This narrative writes fractions as a/b rather than the standard LaTeX fraction form.

Middle Kingdom Egyptian scribes created rational number conversions to unit fraction series methods within a numeration system around 2,000 BCE. Numerals were ciphered 1:1 onto hieratic sound symbols, thereby replacing the Old Kingdom's hieroglyphic many-to-one numeral system. A line was drawn over a hieratic sound symbol to denote a unit fraction. Carl B. Boyer was the first scholar to report the cultural significance of 1:1 ciphered numerals. Egyptian fraction series represented rational numbers written in ordered, from largest to smallest, unit fraction series.

One purpose of the Egyptian fraction notation involved arithmetic operations within a finite weights and measures systems, likely first for money and trade. Egyptian multiplication and division was undervalued by historians for over 100 years. Multiplication contained two sides, an additive form that dates to the Old Kingdom, and a modern form that dates to the Middle Kingdom. Math historians have parsed aliquot part fragments since 1895, suggesting 2/n tables were built with aliquot parts. In 2002, the EMLR was validated containing aliquot parts of LCM multipliers, m/m. The multipliers were used by scribes to scale vulgar fractions optimal unit fraction series. In 2006 the Akhmim Wooden Tablet was validated containing quotient and remainders as its primary division division method.

Over the 3,600 years of Egyptian fractions used the Akhmim Wooden Tablet form of division, creating international trading systems that drew upon rational number notations by creating exact remainder weights and measures systems. The notation exactly converted rational numbers to quotients and unit fraction remainders in hieratic script. The notation had solved an Old Kingdom binary problem. The Old Kingdom problem had thrown away 1/64 units within a 6-term binary notation named Horus-Eye. This narrative writes fractions as a/b rather than the standard LaTeX fraction form and solves the ancient problem near the historical context.

Volume units were partitioned into exact binary quotients and scaled Egyptian fraction remainders. One unit (hekat) and a 1/320 sub-unit (ro) appeared around 2400 BCE in hieroglyphic and exploratory hieratic scripts. Around 1950 BCE the Akhmim Wooden Tablet (AWT) reported in the Egyptian fraction numeration system by solving an Old Kingdom binary round-off problem in hieratic script. The notation exactly converted rational numbers to quotients and unit fraction remainders in hieratic script. The notation was used to solve an Old Kingdom binary problem, worked on for about 400 years. The Old Kingdom problem had thrown away 1/64 units within a 6-term binary notation named Horus-Eye.

Length and area units (cubits, khet, setat and mh) were partitioned into 1/8 setat quotients and mh (1 cubit by 100 cubit strips) remainders. RMP 53, 54, and 55 discuss Ahmes' geometry and associated arithmetic.

Concerning hekat arithmetic details, the AWT defined a hekat unity as (64/64). The unity was partitioned by rational number divisors. Answers were written into binary quotients and scaled (5R/n)*1/320 Egyptian fraction remainders. Scribes also scaled hekat units to 1/10 (hinu), 1/64 (dja), 1/320 (ro), and other units within a modified quotient and remainder system. A secondary hekat system converted m/n to integer quotients and non-scaled Egyptian fraction remainders written as m/n "name". For example, Ahmes created 1/10 units writing m = 10 and n= 3 by using the expression 10/3 hin, meaning (3 + 1/3)hin.

The AWT's binary quotient and Egyptian fraction partitions were proven by multiplying each quotient and remainder answer by the initial divisor. The exact proof calculation returned each quotient and scaled remainder answer to an initial (64/64) unity value. The AWT proof may have been the first rigorous proof in math history.

The Reisner Papyrus(RP), circa 1800 BCE, defines a labor efficiency division by 10 quotient and unscaled remainder rate. The RMP's first six problems used the RP division by 10 method. The RMP is one of several texts that confirm the scribal use of a second form of quotient and remainder arithmetic. Modern historians, going beyond 1920's additive transliteration limitations, by considering meta Egyptian fractions, agree with the RP method, and other forms of meta (unified) Egyptian fraction mathematics.

A second purpose of the Egyptian fraction notation exactly solved one and two variable first degree algebra problems. Rational number answers were written into integer quotients and Egyptian fraction remainders. The Rhind Mathematical Papyrus (RMP) and the Berlin Papyrus cite several problems and solutions, each with hard-to-read intermediate steps reaching Egyptian fraction answers.

A third purpose of the Egyptian fraction notation created a commodity and metal based monetary system. The system is outlined by Mahmoud Ezzamel using modern accounting methods. Yet, Ezzamel fairly parses the four Heqanakht Papers by discussing two absentee landlords' production and management considerations of profit written in ancient Egyptian fractions.

A fourth purpose of the Egyptian fraction notation generally converted rational numbers to optimal and elegant unit fraction series. The RMP 2/n table written by Ahmes in 1650 BCE intuitively and by rule used 'red auxiliary' multiples to write nearly optimal unit fraction series. With the RMP published in 1879, historiamns tried to break the 2/n table code. The earliest successful code breaking effort, in retrospect, was published by F. Hultsch in 1895, honored by the Hultsch-Bruins method.

Returning to the 1650 BCE and the Rhind Mathematical Papyrus it began with a 2/n table. A 200 year older text, the Egyptian Mathematical Leather Roll (EMLR) was a student's introduction to the 2/n table subject. A student converted 26 rational numbers to non-optimal Egyptian fraction series. The EMLR converted rational numbers such as 1/8 by using multiples 3/3, 5/5 and 25/25. Two out-of order series are also discussed on EMLR.

Greeks used the hieratic numeration systems by mapping the counting numbers onto Ionian and Dorian alphabets. Unit fractions were denoted by a Greek letter followed by ', or beta' = 1/2. Greeks, and Hellenes fully used the Egyptian fraction method of converting rational numbers to Egyptian fraction series for several purposes altering Ahmes' methods in minor ways.

For over 3,600 years Egyptian fraction methods unitized rational numbers in nearby Mediterranean cultures. Ancient Near East cultures around the time of the Greeks also mapped the counting numbers onto their alphabets.

Around 800 AD a major change took place in the Mediterranean region. Modern base 10 numerals diffused from India and Arab trade began to enter Europe. By 999 AD, a Catholic Pope adopted Arab mathematics with its base 10 numerals and Egyptian fraction arithmetic. By the time of Leonardo de Pisa, Fibonacci, and the 1202 AD Liber Abaci, European weights an measures were also written in Arabic numerals.

In 1585 AD, the beginning of our modern base 10 decimals, modern base 10 decimals literally erased 3,600 years of Egyptian fraction arithmetic history. By 1900 AD European and Arab scholars were unable to read medieval Egyptian fractions texts as well as the older Egyptian mathematical texts.

Ahmes' 84 problems have been slowly read by scholars after the RMP was published in 1877. The 2/n table was key to scholar research. But what method or methods did Ahmes use to create his 2/n table? Ahmes seemingly left few clues to assist scholars in decoding the 2/n table's construction method(s). Post-1877 scholars relied on intuition and personalized mathematical senses to report suspected details, often confusing the subject. Scholarly debates have correctly focused on the 2/n table and its construction methods. Yet, confirmed 2/n table threads continue to be controversial. Egyptologists, and math historians, i.e. Neugebauer, Exact Sciences in Antiquity, had inappropriately proposed that the RMP 2/n table and related Egyptian fraction methods represented forms of intellectual decline.

However, one simple RMP 2/n table construction method, multiples, was reported in 2002 that changed the debate. An advanced idea had created finite unit fraction statements. By 2006 it was shown that Egyptian fraction statements had solved an Old Kingdom infinite series round-off problem. Scholars continue to parse several theoretical fragments of a group of ancient texts by considering related arithmetic patterns. As a consequence 130 years of confusion related to Egyptian fraction arithmetic is lifting an intellectual fog.

For example, Ahmes, the RMP scribe, has gained the majority of scholarly attention since the RMP was published in 1877. Sylvester in 1891 incorrectly suggested that the greedy algorithm was present in the medieval Liber Abaci and by implication the RMP. Hultsch in 1895 began to parse Ahmes' 2/p conversion patterns by using aliquot parts. It took Bruins in 1944 to confirm Hultsch's earlier work, now known as the H-B method.

Ahmes, therefore, converted 2/p rational numbers into optimal or elegant Egyptian fraction series using a form of the H-B method. Ahmes' shorthand indicated that 2/43, and other 2/n table members, were converted to an Egyptian fraction series by selecting optimal multiples, in the 2/43 case the multiple 42. Ahmes use of the multiple 42 allowed 1/42 to become the first partition. The remaining Egyptian fraction were found by considering the divisors (aliquot parts) of 42 or (21, 14, 7, 6, 3, 2, 1). Ahmes' fragmented shorthand indicates:

2/43*42/42 = (42 + 21 + 14 + 6)/(42*43),

such that:

2/43 = 1/42 + 1/86 + 1/129 + 1/301

in clear hieratic script.

Today, 1920's additive scholarly adherents tend not to accept the H-B method, or the multiple method, as used by Egyptians, Greeks, Hellenes, Arabs, medievals and others for over 3,000 years. Yet, it is clear that the multiple method and its use of modern addition, subtraction, multiplication and division operations correctly parses 4,000 year old rational numbers conversions to Egyptian fraction series. Occam's Razor , the simplest method is likely the historical method, is slowly changing scholarly minds.

As a long term verification of Ahmes' arithmetic, Fibonacci's 1202 AD Liber Abaci reports a very old style of writing multiples within the first four of seven rational number conversion methods.

Summary: Today, interdisciplinary studies groups are created by 'invitation only'. As meta aspects of ancient mathematics are randomly studied, without invitation, based on reading entire bodies of mathematical texts, great academic progress in understanding ancient mathematics will take place. Currently, common abstract mathematics themes are deposited in fragmented story lines, often transliterating a small set of the ancient records, not noticing closely related mathematical methods. Moreover, university math history and philology departments often do not professionally share common meta themes (found by parsing numerical data) even when their offices are in the same building.

That is, broadly considering Old Kingdom edicts, several Pharaohs likely had requested exact numerical systems to control and allocate vital inventories of grain and its products, including beer and bread is not often mentioned. Yet, Egyptian scribes began to exactly convert rational numbers to optimal or elegant unit fraction series around 2,000 BCE. The Middle Kingdom innovation exactly scaled weights and measure units to two Egyptian fraction systems, one defined in the Reisner Papyrus and the second in the Akhmim Wooden Tablet.

Since 2002 AD scholarly debates have increasingly corrected Middle Kingdom foundations of Egyptian fraction mathematics. The 2050 BCE point of departure date formalized Egyptian fraction innovations within abstract arithmetic and practical weights and measures notations. Greeks and Arab scribes modified the theoretical arithmetic and weights and measures notations. Medieval scribes by Fibonacci recorded Greek and Arab versions of the oldest Egyptian fraction mathematics. For example, Sigler's 2002 translation of the 1202AD Liber Abaci includes four 1650 BCE conversion methods. Fibonacci's era used 3,000 year old Egyptian fraction conversion methods that modified an ancient multiplication context to a medieval subtraction context. Today, 20th century Egyptian fraction debates, that fragments analyzed mathematical theory and practical statements are being unified. The oldest Egyptian fraction mathematics are being connected to the medieval era in interesting ways. The largest body of ancient texts, the economic and weights and measures texts, link 1650 BCE hieratic texts to the 1202 AD Liber Abaci and common uses rational number conversions to optimized Egyptian fraction methods.

Bibliography

1

Mahmoud Ezzamel, Accounting for Private Estates and the Household in the 20th Century BC Middle Kingdom, Abacus Vol 38 pp 235-263, 2002

2

Milo Gardner, The Egyptian Mathematical Leather Roll Attested Short Term and Long Term, History of Mathematical Sciences, Hindustan Book Company, 2002.

3

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

4

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

5

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

6

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.

7

L.E. Sigler, Fibonacci's Liber Abaci: Leonardo Pisano's Book of Calculation, Springer, 2002.

8

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