Rhind Mathematical Papyrus 2/n table

The Rhind Mathematical Papyrus[1][2] contains, among other mathematical contents, a table of Egyptian fractions created from 2/n. The text reports 51 rational numbers converted to concise unit fraction series. The document was written in 1650 BCE by Ahmes. Aspects of the document may have been copied from an unknown 1850 BCE text.

The following table expresses 2/n (for odd n less than or equal to 101) in terms of sums of unit fractions, an introduction to scribal conversions of rational numbers to concise unit fraction series. In the Rhind Mathematical Papyrus the unit fraction decomposition was spread over 9 sheets of papyrus. Red ink was used regularly to highlight important values and procedures in the computations; the numbers included in the computations written in red ink were known as red auxiliary numbers. [3]

The 2/n table from the Rhind Mathematical Papyrus
2/3 = 1/2 + 1/6 2/5 = 1/3 + 1/15 2/7 = 1/4 + 1/28
2/9 = 1/6 + 1/18 2/11 = 1/6 + 1/66 2/13 = 1/8 + 1/52 + 1/104
2/15 = 1/10 + 1/30 2/17 = 1/12 + 1/51 + 1/68 2/19 = 1/12 + 1/76 + 1/114
2/21= 1/14 + 1/42 2/23 = 1/12 + 1/276 2/25 = 1/15 + 1/75
2/27 = 1/18 + 1/54 2/29 = 1/24 + 1/58 + 1/174 + 1/232 2/31 = 1/20 + 1/124 + 1/155
2/33 = 1/22 + 1/66 2/35 = 1/30 + 1/42 2/37 = 1/24 + 1/111 + 1/296
2/39 = 1/26 + 1/78 2/41 = 1/24 + 1/246 + 1/328 2/43 = 1/42 + 1/86 + 1/129 + 1/301
2/45 = 1/30 + 1/90 2/47 = 1/30 + 1/141 + 1/470 2/49 = 1/28 + 1/196
2/51 = 1/34 + 1/102 2/53 = 1/30 + 1/318 + 1/795 2/55 = 1/30 + 1/330
2/57 = 1/38 + 1/114 2/59 = 1/36 + 1/236 + 1/531 2/61 = 1/40 + 1/244 + 1/488 + 1/610
2/63 = 1/42 + 1/126 2/65 = 1/39 + 1/195 2/67 = 1/40 + 1/335 + 1/536
2/69 = 1/46 + 1/138 2/71 = 1/40 + 1/568 + 1/710 2/73 = 1/60 + 1/219 + 1/292 + 1/365
2/75 = 1/50 + 1/150 2/77 = 1/44 + 1/308 2/79 = 1/60 + 1/237 + 1/316 + 1/790
2/81 = 1/54 + 1/162 2/83 = 1/60 + 1/332 + 1/415 + 1/498 2/85 = 1/51 + 1/255
2/87 = 1/58 + 1/174 2/89 = 1/60 + 1/356 + 1/534 + 1/890 2/91 = 1/70 + 1/130
2/93 = 1/62 + 1/186 2/95 = 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

Proposed explanations for the way that rational numbers were converted to concise unit fraction decompositions have varied since 1895. Suggestions by Gillings included five different techniques. Problem 61 in the Rhind Mathematical Papyrus gives one formula: \frac{2}{3n} = \frac{1}{2n} %2B \frac{1}{6n} [4] which can be stated equivalently as \frac{2}{n} = \frac{1}{2} \frac{1}{n} %2B\frac{3}{2} \frac{1}{n} (n divisible by 3 in the latter equation)[5] Other possible formulas are[5]:

\frac{2}{n} = \frac{1}{3} \frac{1}{n} %2B\frac{5}{3} \frac{1}{n} (n divisible by 5)
\frac{2}{mn} = \frac{1}{m} \frac{1}{k} %2B\frac{1}{n} \frac{1}{k} (where k is the average of m and n)
\frac{2}{n} = \frac{1}{n} %2B \frac{1}{2n} %2B \frac{1}{3n} %2B \frac{1}{6n} 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 denominators.[5] Others have suggested only one method was used by Ahmes which used multiplicative factors similar to least common multiples.

Comparison to other table texts

An older ancient Egyptian papyrus contained a similar table of Egyptian fractions, the Lahun Mathematical Papyri, written around 1850 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.[6]

The Egyptian Mathematical Leather Roll (EMLR), circa 1,900 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. [7]

The Akhmim wooden tablet wrote fractions in the form 1/n in terms of sums of hekat rational numbers, 1/3, 1/7, 1/10, 1/11 and 1/13. In this document a two-part set of fractions were written in terms of Eye of Horus fractions which were fractions of the form \frac{1}{2^k} 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 %2B 1/4 %2B 1/8 %2B 1/16 %2B 1/32 %2B 1/64 %2B (5 \ ro), which equals 1. [8]

References

  1. ^ Chace, Arnold Buffum. 1927-1929. The Rhind Mathematical Papyrus: Free Translation and Commentary with Selected Photographs, Translations, Transliterations and Literal Translations. Classics in Mathematics Education 8. 2 vols. Oberlin: Mathematical Association of America. (Reprinted Reston: National Council of Teachers of Mathematics, 1979). ISBN 0-87353-133-7
  2. ^ Robins, Gay. and Charles Shute (1987) The Rhind Mathematical Papyrus: an Ancient Egyptian Text. London, British Museum Press.
  3. ^ Anthony Spalinger , The Rhind Mathematical Papyrus as a Historical Document, Studien zur Altägyptischen Kultur, Bd. 17 (1990), pp. 295-, Helmut Buske Verlag GmbH
  4. ^ Clagett, Marshall Ancient Egyptian Science, A Source Book. Volume Three: Ancient Egyptian Mathematics (Memoirs of the American Philosophical Society) American Philosophical Society. 1999 ISBN 978-0871692320
  5. ^ a b c Burton, David M. (2003) History of Mathematics: An Introduction. Boston Wm. C. Brown.
  6. ^ A. Imhausen, UC 32159 University College London, 2002
  7. ^ Annette Imhausen, in The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook, Edited by Victor J. Katz, 2007, pp. 21–22
  8. ^ Vymazalova, H. "The Wooden Tablets from Cairo: The Use of the Grain Unit HK3T in Ancient Egypt." Archiv Orientalai, Charles U., Prague, pp. 27–42, 2002.

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