Why Study Egyptian Fraction Mathematics

Old Kingdom (OK) hieroglyphic numeration and math was base 10. Numerals were mapped many-to-one in a cursive binary system. The OK binary system rounded-off numbers throwing away 1/64 units in arithmetic, algebra, geometry and weights and measures problems.

The OK number theory appeared in a second numeration and mathematical system. After 2050 BCE the Middle Kingdom (MK) hieratic script formalized the second numeration system and a second mathematical system into a partially decentralized economy based on grain units. MK scribes mapped numbers one-to-one onto sound symbols. The ciphered MK system recorded arithmetic, algebra, algebraic geometry, and weights and measure by correcting OK hieroglyphic binary round-off errors. Hieratic geometry problems added back rounded off statements. Scribes replaced radius R by semi-diameter D/2, and pi by 256/81 in area and volume problems and scaled units to 1/64 and 1/320 hekat units. MK arithmetic was finite and recorded algebraic geometry and weights and measures methods that scaled hekat units (of grain) as small as 1/320 of beer, bread, domesticated fowl, and other wage payment units.

Several unit fraction notations were developed in the Ancient Near East from 2050 BCE to 1637 AD, a period of 3,700 years. Egyptian scribes scaled rational numbers n/p by LCM m to a mn/mp (in a multiplication context) before recording unit fraction series. The small number of Greek texts also unitized arithmetic in a multiplication context. Arabs mentored Fibonacci (1202 AD) to scale n/p by LCM m in a subtraction (n/p - 1/m) = (mn -p)/mp context that applied an algorithm when (mn -p) was set to unity (1).

After 1585 AD, and the arrival of base 10 decimals, none of Fibonacci's unit fraction notations were used in Europe. Ghobar, and other Arabic scripts, were replaced,in the 17th century by modern Arabic script, causing unit fraction math to die in the Arab world as well. Today, Arab scholar have few written language clues to decode medieval unit fraction arithmetic texts. That is, after the 18th, 19th and 20th century severe memory loss has adversely effected the reading of ancient unit fraction texts.

Yet, fragmented and hard-to-decode medieval, Arab, and Greek unit fraction texts are re-emerging by ancient meta rules found here, and there. Scholars in the 21st century are increasingly finding complete ancient meta rules by replacing missing finite arithmetic steps that 19th and 20th century scholars only discussed in fragmented additive ways.

Three texts, the MMP, Kahun Papyrus and RMP, are being decoded by considering area (A) of a circle and volume (V), adding height (H) of a circle granary per these meta rules:

A = pi(R)(R) = (2567/81)(D/2)(D/2)

A = (8/9)(D((8/9)(D) cubits (algebraic algebra formula 1.0)

replaced radius (R) with diameter (D/2) and pi by 256/81 (an easy to manipulate number) in

V = (H)(8/9)(D)(8/9)(D) cubits (algebraic algebra formula 2.0)

V = (3/2)(H)(8/9)(D)(8/9)(D) Khar (algebraic algebra formula 2.1)

V = (2/3)H(4/3)(D((4/3((D) Khar (algebraic algebra formula 2.2)

derived from scaling algebraic formula 2.1 by 3/2 considering

(3/2)V =(3/2)(3/2)(H)(8/9)(D)(8/9)(D) = (H)(4/3)(D)(4/3)D) and multiplying both sides by 2/3

Generally, translations of Middle Kingdom finite arithmetic texts, published in the 20th century, were incomplete, misleading and often grossly in error. Ahmes' Middle Kingdom notation and Fibonacci's medieval notation were related by LCM m, a number theory concept. Egyptian scribes scaled finite (n/pq)*(m/m) = mn/mpq to concise unit fraction series by finding the divisors of denominator mpq (a GCD) that best added to numerator mn. Medieval scribes in one of its three notations reported:

(n/pq - 1/m)= (mn - pq)/mpq

example: (4/13 - 1/4) = ([16 - 13)/52 - 1/18] = (54 - 52)/936 = 1/468

meant 4/13 = 1/4 + 1/18 + 1/468

statements that employed the same LCM m that Ahmes employed 2850 years earlier. Ahmes employed two rational number conversion methods. The first method was recorded in RMP 36 by a multiplication use of LCM m recorded as (m/m) per:

a. n/pq = n/pq(m/m) = mn/mpq

example: 4/13 = 4/13(4/4) = 16/52 = (13 + 2 + 1)/52 = 1/4 + 1/26 + 1/52

The second method was recorded in RMP 37 by a subtraction context, the form emulated by Fibonacci per:

b. (n/pq - 1/m)= (mn -pq)/mpq

example: (4/13 - 1/4)= (16 - 13)/52 = (2 + 1)/52 = 1/26 + 1/52

meaning 4/13 = 1/4 + 1/26 + 1/52

with the divisors of mnp recorded in red that best summed to numerator mn, created concise unit fraction series in both conversion methods.

As related historical point, medieval units of measures fell into disuse when Europeans found new trade routes, after 1492 AD. Portugal, Spain, and other Europeans used regional weights and measures units after the birth of base 10 decimals in 1585 AD. It took several yeas for the the innovative base 10 decimal notation to birth the modern metric system. With two books, one for science and one for business, base 10 decimals were approved by the Paris Academy. Napier added logarithms, improving the base 10 decimal notation. Other improvement have been added since Napier.

Any era can be studied within the 3,700 year reign of unit fractions, transliterations of Egyptian, Greek, Arab or medieval Egyptian fraction problems must be obtained. The most difficult ancient documents to parse may be the ancient Egyptian hieratic texts. Parsing raw Egyptian fraction data into additive aspects of base 10 decimals was a singular focus of 20th century scholarship. The 20th century transliterations omitted initial and intermediate Egyptian fraction arithmetic statements by reporting fragmented calculations. Improved translations should be expected to report initial problem statements, and connected intermediate, and final answers, as the scribe recorded the data, sometimes within duplation proofs, and sometimes not.

For example, Ahmes includes updates of scribal initial, intermediate, answers, and proof statements. Updated scribal initial, intermediate and final answers followed by confirming duplation proofs that update scribal arithmetic operations. Middle Kingdom 2/n tables converted 2/n to optimized unit fraction series by scaling 2/n by LCMs written as m/m writing 2m/mn such that the additive aliquot parts of mn were selected to write out optimized, but not always optimal, unit fraction series.

To resolve garbled transliteration issues, and create clear translations, rational number decoding themes parse the ancient raw data summarized by Marshall Clagett, Ancient Egyptian Science, Vol III,1999, Joran Friberg, Unexpected links of Egyptian and Babylonian Mathematics, 2005 and Victor Katz, editor "The Mathematics of Egypt, Mesopotamia, China, India an d Islam", 2008. Clagett, Friberg and Katz published 20th century transliterations of the Rhind Mathematical Papyrus, the Kahun Papyrus, and other hieratic texts. Clagett, Friberg and Katz(editor for Imhausen,Robson) published recent books that omitted important discussions of the Akhmim Wooden Tablet and misreported important scribal arithmetic and algebraic geometry recorded in the RMP, MMP and Kahun Papyhrus.

The raw transliterated Egyptian fraction, algebra and algebraic geometry made available by Clagett, Friberg and Katz (i.e. Annette Imhausen) omitted vital discussions of unified scribal methods and other considerations that are being made available by 21st century journal articles (the most recent Aug. 2010, scribal algebraic geometry).

Twenty-first century journal articles are adding back missing 20th century initial and intermediate calculations to correct translation errors. Increasingly, 21st century journal articles, beginning with Hana Vymazalova and Tanja Pemmerening in 2002 have pointed out under valued Egyptian fraction decoding themes. Theoretical and practical decoding theme were reported in a 2006 journal article. The theoretical and practical aspects parsed the 1900 BCE Akhmim Wooden Tablet (AWT) and 1650 BCE Rhind Mathematical Papyrus (RMP). The AWT began and ended with five divisions of a hekat unity written as (64/64). The (64/64) theme corrected Georges Daressy's 1906 AWT transliteration errors by showing that the AWT scribe exactly divided a hekat unity (64/64) by 3, 7, 10, 11 and 13, writing out binary quotient (Q/64) and a scaled 1/320 of a hekat (5R/n)ro remainder. The RMP used the AWT method over 60 times, 10 times in RMP 47.

The RMP stressed shorthand calculation of answers, and at times proofs of the answers, using a well traveled method that was passed down to the Greek era, as mentioned by Plato, and medieval scribes, as mentioned by Fibonacci. Plato's mathematics, in THE REPUBLIC, mentioned shopkeepers applying theoretical unities within Greek era weights and measures units. To study Fibonacci's Egyptian fraction algebra, geometry, finite arithmetic, weights and measures, decoding doors need to be opened. Reading Fibonacci's 1202 AD Liber Abaci, including Sigler's footnotes, it is clear that medieval weights and measures closely followed Platonic, Greeks and 1,500 older Egyptian scribal methods, as did Fibonacci's seven rational number conversion methods, written as (n/pq - 1/m) = (mn -pq)/mpq statements, with m an LCM.

To begin the 4,000 year old discussion, the 800 year old Liber Abaci, the book was Europe's arithmetic book for 250 years, offers clear unit fraction methods that connect to the oldest Egyptian unit fraction methods. Fibonacci gathered base 10 numeral Arabic texts written from 800 AD to 1200 AD. Prior to 800 AD Greek and other ciphered systems were recorded in Arabic and other regional languages. There are a large number of 800 AD to 1200 AD Arabic numeral texts that amplify the Liber Abaci's math foundations. Scholars gather algebraic texts before and after 800 AD. Scholars also gather arithmetic texts from several eras that amplify the first 124 pages of Fibonacci's 500 page book.

The first 124 pages of the Liber Abaci, counted by Sigler's 2002 translation, and footnotes, show that Fibonacci converted rational numbers to unit fraction series by seven subtraction rules (distinctions). Five of the medieval distinctions were written as(n/pq - 1/m) = mn/(mpq) connect to the Egyptian multiplication method. Ahmes' understood(2/n - 1/m) = (2m -n)/(mn) statements as proofs. Four medieval distinctions look and act like Ahmes red auxiliary method, though translating into a subtraction context. Arabic linguists teaming with Classical scholars offer additional resources that further Greek and immediate Arabic sources from which the Liber Abaci was written into Hindu-Arabic numerals.

Considering the small number of Greek mathematical texts (written in Greek), Egyptian language introductions to Euclid, Archimedes' calculus, Plato's mathematics offer interesting clues. For example, on the numeration level Greeks ciphered numerals 1:1 onto Ionian and Doric alphabets as Egyptian scribes ciphered the counting numerals 1:1 onto hieratic symbols. The Greek mathematical texts show that language hints left by Plato, and others, are of value.

Considering the Egyptian mathematical texts, hieratic was the most common. The longest hieratic text is the Rhind Mathematical Papyrus. The first 1/3 of the text is taken up the 2/n table and 51 optimized unit fraction series that converted 2/3, 2/5, 2/7, ..., 2/101 into a table. The 2/n table red auxiliary numbers assisted in creating the table, a topic that has been controversial since the Hultsch-Bruins method was published in 1895.

In 1900 and 1906 Egyptian multiplication and division began to be decoded by Schack.Schackenberg, and Daressy. Peet, Chace, 1920s scholars and Gillings muddled the pre-1906 views by only reporting the additive aspects of the hieratic texts. Sadly the pre-1906 views of Hultsch's 2/n table, Schack-Schackenberg's RMP 69-78, Berlin Papyrus, and Daressy's RMP 81-83, Akhmim Wooden Tablet were downgraded and became controversial topics for 100 years.

Gillings in 1970, "Mathematics in the Time of the Pharaohs", had only considered two additive rules, stated on page 110:

1. "Working only with the same methods, techniques, and notations available to the Egyptian scribe, we now attempt to reproduce some of these unit fraction tables 'ab initio'.

The Latin phrase 'ab initio', "at first he didn't notice anything strange" introduces a legalistic term 'at the beginning' to imply that scholarship started at the beginning. In retrospect insufficient self-awareness had been applied. The error of not finding Ahmes' beginning point, by pointing out an attested calculation, is parallel to the 1799-1825 barrier that hid the three meanings of Rosetta Stone's hieroglyphic writing for 25 years.

2. "We must of course eschew any modern refinements that could lead us to obvious simplifications. It may be a little irksome, but w have to try to think how the scribe would have thought, to imagine we are writing in hieratic, and to be logical only to the extent that we could expect the scribe to have been logical"

Concerning rule 1, it was self-serving for anyone to suggest an 'ab initio' posture before attested classifications of the RMP problems, and the methods, were accepted by independent interdisciplinary teams. A validated 'bi-lingual' context of Ahmes Egyptian fraction has been found. Modern rational number operations were taken from Fibonacci's 800 AD to 1600 AD number theory, methods that been taken Archimedes rational numbers, methods that had been taken from Ahmes rational numbers. Note the unbroken chain of custody, an appropriate legal term.

Concerning rule 2, one aspect of the statement is true, that Ahmes' text must not introduce modern ' broken feelings'. Gillings attempted to offer a rigorous 'outside of the box' rule to avoid modern guesses. Attested decoding paths, found in the 21st century connect 3,600 years of continuous Egyptian fraction use by a third 'outside the box' rule. Gillings got lost in the modern and ancient forests, amidst the trees. Each ancient tree, each red auxiliary number, and so forth, must be identified and validated, as the third box names and validates.

A 'third outside of the box' rule assisted in the decoding of the Egyptian Mathematical Leather Roll (EMLR) vy returning missing scribal steps.. The EMLR used non-optimal LCMs as the 2/n table, RMP 7-20 and RMP 36 used optimized LCMs. RMP 36 identified the red number aspect of Ahmes' thinking. Gillings missed the subtle details of Ahmes thinking by grouping RMP 7-20, and the EMLR data as identities, rather than several types of LCMs. No attempt was made by Gillings to read RMP 7- 20 in the context of the EMLR's implied use of LCMs and RMP 36's actual use LCMs that exposed red auxiliary numbers added to numerators and optimized unit fraction series. The Kahun Papyrus and its arithmetic progression used a meta arithmetic progression formula that was common to RMP 40 and RMP 64, a fact ignored by many. The Moscow Mathematical Papyrus and arithmetic geometry used several unique formulas. All five texts have been re-parsed after 2005 finding like-problems in the RMP, and other texts, following new decoding paths, and classifications. Concerning the AWT over 40 quotient and exact remainder examples were written in the RMP, 29 times in RM9 81, and seven times in Ahmes' bird-feeding rate method (RMP 83), related facts that Gillings and others had garbled. Removing the garbled statements with the simplest reading of Egyptian texts requires Egyptian fractions to be written in vulgar fraction form - thereby revealing the four Egyptian arithmetic operations.

The KP arithmetic progression was read in 1992 and slightly improved upon in 2005 by finding the simplest decoding path that connects to the largest number of ancient texts. For example, the RMP 40 and RMP 64 defines a KP decoding path preferred by John Legon. Legon's approach is superior to RMP 24-27, a path preferred by algorithmic researchers, since RMP 40, RMP 64 and other like data was excluded. Legon's approach is also superior to RMP 39, a path that excluded RMP 40, RMP 64, and Reisner Papyrus quotient and remainder data.

The simplest rule resolves certain controversies and thereby decodes the poorly reported texts. To read hieratic numbers, the numbers must be parsed from parent ciphered words. This step takes practice. For example, ro meant 1/320 of a hekat. The ro's relationship to the hekat had multiple uses. First, ro was a scaled remainder. Second 320 ro replaced one hekat for partitioning by large divisor purposes.

Ciphered numerals, such as Ahmes' 2/n table, and the RMP 40's arithmetic progression applications, several classes of ancient problems should be investigated to deeper levels. Excessive modern abstractions, however, reading sensed pyramid and other numerical text images, should be minimized. Mathematicians have fairly avoided this risk by excessively following Gillings 'ab initio' rules. A balanced decoding path is available by replacing the long unit fraction series with one vulgar fraction. Note that the ancient arithmetic operations are read by modern looking and acting statements.

Note that standard Egyptology 'dictionary' meanings that introduce classes of problems should be set aside, at the outset. Group like-problems. Follow the numbers. Allow grouped problems to assist the ancient numbers to speak for themselves, no more, or no less.

That is, do not prejudge the numerical output until meta parsings take place. Bring in disinterested third parties to confirm your work. It is not rare to find unique 'number' decoding paths. Note that not one decoding path works to decode Ahmes' 2/n table members, and all classes of 84 problems listed in the RMP. There are eight major decoding paths: addition, subtraction, multiplication, division, optimized LCMs, quotients and remainder division, arithmetic progressions, algebra, and geometry formulas.

Ahmes' 2/n table and 87 problems were not written in additive and multiplication operations. Singular additive and multiplication decoding paths, reported in the 1920s, seriously retarded Egyptian fraction research by not reporting non-additive subtraction and divisions methods that Ahmes and other scribes used to solve 2/n tables, many other classes of mathematical methods, another being algebraic geometry. Today, broader decoding paths read over 10 classes of RMP methodologies, all of which used closely related modern-looking arithmetic operations, when properly decoded.

Several meta decoding doors parse hieratic mathematical problems and methods. It is known, for example, that 20th century transliterations, presented as translations, need to be updated. Classifications of Ahmes' math were unique to 20th century scholars. Updated classifications have been attested by interdisciplinary teams have opened new decoding doors. Additive classifications, that dominated 20th century journals publications, are corrected by adding back missing addition, subtraction, multiplication, and division steps and operations.

It is important to independently confirm every step. For example, Ahmes' 2/n table members, and arithmetic progression problms offer unique issues. Double and tripe checking the decoding door used to parse each problem is required. Work in interdisciplinary teams. Resolve conflicting decoding paths in your group. Your group may also come up with better "Occam's Razor" decoding paths than were published in Egyptology and math history journals of the 20th century.

Ahmes loved numerical formulas, though few were fully described. Hence decoding door need to fully parse each scribal formula. Ancient texts are revealing new classes of formulas.

A third class of formula, decoded in 2005, shows that the Akhmim Wooden Tablet (AWT) weights and measures units were used in over 40 RMP examples. The 1923 additive translation of the AWT by Peet (1923) only read the 1/320 hekat aspect, thereby missing its formula. The majority of Peet's oversights are easily corrected. For example, the formula:

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

(6400/64) times 1/n = Q/64 + (5R/n)*ro (RMP 47)

recorded quotient (Q), and scaled (5/5) remainder (R) to 1/320 (ro) parts of the hekat. Ahmes used two-part quotient and remainder over 60 times. The division by n and multiplication by 1/n formula calculated two-part answers that Peet, Gillings and 20th century scholars had not identified. Adding back the scaled remainder arithmetic formula to Ahmes' tool kit began with Hana Vymazalova, a graduate student, Charles University, Prague, by publishing an AWT paper in 2001. In 2006 six AWT and 30 RMP data elements were published that reported a generalize multiplication formula was used 10 times in RMP 47.

Decoding new formulas, including unscaled and scaled arithmetic methods, 'dictionary' sides of the elements can be parsed by teams of mathematicians and linguists. Teams should discuss scopes and details contained in Middle Kingdom economic issues connected to Egyptian fraction discussions. When Middle Kingdom fragmented Egyptian fraction sentences are reconstructed and double checked by attested Middle Kingdom formulas decoding chapters can be closed.

Concerning cubits, RMP 53-55 data reports a few interesting facts. These problems discuss cubit and khet units written in setats, 100 cubit by 100 cubit areas and setats divided into 1/100 setat strips, and mh units. Reading RMP 54 includes Ahmes' implicit use of the LCM 2/n table conversion method. Ahmes scaled a setat to (4/4) and (2/2) before multiplying by 7/10, 14/10 and 28/10

1. (7/10)*(4/4) setat = 28/40 setat = (25 + 3)/40 setat

as the 2/n table LCM conversion method would have written out

2. 5/8 setat + 300/40 mh = 5/8 setat + 7 1/2 mh

Ahmes' answer.

SUMMARY:

Old Kingdom connections to Egyptian fractions' 2050 BCE birth are being fairly translated into modern arithmetic statements in the 21st century. One of the clearest topics shows that Old Kingdom infinite series system and the finite series Egyptian fraction system developed on separate tracks. After 2,050 BCE Egyptian scribes corrected the Old Kingdom cursive round-off errors into an exact Egyptian fraction series. The replacement Egyptian fraction system dominated the ancient Near East, Greece, Arab and medieval and its regional economic systems for 3,600 years until base 10 decimals ended its life in 1585AD.

Elements of modern arithmetic operations, and algebraic geometry, especially multiplication and division, were hidden hieratic scribal notes. Confirming elements of scribal 2/n table methods were discussed in RMP 36. In RMP 36 Middle Kingdom rational number conversion methods detailed four arithmetic operations built upon modern arithmetic operations. Scribes proved unit fraction answers by applying one Old Kingdom multiplication operation, returning answers to beginning numbers, often identities (or unities). Two RMP 38 proofs multiplied 320 by 7/22 (101 9/11) and 101 9/11 by multiplying by 22/7, obtaining 320. RMP 66 did the same thing by Ahmes dividing 10 hekat (3200 ro) by 365 (obtaining 8 + 280/365), and proved the unit fraction answers 8 + 2/3 + 1/10 + 1/2190) times 365 = 3200 ro showing that Middle Kingdom multiplication and division operations were inverse to one another (in the modern sense). In RMP 36 and 37 three discussions of red auxiliary numbers fully expose scribal alignments of a red number numerator to one unit fraction. RMP 41,42 and 43 vividly recorded algebraic geometry with radius (R) replaced by diameter (D/2) and pi buy 256/81, using four formulas. MMP 10 used sqrt(A) = (8/9)D cubits squared, the simplest formula, and the Kahun Papyrus used sqrt (V)= (2/3)H(4/3)D) khar, in RMP 43; multiplied 1500 khar by 1/20 into (75) 400-hekat in RMP 44; 400-hekat and 100-hekat multiplications by 1/n into quotient (Q/64) 4-hekat and 1-hekat quotients and remainder (5R/n) 4-ro and 1 ro remainder in RMP 47.

Concerning the 3,700 year life of Egyptian fraction arithmetic, when stripped of Greek, Arabic and medieval cultural differences, a 2,800 year era of Egyptian fraction system was translated by Arabs into Hindu-Arabic numerals in 800 AD maintaining ancient and modern arithmetic definitions. Replacement Hindu-Arabic numeration system dominated Latin speaking and writing Europe after 999 AD (urged by Pope Sylvester) until 1454 AD with the fall of Byzantium. Egyptian fraction mathematics formally died when a replacement algorithmic decimal numeration system was published in 1585 AD (approved by the Paris Academy) accepted Simon Stevin's two books, one for business and one for science.