Rhind Mathematical Papyrus 2/n table

The Rhind Mathematical Papyrus,^[1][2] an ancient Egyptian mathematical work, includes a mathematical table for converting rational numbers of the form 2/n into Egyptian fractions (sums of distinct unit fractions), the form the Egyptians used to write fractional numbers. The text describes the representation of 51 rational numbers. It was written in approximately 1650 BCE by Ahmes, the first writer of mathematics whose name we know. Aspects of the document may have been copied from an unknown 1850 BCE text.

The table

The following table gives the expansions listed in the papyrus.

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

This part of the Rhind Mathematical Papyrus was spread over 9 sheets of papyrus.^[3]

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: $\frac{2}{3n} = \frac{1}{2n} + \frac{1}{6n}$^[4] which can be stated equivalently as $\frac{2}{n} = \frac{1}{2} \frac{1}{n} +\frac{3}{2} \frac{1}{n}$ (n divisible by 3 in the latter equation)^[5] Other possible formulas are:^[5]

$\frac{2}{n} = \frac{1}{3} \frac{1}{n} +\frac{5}{3} \frac{1}{n}$ (n divisible by 5)

$\frac{2}{mn} = \frac{1}{m} \frac{1}{k} +\frac{1}{n} \frac{1}{k}$ (where k is the average of m and n)

$\frac{2}{n} = \frac{1}{n} + \frac{1}{2n} + \frac{1}{3n} + \frac{1}{6n}$ 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 $1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + (5 \ ro)$, 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 (2 vols.), Classics in Mathematics Education 8, Oberlin: Mathematical Association of America. Reprint, Reston: National Council of Teachers of Mathematics, 1979, ISBN 0-87353-133-7.
2. Robins, Gay; Shute, Charles (1987), The Rhind Mathematical Papyrus: an Ancient Egyptian Text, London: British Museum Press.
3. Spalinger, Anthony (1990), "The Rhind Mathematical Papyrus as a Historical Document", Studien zur Altägyptischen Kultur 17: 295–337, JSTOR 25150159.
4. Clagett, Marshall (1999), Ancient Egyptian Science, A Source Book. Volume Three: Ancient Egyptian Mathematics, Memoirs of the American Philosophical Society, American Philosophical Society, ISBN 978-0-87169-232-0.
5. Burton, David M. (2003), History of Mathematics: An Introduction, Boston: Wm. C. Brown.
6. Imhausen, A. (2002), "UC 32159", Lahun Papyri: table texts (University College London)
7. Imhausen, Annette (2007), "Egyptian mathematics", in Katz, Victor J., The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook, Princeton, NJ: Princeton University Press, pp. 1–56. See in particular pages 21–22.
8. Vymazalova, H. (2002), "The wooden tablets from Cairo: The use of the grain unit HK3T in ancient Egypt", Archiv Orientální (Charles U., Prague) 70 (1): 27–42.