
| 2/69 = 1/46 + 1/138 || 2/71{{figure space}} = 1/40 + 1/568 + 1/710         || 2/73 = 1/60 + 1/219 + 1/292 + 1/365
|-
| 2/75 = 1/50 + 1/150 || 2/77{{figure space}} = 1/44 + 1/308                 || 2/79 = 1/60 + 1/237 + 1/316 + 1/790   
|-
| 2/81 = 1/54 + 1/162 || 2/83{{figure space}} = 1/60 + 1/332 + 1/415 + 1/498 || 2/85 = 1/51 + 1/255 
|-
| 2/87 = 1/58 + 1/174 || 2/89{{figure space}} = 1/60 + 1/356 + 1/534 + 1/890 || 2/91 = 1/70 + 1/130 
|-
| 2/93 = 1/62 + 1/186 || 2/95{{figure space}} = 1/60 + 1/380 + 1/570         || 2/97 = 1/56 + 1/679 + 1/776   
|-
| 2/99 = 1/66 + 1/198 || 2/101 = 1/101 + 1/202 + 1/303 + 1/606      ||
|}

This part of the Rhind Mathematical Papyrus was spread over nine sheets of papyrus.<ref>{{citation
 | last = Spalinger | first = Anthony
 | journal = Studien zur Altägyptischen Kultur
 | jstor = 25150159
 | pages = 295–337
 | title = The Rhind Mathematical Papyrus as a Historical Document
 | volume = 17
 | year = 1990}}.</ref>

==Explanations==
Any rational number has infinitely many different possible expansions as a sum of unit fractions, and since the discovery of the Rhind Mathematical Papyrus mathematicians have struggled to understand how the ancient Egyptians might have calculated the specific expansions shown in this table.

Suggestions by Gillings included five different techniques. Problem 61 in the Rhind Mathematical Papyrus gives one formula:
: <math> \frac{2}{3n} = \frac{1}{2n} + \frac{1}{6n} </math>,<ref name="Clagett">{{citation
 | last = Clagett
 | first = Marshall
 | isbn = 978-0-87169-232-0
 | publisher = American Philosophical Society
 | series = Memoirs of the American Philosophical Society
 | title = Ancient Egyptian Science, A Source Book. Volume Three: Ancient Egyptian Mathematics
 | year = 1999
 | url-access = registration
 | url = https://archive.org/details/ancientegyptians03clag
 }}.</ref> which can be stated equivalently as <math>\frac{2}{n} = \frac{1}{2} \frac{1}{n} +\frac{3}{2} \frac{1}{n}</math> (''n'' divisible by 3 in the latter equation).<ref name="Burton"/>
Other possible formulas are:<ref name="Burton"/>
: <math>\,\frac{2}{n}\; = \;\frac{1}{3} \frac{1}{n} +\frac{5}{3} \frac{1}{n}</math> (''n'' divisible by 5)

: <math>\!\frac{2}{mn}\! = \frac{1}{m}\!\frac{1}{k} +\frac{1}{n} \frac{1}{k}</math> (where ''k'' is the average of ''m'' and ''n'')

: <math>\,\frac{2}{n}\; = \,\frac{1}{n} + \frac{1}{2n} + \frac{1}{3n} + \frac{1}{6n}</math>. This formula yields the decomposition for ''n'' = 101 in the table.

Ahmes was suggested to have converted 2/''p'' (where ''p'' was a [[prime number]]) by two methods, and three methods to convert 2/''pq'' [[composite number|composite]] denominators.<ref name="Burton">{{citation
 | last = Burton | first = David M.
 | location = Boston
 | publisher = Wm. C. Brown
 | title = History of Mathematics: An Introduction
 | year = 2003}}.</ref> Others have suggested only one method was used by Ahmes which used multiplicative factors similar to [[least common multiple]]s. A detailed and simple explanation of how the 2/''p'' table may have been decomposed was provided by Abdulrahman Abdulaziz. <ref>Abdulrahman A. Abdulaziz,
On the Egyptian method of decomposing 2/n into unit fractions,
Historia Mathematica,
Volume 35, Issue 1,
2008,
Pages 1-18,
ISSN 0315-0860,
https://doi.org/10.1016/j.hm.2007.03.002.</ref>

==Comparison to other table texts==
An older ancient Egyptian papyrus contained a similar table of Egyptian fractions; the [[Lahun Mathematical Papyri]], written around 1850&nbsp;BCE, is about the age of one unknown source for the Rhind papyrus. The Kahun 2/''n'' fractions were identical to the fraction decompositions given in the  Rhind Papyrus' 2/''n'' table.<ref>{{citation
 | last = Imhausen | first = A. | authorlink = Annette Imhausen
 | publisher = University College London
 | title = UC 32159
 | url = http://www.digitalegypt.ucl.ac.uk/lahun/uc32159.html
 | work = Lahun Papyri: table texts
 | year = 2002}}</ref>

The [[Egyptian Mathematical Leather Roll]] (EMLR), circa 1900&nbsp;BCE, lists decompositions of fractions of the form 1/''n'' into other unit fractions. The table consisted of 26 unit fraction series of the form 1/''n'' written as sums of other rational numbers.<ref>{{citation
 | last = Imhausen | first = Annette | authorlink = Annette Imhausen
 | editor-last = Katz | editor-first = Victor J.
 | contribution = Egyptian mathematics
 | location = Princeton, NJ
 | pages = 1–56
 | publisher = Princeton University Press
 | title = The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook
 | year = 2007}}. See in particular pages 21–22.</ref>

The [[Akhmim wooden tablet]] wrote difficult fractions of the form 1/''n'' (specifically, 1/3, 1/7, 1/10, 1/11 and 1/13) in terms of [[Eye of Horus]] fractions which were fractions of the form {{sfrac|1|2<sup>''k''</sup>}} and remainders expressed in terms of a unit called ''ro''. The answers were checked by multiplying the initial divisor by the proposed solution and checking that the resulting answer was 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + 5 ''ro'', which equals 1.<ref>{{citation
 | last = Vymazalova | first = H.
 | issue = 1
 | journal = Archiv Orientální
 | pages = 27–42
 | publisher = Charles U., Prague
 | title = The wooden tablets from Cairo: The use of the grain unit HK3T in ancient Egypt
 | volume = 70
 | year = 2002}}.</ref>

==References==
{{reflist}}

{{DEFAULTSORT:Rhind Mathematical Papyrus 2 n table}}
[[Category:Egyptian fractions]]
[[Category:Papyri from ancient Egypt]]
[[Category:2nd-millennium BC manuscripts]]
[[Category:Mathematics manuscripts]]
[[Category:Ancient Egyptian objects in the British Museum]]
[[Category:Egyptian mathematics]]