Egyptian fractions were rigorously defined within an arithmetic notation created by 2,000 BCE Egyptians, a first in Western math history. The notation exactly represented rational numbers as quotients and unit fraction remainders scaled in optimal and elegant ways. One purpose of the rational number notation solved a binary Horus-Eye rounding-off problem. The Horus-Eye problem had thrown away 1/64 units by using only the first 6-terms of a binary series. This narrative writes fractions as a/b rather than the standard LaTeX fraction form and solves the ancient problem.

One solution to the binary (Horus-Eye) round-off problem partitioned volume units into binary quotients and scaled Egyptian fraction remainders. A volume unit named hekat was reported in the 1900 BCE Akhmim Wooden Tablet (AWT) within a version of a new rational number numeration system. The Egyptian fraction numeration system de facto replaced the Old Kingdom's binary Horus-Eye fraction numeration system for scientific and business purposes by removing the round-off error.

Concerning interesting details, the AWT defined a hekat unity as (64/64). The unity was partitioned by rational numbers 3, 7, 10, 11 and 13. Answers were written in Q/64 binary quotients and (5R/n)ro Egyptian fraction remainders, with ro equaling 1/320. 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. The secondary hekat system converted m/n to integer quotients and non-scaled Egyptian fraction remainders. Scaling was created by the numerator m. For example, Ahmes wrote in 1/10 units, with m = 10 and n= 3, by using the expression 10/3 hin to write (3 + 1/3)hin.

Concerning the AWT's binary quotient and Egyptian fraction series, the n = 3, 7, 10, 11 and 13 partitions were proven by multiplying each quotient and remainder answer by the initial divisor. The proof calculation returned each answer to the initial (64/64) unity value. The AWT proof may have been a first in math history.

A second purpose of the Egyptian fraction system exactly solved one and two variable first degree algebra problems by writing rational number answers as integer quotients and Egyptian fraction remainders. The Rhind Mathematical Papyrus (RMP) and the Berlin Papyrus cite several algebra problems and solutions, each with incomplete intermediate steps reaching Egyptian fraction answers.

Egyptian fractions also represented a numeration system by mapping 1:1 the counting numbers to hieratic sound symbols. This feature was noted by Carl Boyer, the math historian, as the first ciphered numeration system. Unit fractions were created by a line over hieratic sound symbols. Secondarily, Egyptian fractions were applied within unitized weights and measures systems. Over the 3,600 years of Egyptian fractions use international trading systems drew upon its exact rational number notations by creating a wide range of Egyptian fraction remainder weights and measures systems.

Greeks also mapped the counting numbers to 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 writing rational numbers for several purposes altering their Egyptian predecessors' methods in minor ways.

Egyptian fraction methods of writing unitized rational numbers were used by nearby Mediterranean cultures. Ancient Near East cultures first mapped the counting numbers to their alphabets creating numeration systems. After modern base 10 numerals diffused through Arab culture from India around 800 AD medievals scribes re-wrote Arab mathematics using Arabic numerals. Leonardo de Pisa, aka Fibonacci, wrote the 1202 AD Liber Abaci, and other texts, in the new numerals and the older Egyptian fraction arithmetic. Leonardo's weights an measures system was also written in Arabic numerals. Mediterranean trade run by Arab businessmen was documented in Arabic numerals by Fibonacci within traditional Egyptian mathematical threads. Four of Fibonacci's threads connect to 1,900 BCE and the Akhmim Wooden Tablet and the 1650 Rhind Mathematical Papyrus.

In 1585 AD, the date most often mentioned as the beginning of our modern base 10 decimals, the European base 10 decimal system nearly erased 3,600 years of Egyptian fraction arithmetic and weights and measures history. The Arab world also erased large chunks of Egyptian fractions' 3,600 year history. By 1900 AD European and Arab scholars were unable to accurately read medieval and ancient Egyptian fraction texts.

Ahmes' 2/n tables shorthand and arithmetic could not be accurately read since the shorthand had omitted critical intermediate steps apparently garbling the text. Scribes like Ahmes had left few clues to decode their earliest Egyptian fraction methods. Post-1877 scholars have often relied on intuition and personalized mathematical senses while reporting suspected details of Ahmes' and other scribal arithmetic methods. Scholarly debates for 130 years have focused on the 2/n table and its construction methods. This class of Egyptian fraction debate has often been controversial. For example, Neugebauer, Exact Sciences in Antiquity, proposed that the 2/n table and its methods represented an intellectual decline. The historical role of the RMP 2/n table, parsed over the last 130 years, confirms that advanced scribal ideas had created ciphered numerals, abstract numbers, and finite Egyptian fraction arithmetic, as well as solving the Horus-Eye problem.

As a consequence 130 years of confusion is slowly lifting an intellectual fog. The once dominant 3,600 life of Egyptian fractions is slowly re-emerging. Scholars are increasingly parsing one arithmetical or algebraic fragment at a time, often considering one text as unrelated to another like text. For example, scholars are focusing on scribal methods that created 2/n tables and closely related arithmetic progressions in the Kahun Papyrus and the RMP. Confirmation occurs when at least two ancient texts report the same arithmetic or algebraic detail.

Ahmes, The RMP scribe, has gained the majority of the attention since the RMP was first published in 1877. Several mis-steps have been made along the way. Sylvester in 1891 incorrectly suggested that the greedy algorithm was present in the medieval Liber Abaci and by implication the RMP. Yet, Hultsch in 1895 fairly parsed Ahmes' 2/p Egyptian fraction patterns by using a form of number theory. Mis-steps had also been associated with 1920's scholars working from an additive point of view. It took Bruins in 1944 to confirm Hultsch's earlier work even though it was shown in 1895 that Ahmes had likely converted 2/p rational numbers into optimal or elegant Egyptian fraction series. Today, it is clear (to many researchers) that once an optimal first partition (1/A), or an optimal multiple, was selected by Ahmes, aliquot parts (divisors of A) were inspected to partition the numerator of the remainder

(2/p - 1/A) = (2A - p)/Ap

and the ancient expression:

2/p= (2/p)*(A/A) = 1/A + (p1 + p2 + .. + pn)/(pA)

with p1, p1, ..., pn being selected divisors of A

Ahmes converted by his own shorthand 2/43 to an Egyptian fraction series by selecting 1/42 as the first partition, 1/A, within a modern range 2/p < 1/A < 1/p and the ancient statement 2/43 = 2/43*(42/42). Ahmes inspected the divisors of 42 seeing 21, 14, 7, 6, 3, 2, 1. Ahmes compared prime and composite divisors to the numerator, 41, taken from the modern expression (84 - 43)/(42*43) and the ancient expression 84/(42*43). Finally, Ahmes additively computed the numerator 41 from the set of divisors, (21 + 14 + 6) = 41 such that:

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

and by Ahmes' shorthand:

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

Ahmes' method looked slightly different when compared to Hultsch's RMP analysis. Ahmes' shorthand indicated that 2/43 had been restated to a 42/42 multiple. Ahmes allowed the divisors of 42 to calculate an optimal Egyptian fraction series. Ahmes' shorthand left out several steps, hence scholars had often misread the information.

In 1944 Hulstch's proto-number theory view was independently confirmed by E.M. Bruins. Today the Hultsch proposal is known as the Hultsch-Bruins (H-B) method. Bruins proved the reliability of this method to the math history community by publishing several papers. Today, many 1920's additive scholarly adherents tend not to fully accept the H-B method as the final word on ancient rational number methods used by Egyptians, Greeks, Hellenes, Arabs, medievals and others that implemented the notation.

Initial none acceptance of the H-B method was related to the 49 year gap between's Hultsch's 1895 proposal and Bruins 1944 confirmation. Intermediate 1920's scholars (i.e. Peet, Neugebaur, Chace, et al) had taken different approaches to attempt to rigorously decode Ahmes' rational number series, and recreate its arithmetic and algebraic applications. The 1920's proposals had been built upon additive transliteration considerations, often introducing a range of assumptions. One assumption proposed to decode Ahmes first degree algebraic steps. The proposal was ironically named 'false position', short for false supposition. Robins-Shute mention the method in an analysis of the RMP (in a 1987 book published by the British Museum). 'False position' had been thought to have been named after an 800 AD Arab method that formally guesses within second degree algebraic roots calculations, a historical link that never existed. That is, Ahmes' first degree algebraic solution did not employ a single 'false position' process, even though several math history books continue suggesting the contrary.

As proof that Ahmes' rational number arithmetic did not employ guess work, as required by single 'false position' suggestions. RMP 33 and RMP 31 data is exactly parsed by following Ahmes thinking.

RMP 33 requested Ahmes to solve for x:

which he did using these steps:

x + (2/3 + 1/2 + 1/7)x = 37

x's were collected, as we do today by:

x + ((28 + 21 + 6)/42)x = 37

finding,

(97/42)x = 37

such that

x = 37*(42/97) = 1554/97

a vulgar fraction, that was written then, as it is today in modern algebra.

One difference in the ancient version of 1554/97 was that Ahmes' value for (x),

It should be noted that Ahmes re-wrote 2/97 using a 2/n table look up, or by computing 2/97 by selecting a first partition 1/56, such that:

(2/97 - 1/56) = (112 - 97)/(56*97) = (8 + 7)/56*97)

allowed Ahmes' quotient (16) and Egyptian fraction remainder (2/97) answer to be written as:

x = 16 + 1/56 + 1/679 + 1/776

As a second confirmation of Ahmes' use of remainder arithmetic and exact Egyptian fraction thinking, rather than the suggested 'false position' method a closely related problem, RMP 31, asked Ahmes to:

RMP 31: solve for x

which Ahmes did using these steps:

x + (2/3 1/2 + 1/7)x = 33

(collecting terms as noted above, and so forth) such that

(97/42)x = 33

x = 1386/97

It is therefore proven that 'false position' had not been used by Ahmes to write out his 1386/97 quotient and remainder answer. Hence Ahmes' traditional Egyptian fraction methods, written in remainder arithmetic, inspected the intermediate step:

x = 14 + 28/97

in a manner that solved for 28/97 re-written as 26/97 + 2/97 by considering:

(26/97 - 1/4) = (104 - 97)/(4*97)

restated as:

26/97 = 1/4 + (4 + 2 + 1)/(4*97) = 1/4 + 1/97 + 1/194 + 1/388

and,

2/97 = 1/56 + 1/679 + 1/776,

as calculated in RMP 33, and the 2/n table,

such that, Ahmes' set of unit fraction series represented 28/97 (x) as:

x = 14 + 1/4 + 1/56 + 1/97 + 1/194 + 1/679 + 1/776

thereby fully disclosing Ahmes arithmetic and algebraic thinking.

That is, Ahmes' Egyptian fraction conversions of 1554/97 and 1386/97 followed standard scribal rules (described above), and did not follow the guessing steps proposed in the 1920s, nor connected in any direct way to 800 AD Arab double false position as fairly decoded in Arab and medieval texts.

Today it is clear that modern definitions of addition, subtraction, multiplication and division can be appropriately used to parse ancient Egyptian fraction versions of rational numbers. For example, modern multiplication of ancient Egyptian fractions simplify certain problems in history of math classes by disregarding false position suggestions. This point of view shows that 'false position' was a misleading 1920's view of Egyptian mathematics. Modern arithmetic operations fairly parses RMP problems, and other arithmetic progression problems described in Kahun Papyrus, once the ancient quotient and remainder arithmetic is understood from the scribe's point of view.

In conclusion, it is important to note that Ahmes' modern algebraic arithmetic was recorded in Fibonacci's Liber Abaci. Modern arithmetic, written in Fibonacci's base 10 numerals fairly parses scribal multiples used in four Egyptian fraction distinctions recorded on pages 124-125 of Sigler's 2002 translation of the Latin Liber Abaci. For example Ahmes' 1650 BCE thinking is clearly shown to follow these steps:

2/n conversions to Egyptian fractions series, when n was not prime,

Fibonacci and Ahmes selected a multiple of 2/n in a manner that solved the conversion.

An older Egyptian text, the Egyptian Mathematical Leather Roll(EMLR), written around 1850 BCE, records 26 Egyptian fraction series created by finding multiples in a manner that Ahmes used in 1650 BCE for his 2/n conversions. Fibonacci knew this method as well, as cited as his first distinction.

For example, the EMLR student scribe's work converted binary and prime rational numbers, such as 1/8, using several multiples (choosing 3/3, 5/5 and 25/25). The EMlR can therefore be seen as as a practice document, as noted by:

(1/8)* 3/3 = 3/24 = (2 + 1)/24 = 1/12 + 1/24

(1/8)* 5/5 = 5/40 = (4 + 1)/40 = 1/10 + 1/40

(1/8)* 25/25 = 25/200 = (1/5)*(25/40) = (1/5)*(24 + 1)/40 = (1/5)*(3/5 + 1/40) = (1/5)*(1/5 + 1/3 + 1/15 + 1/40), or

1/8 = 1/25 + 1/15 + 1/75 + 1/200

an out-of order series, as further discussed on: http://emlr.blogspot.com .

Scholarly debate on the theoretical and practical foundations of the 3,600 year longevity of Egyptian fractions had stressed its practical aspects during most of the 20th century. However, when an English translation of the Liber Abaci in 2002 AD parsed Egyptian fraction data bases including its older theoretical threads the debate changed. Scholars are taking note of Egyptian fractions theoretical issues that were connected to weights and measures systems.

Summary: Egyptian scribes working in Middle Kingdom hieratic script generally converted rational numbers to exact unit fraction series in optimal and elegant ways. The Egyptian fraction innovation exactly scaled weights and measure units. Pharaoh had requested the system to control and allocate vital inventories of grain and its products, including beer and bread.

The Egyptian fraction method of weights and measures was popularized in international trade after 1,000 BCE by altering Middle Kingdom unit values for a hekat control unit named dja by a factor of four. Greeks and other cultures modified the system with the latest cultures being documented in medieval times by Fibonacci writing in the Liber Abaci.