INTRODUCTION: The 143 year old body of literature suggests decodings aspects of Egyptian fraction mathematics. The literature touches on aspects of 4,000 year old Egyptian fractions, one focus of this post. Modern research on 1650 BCE Egyptian mathematics began in 1879, 15 years after the Rhind Mathematical Papyrus(RMP), and the Egyptian Mathematical Leather Roll (EMLR) were deeded by the family of Henry Rhind to the British Museum.

The 1879 debate began after a bootleg copy of RMP was taken from the British Museum and published in Germany. The RMP contents initiated a wide ranging debate. British, German, European, USA and Arab scholars have debated the RMP's themes beginning with its 2/n table. The arithmetic debate reached an additive acme in 1927 with Chace's view of the RMP.

Scholars continued to discuss conflicting RMP themes well after after Chace's publication. Additional Egyptian fraction texts were brought into the debate. In 1927 scholars hoped that the EMLR would shed light on the RMP and its Egyptian fraction methods. The 26 line 1800 BCE EMLR was unrolled and additively read by British Museum scholars. An anticipated deeper understanding of Egyptian arithmetic was not reported. Several early 1930's German scholars suspected that the EMLR, at some point, would provide deeper insights.

In 1933 the events of World War II virtually stopped Egyptian fraction research. Research started up again in 1945. Research slowly progressed over the next 50 years. Gillings published an excellent summary of the available Egyptian fraction texts in 1972 and indirectly mentioned the Akhmim Wooden Tablet. Gillings accepted the majority of the 1920's additive views of Peet, and Chace while adding several minor suggestions as aids to read the most popular texts.

In 2002 three publications jump-started several scholarly views on Egyptian fraction arithmetic onto new arithmetic tracks. The first was an algebraic version of 22 EMLR conversions of rational numbers created by six multiples. The multiples may have been non-additive multiples. They hinted at a deeper arithmetic likely used by Ahmes. The EMLR multiple method was soon connected to the second publication, the 1202 AD Liber Abaci's and seven rational number conversion methods published by Sigler. The Liber Abaci had been read for years in fragmentary ways. Finally the full text was translated from Latin to English. The third publication was the 1900 BCE Akhmim Wooden Tablet. This text hinted at an abstract form of Egyptian remainder arithmetic. The paper was published by Vymazalova, a Charles U. graduate student.

In 2007 a multiple method connected the first 2002 publication to the second 2002 publication. The newly reported link allowed EMLR and RMP Egyptian fraction data to be computed by an identical multiple method. It is clear by considering the entire scope of Egyptian fraction literature that six non-optimal EMLR multiples had been adapted by Ahmes, the RMP scribe, into a single optimal multiple method. Ahmes seemed to easily convert 51 2/n table rational numbers to optimized Egyptian fraction series by selecting an optimal multiple. Research continues to parse Ahmes' math specifics. All that is known for sure is that Ahmes used 'red auxiliary' numbers, an LCM method. Ahmes' selection of an optimal multiple may have also considered Akhmim Wooden Tablet and RMP remainder arithmetic.

BACKGROUND: Using Webster's new collegiate dictionary, an Egyptian is defined by: 1. a native or inhabitant of Egypt; 2. the Afro-Asiatic language of the ancient Egyptian from the earliest time to the 3rd century A.D.

By adding the word fraction to the word Egyptian, creates the phrase: Egyptian fraction. This narrative will show that Egyptians living before the 3 century A.D. wrote Egyptian fractions in ways that took over 115 years of debate to 'break the ancient scribal code' of Egyptian fraction arithmetic. The 115 year narrative's definition of an Egyptian fraction disallows the interjection of post-300 AD non-Egyptian fraction ideas and methods, such as represented by the modern greedy algorithm, and other modern decoding attempts that had hidden the ancient scribal methods from full view.

For example, by removing the modern idea of algorithm, and its 800 AD birth, as a RMP decoding possibility, the 2/n table and the EMLR methods began to be fairly decoded in other ways. That is, the possible greedy algorithm's use in the Liber Abaci (as noted by Sylvester in 1891 in the last of its Egyptian fraction methods) only included the use of a second subtraction step, and not an n-step algorithm. In other words, by placing algorithms, and other none scribal arithmetic suggestions (like false position) aside, the central outline of 2,000 BCE scribal arithmetic come into view.

Three of the four scribal arithmetic operations look much like our own modern arithmetic operations. Of course, the oldest duplation multiplication operation was unique to Egyptian mathematics, a form of arithmetic that was disliked by Greeks, and Arabs. It should be noted, by the time of the Liber Abaci (1202 AD), Greek and Arab lattice multiplication came into dominance, thereby fully replacing scribal duplation methods. Yet, the oldest Egyptian arithmetic's use of addition, subtraction and division operations, looked the same in 1650 BCE, as they did in 1202 AD, as the three arithmetic operations look and perform today.

Generally ancient scribes wrote rational numbers as exact unit fraction series in optimal ways. The scribal methods for converting rational numbers has been a murky subject, in several respects. Hence few modern scholars have ventured into the deeper aspects of all four of the ancient scribal arithmetic topics, as they relate to the 4,000 year time period, since their first appearance. In other words, this summary is intended to high-light a few of the murky aspects of this longer arithmetic subject reporting for the first time a unified definition of the oldest Egyptian fractions.

Returning to a broader Egyptian fraction decoding topic, the first chapter stresses that the Egyptian Mathematical Leather roll and its conversion of 1/p and 1/pq unit fractions to Egyptian fractions was an ancient teaching tool for anyone wishing to become a scribe. The EMLR student raised its simple set of unit fractions to multiples of 2, 3, 4, 5, 7, and 25, as needed, and then parsed his/her denominators by multiples of the denominator. For example 1/3 was raised 2/2 = 2/6, allowing 1/3 + 1/3, a non-Egyptian fraction looking definition to be stated. Next 1/4, one of the binary numbers was raised to 4/4= 3/12, allowing (2 + 1)/12 to write 1/6 + 1/12. In total, the EMLR converted 26 lines of Egyptian fractions, several repeated 1/p or 1/pq unit fractions, converted by a different multiple of 2, 3, 4, 5, 7 and 25, and finding not-so-elegant Egyptian fraction series.

The second modern decoding chapter begins by finding elegant two-term Egyptian fraction series for 2/pq vulgar fractions. Ahmes found the optimal 2/nth table series by following the same pattern, raising 2/pq to (p + 1), used as a splitting method. The method looked like this, when first found in 1995,

For example, to convert 2/21, with p = 3, and q = 7, let (p + 1)= (3 + 1)= 4, such that

or simply using the multiple of

as the Liber Abaci suggested in 1202 AD, as found in 2007 per:

However, a general solution that covered all but three of the 2-term series (2/35, 2/91, 2/95) was not published until 2002.

These two-term series may have been created by either using a subtraction or a division method. Evidence from the EMLR, first unrolled in 1927, and not fully read until 2002, points toward scribal division being the primary method used by Ahmes to compute the 2-term series, and several other 2/nth table members.

The final multiple method is connected to the EMLR 1/pq conversion method. Both method raised its vulgar or unit fraction to a multiple of itself, and wrote out the answer by decomposing the numerator in terms of divisors of the denominator:

with A = 4, 5, 7, 25

example: A = 25

or, using Occam's Razor, scribes raised 1/8 to a multiple of 25 of itself,

such that,

or, as the 1202 AD Liber Abaci suggests

and so forth, following the above steps, reaching

note the out of order series as a numerical fact that impliwa a historical link to the EMLR method.

A possible 2/nth table decoding chapter began in 1891. This chapter was associated with J.J. Sylvester, a British mathematician. He suggested that a greedy algorithm had created certain ancient Egyptian fractions. Sylvester improperly reported a greedy algorithm in Fibonacci's 1202 AD Liber Abaci. Fibonacci's potential greedy algorithm was a second subtraction method and not an n-degree algorithm. Fibonacci had converted a small vulgar fraction with a first partition and obtained an intermediate vulgar fraction that could not be written as a unit fraction series. Fibonacci introduced a second partition to find a final Egyptian fraction series, completing the problem. In 1993 Heinz Leuneburg published "Leonardi Pisani Liber Abbaci oder lesevergeugen eines mathematikers" de facto resolving the greedy algorithm and related issues, for those that read German reporting the details of the problem, listed as the seventh method.

In 1895 F. Hultsch broke a RMP 2/p code. In 1945 E.M Bruins independently confirmed Hultsch's code breaking method, and published it in 1948. Peet's 1923 additive views was therefore formally refuted by the Hultsch-Bruins method. Peet's view had concluded that Ahmes used additive methods to create the 2/th table. Peet also improperly suggested that scribal division was based on an inverse multiplication method, an incomplete and misleading conclusion. A modern algebraic analysis of the EMLR, the RMP, the Akhmim Wooden Tablet, and the Kahun Papyrus confirm that the historical scribal division operation looked very much like modern division.

Considering the RMP, Ahmes reported division within remainder arithmetic, and simple algebra problems is processed in modern ways. That is scribal subtraction was not historically linked to Egyptian duplation multiplication, within 'Falss Position" as Peet and others had suggested in 1923. "False position", as reported on Wikipedia was not invented by Arabs until 800 AD. Seen from Peet's additive views scribal arithmetic was improperly disallowed from using any property of number theory, such as p and q being sensed as prime numbers.

Occam's Razor is an important consideration in all of these 2/nth table discussions. The scribal texts must determine the scope and details of scribal division, and the other arithmetic operations. Modern logical assumptions should provide little or no light to this class of ancient search for scribal arithmetic thinking. Ancient scribal thinking should therefore be the singular focus when researching the contents of any ancient mathematical text.

A practice of reading an Egyptian mathematical text independent of other mathematical texts (reading only the RMP, without confirming issues that are contained in other texts) seems to have allowed Peet's incomplete views to go virtually unchallenged for 80 years. Bringing in the EMLR, AWT, Reisner Papyri, and other texts greatly assists in decoding the RMP 2/nth table, Kahun Papyrus, and other major aspects of the scribal arithmetic, algebra and geometry. Peet's 80 year old views are now being challenged openly, in serious ways

Returning to another central aspect of this subject, By 1906 G. Daressy, a French Egyptologist, had added the Akhmim Wooden Tablet (AWT), housed in Cairo's Egyptian National Museum, to this arithmetic debate. Daressy was unable to read all the fractions in the text. Typographical errors in transcribing the hieratic text had hindered Daressy's reading of the exact aspects of two divisions of the five problems. The problems had divided a hekat by 3, 7, 10, 11 and 13. Daressy was unable to read the 11 and by 13 divisions as inexact, for those that read French, thereby slightly throwing his research off track.

Interestingly, by 2002 Hana Vymazalova, a Charles U, Prague grad student in Egyptology, gained a fresh copy of the AWT from the Cairo Museum, and re-read it, and found that all five AWT division answers had been exact, after all. Vymazalova's work stressed the multiplying of the hekat partitioning answers by their initial divisors, and exactly finding 64/64, the starting hekat unity (identity) term, a very important discovery of one scribal consideration.

However, on another level Vymazalova had accepted Peet's awkward and incomplete definition of Egyptian division as properly explaining the partitioned two-part statements appearing in the AWT and the RMP. This was one of Vymazalova's oversights, that has been corrected by others. By 2005 others had shown that Peet had only discussed the 1/320th unit named ro, and not the division of a hekat content from which it was calculated, while totally omitting any mention of remainder arithmetic.

Returning to 1923, Peet's incorrect views of Egyptian arithmetic and its ancient division methodology had argued strongly against the partial results published by Daressy in 1906, related to n = 3, 7 and 10. Peet had reported that all scribal divisions, in every available ancient mathematical texts, had followed an inverse multiplication method, and that ro was only a volume unit, one that was disconnected from the division operation.

For those that read French, Daressy had reported a division method that did not look or act like an inverse operation of scribal multiplication.

Considering the 111 year old debate, in several European languages, in terms of J.J. Sylvester, Gillings and Hultsch-Bruins, and the 80 year debate between Peet and Daressy, has now been effectively resolved in favor of Hultsch-Bruins and Daressy. Vymazalova, and others have shown that Egyptian division can now be freshly reported as having being based on scribal remainder arithmetic using a division methodology that had very little to do with Peet's 1923 views of Egpytian multiplication.

Ahmes and the other ancient scribes had arguably used aliquot parts, by first inspecting the divisibility of its first partition denomintors, as first explained by the Hultsch-Bruins method. The H-B method writes all RMP 2/p vulgar fractions into Egyptian fraction series and several 2/pq vulgar fractions into Egyptian fractions by analyzing the numerator's divisors of the first partition. Taking all of Ahmes' division and vulgar fraction conversion methods together, using remainder arithmetic as a unifying theme. Egyptian division can now be seen as an operation on its own, unique in several important ways.

After reading the corrected version of the AWT in 2003, where a hekat division was substituted in the form of 64/64, as a hekat unity, the one example opens most of the central points of the ancient method of Egyptian division to modern analysis:

Let n = 11

such that:

and writing the data in the ancient final form:

as the scribe introduced by shorthand notes, by finally reaching

This two-part answer was proven by multiplying it by 11, or

became

and

in a manner that Peet had not explained. Peet had only seen the 1/320 = ro aspect, otherwise holding onto his rigorous and incorrect view of Egyptian division as being an inverse process linked to the ancient form of Egyptian multiplication.

Seen in this 2007light, free from confusing pseudo-connections to Egyptian multiplication, Ahmes' thinking was straight forward, using either modern notation or the ancient hieratic script.

It should be noted that the Egyptologists considers these 2/nth table fractions as hard to read, and do not consider the scribal use of proto-number theory.

Typographical, and other errors, hinder Egyptologists. Hieratic texts are misread for several reasons. One error takes place when scholars inadvertently introduce typos, the class of error that Daressy faced when he read the Akhmim Wooden Tablet in 1906. Another error is introduced by Egyptologists stressing hieroglyphic arithmetic as pedagogically dominant over hieratic arithmetic. Hieratic math, and its arithmetic, can only be read by deciphering its numerals, and reading its exact finite arithmetic. My view is that hieratic arithmetic should have always been seen as independent from hieroglyphic arithmetic. That is, hieroglyphic arithmetic used many-to-one numerals, and an infinite series form of arithmetic. Hieroglyphic arithmetic can not exactly be translated into hieratic numerals, and exact Egyptian fraction series.

Returning to Daressy, Egyptologists often under valued his 1906 analysis. Daressy had introduced misinterpretations, thereby not completely reading the hieratic text. However, Daressy was on the right track, compared to Peet and others that only read the ro aspect of the Akhmim Wooden Tablet (AWT).

Commenting on the AWT remainder arithmetic structure, it used a method of division that is now easy to read:

for a hekat unit,

for a hinu unit.

Interestingly, Ahmes wrote the hin unit in relation to the hekat, by using the fact:

10/n hin

with n being the divisor used in

(64/64)/n = Q/64 + (5R/n)*ro

The hinu system eliminated the need for the remainder term including ro.

It is suspected that other hekat sub-units followed the hin relationship. Dja has been suggested by Tanja Pommerening in 2002 and 2005.

Note that the hekat and hinu Q/64 term was always written as a binary (or Horus-Eye) series. Of equal importance was the remainder term, 5R/n as used in the hekat system, and 10/n hin in the hinu system. The remainder vulgar fraction was written as an Egyptian fraction series. The hekat use of a scaled down factor of 320 was selected for a very good reason. The use of 1/320 = ro reduced the size of the vulgar fraction that was required to write the associated Egyptian fraction series. Both the hekat 2-part answers and the hinu two-part answers were considered as single numbers by ancient scribes.

CONCLUSION: Scribal and medieval Egyptian fractions methods recorded general conversions of rational numbers to optimal and elegant series. Ahmes, for example, wrote exact unit fraction series by a single multiple method. Over a longer period of time, seven conversions methods were summarized in the 1202 AD Liber Abaci. Four Liber Abaci methods had been used by earlier scribes writing 2/n tables. Scribal methods in 1202 AD and 1650 BCE had raised rational numbers to desired multiples to compute optional and elegant Egyptian fraction series.

Early scribal methods had also partitioned a volume unit named the hekat to hin, oipe, dja and ro sub-units. Scribal weights and measures system scaled units and sub-units by exact Egyptian fraction remainders based on royal edits. Pharaohs were interested in controlling beer, bread, grain and other vital national inventories. Egyptian fraction volume controls were recorded in the Akhmim Wooden Tablet using a remainder arithmetic system. Remainder arithmetic units employed exact Egyptian fraction remainders. Hekat units included the hin, oipe, dja and ro. Hekat units partitioned vulgar fractions in weights and measures applications. Several classes of remainder arithmetic practices were cited in Egyptian fraction texts. One class was recorded in the Reisner Papyri and the RMP. The Reisner and RMP reported worker production rates measured in units of 10, production outputs of 10 workers days, likely in 10 hour days.

Scholars will continue to parse aspects of Egyptian fraction texts likely pointing out previously unreported scribal methods. Scribal algebra offers one of three math windows that parse ancient Egyptian fraction texts. Modern and scribal algebra offers a window that parses misunderstood scribal arithmetic operations, and scribal connections to Egyptian fractions and scribal remainder arithmetic. Scribal 2/n tables offers a second window that parses optimal methods. Remainder arithmetic offers a third window parsing scribal weights and measures systems. Considering all three windows, scholars will debate aspects of deeper forms of scribal Egyptian fraction methods for years to come. One of the newer, and more interesting aspects is remainder arithmetic.