An unproved 20th century hypothesis, that all Egyptian fraction arithmetic was additive, severely muddled translations of Egyptian mathematical texts. The additive suggestion limited the reading of the math texts to practical topics. Other well defined points of view were de facto forbidden. For example, the 1,900 BCE Akhmim Wooden Tablet (AWT) and the 1650 BCE Rhind Mathematical Papyrus (RMP), with 40 examples, contained a non-additive pattern. Chace in 1927 reported RMP 83, Ahmes' bird feeding rate problem, by considering an additive pattern, thereby misleading the academic community. Ahmes' RMP 83 listed seven non-additive grain (hekat) portions, dividing a hekat unity (64/64) seven times, within the following bird-feeding rate context:

1. 2 geese, and a crane each eating $$(1/8 + 1/32) hekat + (3 + 1/3) ro$$ ;

2. a set-duck eating $$(1/32 + 1/64) hekat + 1 ro$$ and,

3. a set-goose, dove, and quail each eating $$(1/64) hekat + 3 ro$$

Ahmes was asked: How much grain did the seven birds eat in one day?

Ahmes' seven daily grain portions divided a hekat unity by 6 (3 times), by 20 (once) and 40 (three times) such that

a total of 5/8 hekat of grain was obtained by:

$$3/6 + 1/20 + 3/40$$

$$20/40 + 2/40 + 3/40$$

Ahmes' requested sum was not reported by Chace.

Scaled valuations of geese, crane, set-ducks and other fowls was detailed in the Kahun Papyrus (KP), showing fixed prices for buying and selling the birds. The KP text was transliterated by Griffith and published in Marshall Clagett, Ancient Egyptian Science, Vol III,1999. Clagett published standard 1920s KP and RMP transliterations suggesting the information provides complete translations. Initial and intermediate statements were omitted by the Middle Kingdom scribes, a situation that Clagett and the 1920s transliterations did not correct.

To achieve complete translations from grabled transliterations, for example, from the 1900 BCE Akhmim Wooden Tablet (AWT), Hana Vymazalova parsed and proved an aspect of AWT arithmetic in 2002. Ahmes bird-feeding method used the same arithmetic in 1650 BCE, which we now know, began and ended with (64/64), a hekat unity. The definition of scribal division context of five hekat division problems was not mentioned by Clagett. Hana Vymazalova reported complete translations of the five AWT division of a hekat proofs in 2002. Vymazalova corrected Daressy's 1906 AWT analysis.

Vymazalova's hekat unity (64/64) opened an important door to solving the entire bird feeding rate problem, and like problems reported in other MK texts. Vymazalova corrected two of Daressy's 1906 errors, the divisor n = 11 and n =- 13 cases. Vymazalova showed that all five divisions of a volume unit name hekat ended with a hekat unity written as (64/64).

The AWT hekat unity method was used by Ahmes 250 years later. Ahmes also solved a 1/6 of a hekat problem by writing:

$$(64/64)/6 = 10/64 + 4/384$$ rescaling the remainder 4/384 to 20/2220 by writing 20/6* (1/320), and naming 1/320 as ro.

The RMP 1/6 portion of the hekat steps looked like this:

$$(8 + 2)/64 + (20/6)ro$$

and,

$$(1/8 + 1/32)hekat + (3 + 1/3)ro$$

The non-additive aspects of the AWT and RMP division problems were first parsed in a 2005 study, and published in 2006. The study contrasted 40 hekat volume unit measures and found a generalized quotient and remainder hekat unity division method that limited n to the range of 1/64 < n < 64. The AWT and RMP method defined a hekat unity as (64/64), per Hana Vymazalova's 2002 paper, and divided the unity by several rational numbers n. Answers were written in binary quotients and scaled Egyptian fraction series when n was limited to the range 1/64 < n < 64. For general n values a simple quotient and remainder answer was written (namely defining hekat subunits, i.e hin = 1/10, such that 10/n hin, and dja = 1/64, such that 64/n dja, per Tanja Pemmerening's 2002 paper).

Exact binary quotient and scaled ro remainders were common in the Egyptian Middle Kingdom, 2000 BCE to 1500 BCE. RMP 83 and its division method was discussed by Chace, concluding "... the author (Ahmes) does not specify explicitly how he performed them". The broader context of the Old Kingdom Horus-Eye aspect of Ahmes' bird-feeding rate problem were jointly published in 2006 by Gardner.

In summary, theoretically expected usage rates were calculated by Ahmes and Egyptian scribes to control major commodity inventories. Knowing expected usages allowed decentralized managers to double check actual inventory usages, calculating profits, losses, paying taxes, and creating contracts of many types within a double entry accounting system. Overall, the economic context of Egyptian fractions contained theoretical elements, one being Ahmes' bird-feeding rate division method. In total, innovative Egyptian fraction mathematics facilitated centralized and decentralized business practices, including land rental contracts operated by absentee landlords.

Bibliography

1

A.B. Chace, Bull, L., Manning, H.P. and Archibald, R.C. The Rhind Mathematical Papyrus, Mathematical Association of America, Vol 1, 1927, vol 2, 1929, and reprint 1979 (NCTM).

1

A.B. Clagett, Marshall Ancient Egyptian Science, Vol 3, American Philosophical Society, Philadelphia 1999.

2

Georges Daressy, "Calculs Egyptiens du Moyan EmpireÃÃÃÃÃÃÃÃÃÃÃÃÃÃâ, Recueil de Travaux Relatifs De La Philologie et al Archaelogie Egyptiennes Et Assyriennes XXVIII, 1906, 62ÃÃÃÃÃÃÃÃÃÃÃÃÃÃâ72, Paris, 1906.

3

Milo Gardner, An Ancient Egyptian Problem and its Innovative Solution, Ganita Bharati, MD Publications Pvt Ltd, 2006.

4

Milo Gardner, The Egyptian Mathematical Leather Roll, Attested Short Term and Long Term, History of the Mathematical Sciences, Editors: Ivor Gratton-Guinness, and B.S. Yadav, Hindustan Book Agency, 119-134, 2002.

5

Richard Gillings, Mathematics in the Time of the Pharaohs, Dover Books, 1992.

6

T.E. Peet, Arithmetic in the Middle Kingdom, Journal Egyptian Archeology, 1923.

7

Tanja Pommerening, "Altagyptische Holmasse Metrologish neu Interpretiert" and relevant phramaceutical and medical knowledge, an abstract, Phillips-Universtat, Marburg, 8-11-2004, taken from "Die Altagyptschen Hohlmass, Buske-Verlag, 2005.

8

Hana Vymazalova, The Wooden Tablets from Cairo:The Use of the Grain Unit HK3T in Ancient Egypt, Archiv Orientalai, Charles U Prague, 2002.