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In the study of ancient Egyptian mathematics, red auxiliary numbers were additive numbers that summed to a numerator used in Middle Kingdom arithmetic problems. In the 1650 BCE Rhind Mathematical Papyrus (RMP), the additive set of divisors were written in red ink. Red auxiliary numbers were used for several purposes The red auxiliary numbers were divisors of a greatest common divisor (GCD) that was used to generally convert 2/n to optimized, but not optimal, unit fraction series. The main purpose created 2/n tables by hi-lighting scribal skills in converting rational numbers to optimized, but not optimal, unit fraction series. The RMP 2/n table reported Egyptian fraction series by multiplying 2/n by a LCM m, written as a unity (m/m), that obtained 2m/mn. Ahmes the RMP author, practiced the selection of red auxiliary numbers in RMP 21, 22, and 23 and detailed other aspects of the red number method in RMP 36. The scaled 2m/mn rational number was converted to a unit fraction series by finding the best set of divisors of (GCD) mn, as discussed below.

For example, Ahmes converted 2/43 by LCM 42, thinking:

2/43×(42/42) = 84/1806

To parse 84/1806 Ahmes selected the best divisors of 1806 from {43, 42, 21, 14, 7, 6, 4, 3, 2, 1}, denoting the divisors that best summed to numerator 84 in red ink.

Ahmes chose 43 + 21 + 14 + 6 rather than 43 + 21 + 14 + 4 + 2, or 43 + 41 + 14 + 3 + 2 + 1,

such that:

2/43 = 84/1806 = (43 + 21 + 14 + 6)/1806 = 1/42 + 1/86 + 1/129 + 1/301

The 2/n table method that was validated in RMP 36 with an explicit conversion of 3/53 scaled to 60/1060. The best divisors of 1060 were summed to 60 within (53 + 4 + 2 + 1)/1060 and 1/20 + 1/265 + 1/530 + 1/1060. The red number method was used over six times in RMP 36 and over seven times in RMP 37.

Scribal red numbers, LCMs and GCDs were known for over 130 years before being rigorously parsed. Math historians failed to parse aspects of red numbers seen in the Rhind Mathematical Papyrus and the (RMP 2/n table) in the manner recorded by Ahmes. Math historians recognized red numbers were connected to LCMs, but few explicit details were described in the 2/n table. In the 21st century math historians began to parse Middle Kingdom origins and applications of red numbers. Ahmes practiced finding LCMs and its red divisor factored aspect of GCDs in RMP 21, 22 and 23, a topic introduced by George G. Joseph, "Crest of the Peacock" in 1991:

On page 37, example 3.7 Joseph reports:

Complete 2/3 + 1/4 + 1/28 to 1.

This meant: solve 2/3 + 1/4 + 1/28 + x = 1 (example 3.7)

The lowest common denominator (LCM) is not 28, but rather 42. Modern students would likely multiple 3 times 28 finding an LCM of 84. But 42 was sufficient for Ahmes, and Egyptian scribes, as noted by:

84/3 + 42/4 + 42/28 + 42x = 42 (example 3.7.1)

was written out in fractions

28 + (10 + 1/2) + (1 + 1/2) + 2 = 42 (example 3.7.2)

with 42 marked in red, and not our modern algebra form that would have written 42x.

The unknown fraction x is found by solving

42x = 2, or x = 2/42 = 1/21 (example 3.7.3)

meant that

2/3 + 1/4 + 1/28 + 1/21 = 1 (example 3.7.4)

Additional RMP problems asked Ahmes to complete a series of fractions that added up to a given number include:

RMP 21: Complete 2/3 + 1/15 + x = 1

using Least common multiple (LCM) 30 to find

60/3 + 30/15 + 30x = 30
20 + 2 + 8 = 30
30x = 8
x = 8/30 = 4/15 = (3 + 1)/15 = 1/5 + 1/15

such that:

2/3 + 1/5 + 2/15 = 1

was rewritten as:

2/3 + 1/5 + 1/10 + 1/30 = 1

and,

RMP 23: Complete 1/4 + 1/8 + 1/10 + 1/35 + 1/45 + x = 3

using LCM 45 to compute x = 1/9 + 1/40

(certain details have been left for the reader)

It is important to note that the Egyptian Mathematical Leather Roll (EMLR) and the RMP 2/n table employed LCMs within modern-like multiplication and division operations. Decoding Red numbers arithmetic and its applications have surprisingly decoded modern-like arithmetic operations that were hidden in scribal shorthand notes.

In RMP 37 LCM 72 scaled 1/4 to 72/288 and 1/8 to 72/576. Red numbers parsed 72/88 by (9 + 18 + 24 + 3 + 8 + 1 + 8 + 1)/288 and 72/576 by (8 + 36 + 18 + 9 + 1)/576, both non-optimal unit fraction series. Ahmes aligned each red number below a unit fraction showing that red denoted an inverse relationship to a unit fraction, even to numerators that were not integers.

The optimized Egyptian fraction series was an implicit topic of the RMP 2/n table. The RMP began with 1/3 of the text reporting 51 2/n optimized, but not always optimal, Egyptian fraction series by implicitly selecting optimized, but not optimal, red auxiliary numbers from the divisors, aliquot parts, of scaled denominators mn.

Red numbers defined a core method that math historians pondered for over 130 years finally parsing the method in the 21st century. Aspects of the method were exposed in a 2002 AD EMLR paper, 2006 with an Akhmim Wooden Tablet paper, and an Ebers Papyrus paper. Non-additive numerical methods connected Egyptian weights and measures, 2/n tables, and other ancient scribal methods to red numbers, with RMP 36 and RMP 37 providing the final abstract aspects of the method.

In summary, after 2002 alternative views of Middle Kingdom red numbers exposed LCMs and GCDs in updated translations of the scribal raw data. Several scribal notes of RMP problems included modern-like multiplication and division operations. Recent journal papers report Ahmes' use of red numbers, one being origin issues related to the factoring of rational numbers into its unique prime factors. Scribal division operations and applications employed modern-like quotients and remainders in a finite arithmetic system. Ahmes divided 2 by n 51 times to create a 2/n table. Ahmes also divided a hekat unity (64/64) and 320 ro, by n in two different volume based weights and measures systems. The traditional Old Kingdom duplation multiplication operationally proved the arithmetic accuracy of unit fraction answers, and was not the primary Middle Kingtdom multiplication operation.

## References

• Daressy, G. "Cairo Museum des Antiquities Egyptiennes." Catalogue General Ostraca, Volume No. 25001-25385, 1901.
• Daressy, Georges, “Calculs Egyptiens du Moyan Empire”, Recueil de Travaux Relatifs De La Philologie et l'Archéologie Egyptiennes Et Assyriennes XXVIII, 1906, 62–72.
• Gardner, Milo, "An Ancient Egyptian Problem and its Innovative Arithmetic Solution", Ganita Bharati, 2006, Vol 28, Bulletin of the Indian Society for the History of Mathematics, MD Publications, New Delhi, pp 157–173.
• Gillings, R. Mathematics in the Time of the Pharaohs. Boston, MA: MIT Press, pp. 202–205, 1972. ISBN 0-262-07045-6. (Out of print)

red auxiliaries pages:78,81,85,87.97,99,102–103,160–161,251–252

• Joseph, George G. "Crest of the Peacock", Penguin Books, 1991 (pages 71: Egyptian Division the use of 'red auxiliaries to page 73
• Peet, T. E. "Arithmetic in the Middle Kingdom." J. Egyptian Arch. 9, 91–95, 1923.
• Pommerening, Tanja, "Altagyptische Holmasse Metrologish neu Interpretiert" and relevant pharmaceutical and medical knowledge, an abstract, Phillips-Universtat, Marburg, 8-11-2004, taken from "Die Altagyptschen Hohlmass" in studien zur Altagyptischen Kulture, Beiheft, 10, Hamburg, Buske-Verlag, 2005
• Robins, Gay and Shute, Charles "Rhind Mathematical Papyrus", British Museum Press, 1987
• 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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