Wednesday, August 07, 2013
Wednesday, May 10, 2006
Egyptian Fractions
ABSTRACT
The history of Egyptian fractions is outlined by three documents, two ancient Egyptian and one medieval. The older of the two documents, the EMLR, was written by an unknown student scribe before 1800 BCE.The text describes 2/2, 3/3, 4/4, 5/5, 6/6, 7/7 and 25/25 multiples of 1/p and 1/pq. scaled to vulgar fractions in alternatives ways. For example 1/8 was scaled by multiplication three ways; the most complicated: 1/8(25/25) = 25/200 = (8 + 17)/200 = 1/25 + [(17/200)(6/6) = 102/1200 = (80 + 16 + 6)/1200] = 1/25 + 1/15 + 1/75 + 1/200, one of two out-of-order series cited in the text.
The second document is the Rhind Mathematical Papyrus, written by Ahmes in 1650 BCE, and a hard-to-decode 2/n table. The decoded 2/n table extended two EMLR conversion methods converted 2/p and 2/pq fractions to concise Egyptian fraction series, also in a multiplication context in 2011. Ahmes used third and forth conversion methods. The third was re-discovered by F. Hultsch in 1895 and confirmed by E.M.Bruins in 1944 . A generalized fourth method appears in RMP 36, scaled 53/53 = 2/53(30/30) + 3/53(20/20) + 5/15(12/12) + 15/53(4/4) + 28/53(2/2).
L.E. Sigler in 2002 reported 1202 AD medieval rational number conversion methods recorded in a new subtraction context devised by Arabs in 800 AD . Silger's report includes seven conversion methods, written down by Leonardo de Pisa (Fibonacci) Liber Abaci. in 1202 AD. Sigler's seven methods, or distinctions, converted vulgar fractions to elegant and not-so-elegant Egyptian fraction series. Methods one and two factor and convert vulgar fractions following rules developed in the EMLR and RMP 2/n table. Method three parses 8/11 to 2/11 + 6/11 and converts each vulgar fraction to an Egyptian fraction series. Methods four, five, and six detail Egyptian and medieval versions of the Hultsch-Bruins method. Method six is an exclusive medieval method. Method seven includes two methods, the first being an extension of Ahmes' H-B method. Leonardo allowed a second subtraction, a method that modern algorithms have extended to n-subtractions, creating no-so-elegant Egyptian fraction series. The second half of the seventh method followed the pattern set down by the first of the seven methods. It shows that Leonardo generally factored vulgar fractions before choosing two or more component vulgar fractions, and calculating elegant Egyptian fraction series from the smaller vulgar fractions.
BACKGROUND
Modern scholars have had great difficulties in formally pinning down, and confirming, a clearly defined historical context in which Egyptian fractions were first understood by ancient scribes. Scholars had often assumed that scribal arithmetic had only been been additive, based on aspects of the hieratic Egyptian fraction notation being suspected to have been linked to the Old Kingdom and hieroglyphic binary arithmetic. Modern scholars have often read Egyptian fraction documents, like the Liber Abaci, and skipped over its easy to read 1/p and 1/pq series and closely related quotient and remainder threads, two subjects that connect a subject that dominated Western mathematics for at least 3,6000 years. The EMLR , Reisner Papyri , Akhmim Wooden Tablet, and the Kahun Papyrus, a medical text, document the beginning era of Egyptian fractions, while the Liber Abaci documents its last chapter, as an obituary.
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It should also be noted that the early cursive hieratic form of writing numbers through its binary fraction (Horus-Eye) math statements, has been oddly seen by scholars (ie Neugbauer), as having gone into a form of 'intellectual decline' when reaching the Middle Kingdom and it hieratic numeration and hieratic math.
One aspect of the long proposed, and unconfirmed intellectual decline, tends to stress comparisons of Egyptian astronomy and weights and measures to its Babylonian counter parts. The older Babylonian numeration, astronomical and certain math facts have bee stressed as intellectually superior (by Neuebauer), while at the same time scholars (like Neugebauer, writing in Exact Sciences in Anitiquity) have overlooked and minimized the certain aspects of the body of mathematical knowledge contained in the Egyptian hieratic math texts, beginning with the unified and set of generalized methods for writing rational numbers in the RMP 2/nth table, as decoded by modern arithmetic.
One proposed positive suggestion made by scholars noted that Horus-Eye binary numeration formed a basis of additive views of Egyptian fractions, a view that clouded any potential suggestion of hieratic numeration or math containing any abstract properties.
Another less frequentlly formalized suggestion was that cursive binary numeration formed an algorithmic method utilized by Egyptian scribes, a proposal that may mean that Old Kingdom math contained abstract features at one time, most of which were lost by the time of the Middle Kingdom and hieratic math.
The modern Egypotology community has often stressed a proposed hieroglyphic and Horus-Eye link to such an extent that the hieratic math texts have not been independently read by many Egyptologists and math historians to rigorously test if other intellectual foundations may have existed during the Middle Kingdom. Otto Neugebauer, Exact Sciences in Antiquity, one of the most often quoted experts, has written the definitive minimalist treastise. Neugebauer concluded, without citing specific examples or attested references, that the 2/nth table table method of writing Egyptian fractions followed a mathematical form that appeared in a 'intellectual declined' state.
Thus, a problem of scholars not frequently or not fairly attempting to look for an 'original' first context from which Egyptian fractions may have first been used, beyond the hieroglyphic paradigm, considering only the hieratic texts, has been a major translating, decoding, 'blind- in-one-eye' problem since the RMP was published in 1879. The blind- in-one-eye metaphor is pertinent because scholars have tended to overstress unproven hieroglyphic foundations of hieratic math, while, at the same time, not attempting to independently read the hieratic texts as a potentially separate body of knowledge.
Cleary early 1920's attempts to fairly and fully decode read the Middle Kingdom Egyptian fractions texts, often used 'one additive eye' beginning with the Rhind Mathematical Papyrus (RMP) and its 2/nth table, 8 pages of the RMP's 25 pages. The 2/nth table has proven to be filled with several types of modern logical dead-ends, and conflicting opinions., reported by modern scholars. Conflicting opinions have been offered by members of both of the history of mathematics and the Egyptology communities. Opinions have ranged from the overly 'restrictive' minimalist to several 'far out' pedagogical positions, most often based on very small samples of scribal math ematical texts, and then making overly broad a priori generalizations (i.e. Neugebauer's intellectual decline proposal).
J.J. Sylvester was the first noted mathematician that provided serious comments on the subject, taking a very small slice of Liber Abaci data. Sylvester first wrote on Egyptian fractions in 1883, consulting with Cantor, suggesting a contrived analysis and basis of the 2/nth table. Later, in 1891, Sylvester suggested that an algorithmic pattern may have been used by Leonardo in creating n/p and n/pq Egyptian.fraction series. Sylvester offered his 1891 suggestion by fairly noting that Leonardo (Fibonacci) had reported a recreational use of a proto-greedy algorithm in 1202 AD. Leonardo had revised the traditional Hultsch-Bruin method, a one subtraction method that calculated elegant Egyptian fractions, to a two step subtraction method that calculated not-so-elegant Egyptian fractrions.
Among Sylvester's minor oversights was one where he had unfairly omitted Fibonacci's two parametric methods that converted vulgar fractions into Egyptian fraction series. One Fibonacci method converted n/p vulgar fractions, and the second method converted n/pq vulgar fractions. Heinz Leuneburg detailed this two-part subhject in his 1993 review on the Liber Abaci, as did L.E. Sigler in his 2002 complete translation of the Liber Abaci. Sylvester may have only offered a 'far out' proposal, one that several researchers have refuted by noting that Fiboancci's method was limited to a second subtraction, possibly not using any form of algorithm. Ahmes may have also only used one subtraction to build his 2/p series.
Given that the general class of algorithms, in which greedy algorithms are a member, are confirmed to have been formalized by Islamic mathematicians (around 800 AD), any potential greedy algorithm use by Ahmes should have been given a very small historical chance to have been confirmed, by Sylvester or anyone else, unless a large number of 2/nth table members can be computed by it. Given than only four of 51 fall into this 'possible' category, Ahmes, therefore, most probably had not gained significant insights from algorithmic thinking, The EMLR shows that multiiples of 2, 3, 4, 5, 7, and 25 were generalized in the RMP 2/nth table to convert 2/pq series by selecting the a multiple (p + 1)/(p + 1).
That is, Fibonacci's recreational work that took place 3,000 years after Ahmes' work, most likely had not been based in Islamic suggestions, but a simple extension of the traditional Hultsch-Bruins method. Therefore, Sylvester's algorithmic method should be seen unrelated to the Liber Abaci unit fraction statements, several of which (six of ten) that had been passed down from Middle Kingdom scribes writing 2/nth tables.
Yet, despite the well defined historical 3,200 year out-of-sequence facts, related to the first appearance and use of the greedy algorithm, and other discussions related to its possible use by Ahmes and other scribes, continue to appear on otherwise reputable web pages. One is St. Andrew's University, Scotland. St. Andrews cites David Eppstein's modern use of 10 algorithms, as first published in "Intelligencer" in 1991, rather than Sylvester's 100 year older views. Why does St. Andrews and other reputable universities cite the greedy algorithm when discussing Egyptian fractions?
Are these web pages trying to cloud what form of mathematics that the scribes like Ahmes actually used, reading only the hieratic texts, use to build their 2/nth tables? Or, are the web pages reporting a little of ancient history to justify a modern mathematical subject? Readers of this blog can make up their own minds if the greedy algorithm, and its many clones, had anything to do with scribal unit fraction arithmetic statements. For myself, I say none, or a very small chance (even in the Liber Abaci). But others are free to disagree, provided that hard scribal data is attached to a fair discussion.
Other scholars have differed with Sylvester. In 1895, F. Hultsch, an Egyptologist, suggested that an abstract solution to an aspect of the 2/nth table solved the majority of 2/nth table members by touching on a 'why' side of the scribal arithmetic. Hultsch's suggestion pointed out that divisors of the first partition of 2/nth table members could be further sub-divided, opening a divisor door to create members of the 2/nth table, without using an algorithm. This method has been confirmed several times, the first independent work being completed by E.M. Bruins in 1944 and published in 1952.
E.M. Bruins, working 50 years later in 1944 reached the same 'aliquot part', a non-algorithm, conclusion that Hultsch had discusssed. Today the method is named after both men, or the Hultsch-Bruins' Method. Oddly the Hultsch-Bruins method is not widely discussed in either the Egyptology or history of math communities in 2006. Yet, its main thesis still serves as a central method used by many that directly and easily decode many 2/p members of the 2/nth table and other Egyptian fraction series published elsewhere in other hieratic texts, especially in the subtraction mode, knowing the first partition derived by some previous study. Ahmes left a clue to his previous study of first partitions. It was 'red auxiliary' numbers, an LCM sorting routine.
G. Daressy, another Egyptologist, translated and analyzed the Akhmim (Cairo) Wooden Tablet in 1906. He pointed out a possible abstract link to the oldest form of Egyptian fractions. Daressy had shown that three of the five divisions of a hekat, using the divisors 3, 7, 10, and not 11 and 13 were exact two-part form statements. All five divisions were written such that the first-term used binary fractions and the second term included Egyptian fractions (followed by ro).
Daressy had been unable to directly prove that the division by 11 and 13 problems were also exact. Given no general rule for all five AWT division statements, his translation project gained little traction in the Egyptology or history of math communities. Sadly his raw data and analysis were soon forgotten, as the Hultsch-Bruins Method had also been often forgotten. Possibly by chance, in 2002, Daressy's 1906 work began to resurface, per the work of Hana Vymazalova, a Charles U., Prague, graduate student.
Before discussing Daressy in the 21th century, let's continue in the 20th century. T. Eric Peet, a mathematician, in 1923, attempted to unify the conflicting practical views concerning the contents of Egyptian fraction arithmetic. Peet's writings disputed Daressy and Hultsch's analyses and conclusions. Peet cited Daressy's incomplete logic and inconclusive data as insufficent, while (at the same time) totally ignoring Hultsch's earlier work. Peet's form of Egyptian arithmetic came down on the side of the additive side of the discussion, prematurely concluding, as a minimalist, that scribes had not used any form of abstract mathematics, or had not experimented or developed any abstract definitions of number. Yet it is clear that a hekat unity, 1 = 64/64, was used in hekat division, as well
as several additional examples of abstract definitions of number are included in the Egytptian Mathematical Learther Roll and the RMP and its 2/nth table.
One of Peet's proposals attempted to show that scribal division had been closely linked to scribal multiplication by using an inverse method, all additively based. 'False position', a second century AD East Indian innovation, possibly linked to Diophantus and 100 AD Chinese texts. False position, an algebraic method of finding roots, entered the hieratic Egyptian math literature at the time when Peet and others discussed Egyptian algebra dated to 1650 BCE. How could have 'false position' been found in Egypt, 2,000 years before it was discovered in China, Greece or India? The 1202 AD Liber Abaci covers this topic very well, as related to this history of Egyptian fractions subject.
Formal academic literature, published by the history of mathematics community in the 1920's, shows that Peet's analysis and conclusions only related to additive scribal arithmetic. Even though the proposed use of false position was widely accepted by DE Smith, Chace, Neugebauer any many members of the history of math community, this researcher has not been able to find a published hieratic text that expliciitly defined the method. Later Gillings and the bulk of the Egyptology community also signed-on to Peet's additive and related minimalist points of view. Minimalist thought is a sad situation that continues to cloud the historical record related to the actual scribal methods used by Ahmes, et al.
In 1987 Robins-Shute wrote their view of the RMP, the 2/nth table and Hultsch-Bruins, mentioning products (and not quotients) and other bits of information that they properely called remainders. Yet, in their otherwise
interesting book Robins-Shute explicitly denied the general use of quotients and remainders, as well as the Hultsch-Bruins Method, by Ahmes.
In 1995 Kevin Brown posted a clearer statement, citing a wide ranging view of Egyptian fractions on his web site. Brown suggested that Egyptian fractions, as a notation, may have been birthed from Egyptian gambling
practices. Math Cats, a grammar school based web site out of Maryland may have said it best, " We know they (Egyptians) used this system for over 2,000 years .. but we don't know why!".
That is, confusion related to the minimalist reading of Egyptian fraction arithmetic continued, infrequently challenged, until 2002. In 2002 a clear publication of the bottom half of the Akhmim Wooden Tablet by Hana
Vymazalova changed everything.
THE 'WHAT' AND 'WHY' SIDE OF HIERATIC TEXTS
The 'why' side of Egyptian fraction began to be reported in 2002. A clear picture of the scribal context of Egyptian fraction was published in 2005 on the internet follows these facts:
A. Akhmim Wooden Tablet, 2,000 BCE, began to 'decoded' in 2002
1. The 'what' side of this text shows that a scribe partitioned a hekat by five divisors, (3, 7, 10, 11 and 13). This was done by beginning and ending with a hekat unity 1 = (64/64), as Hana Vymazalova, the Charles U, Prague grad student, clearly detailed in "Wooden Tablets from Cairo ..., Archiv Orientakni, Vol I, pages 27-42, 2002.
The first modern analysis of the AWT text had been published by G. Daressy in 1906. Daressy had not clearly seen all of the 'what' aspects of the text. What Daressy had seen caused Peet, in 1923, to strongly comment in opposition. Peet attempted to negate Daressy's incomplete analysis, one that tended to show say that Daressy's view of Egyptian division was incorrect, with a short comment concerning ro equalling 1/320th of a hekat. Peet's paper on Egyptian arithmetic excluding the methodology used to exactly divide a hekat by 3, 7, 10, 11 , 13 and
by implication, any n upto and including 64.
Well, it turns out that Peet had correctly viewed ro, a numerical word, equal to 1/320th of a hekat. But little else of Peet's analysis was accurate when he discussed the AkhmimWooden Tablet (AWT) in terms of a Egyptian division. For example, Peet had suggested that scribal division followed an inverse relationship to scribal multiplication. The AWT reports scribal division was closely related to scribal subtraction, a method that looked very much like the Hultsch-Bruins' method, and proof its 2-part division answers by multiplying the answers by the initial divisor (3, 7, 10, 11 and 13) calculating 64/64, the starting hekat unity value (as Hana Vymazalova published in 2002).
2. That is, the 'why' side of the AWT details abstract aspects of remainder arithmetic that Peet had not expected, nor reported in any way. The AWT acrtually reported the hekat unity (64/64) divided by n, by the statement:
(64/64)/n = Q/64 + (5R/n)*ro
with n = 3, 7, 10, 11 and 13.
a form that replaced the remainder 1/64th term with 5/320, thereby creating a constant 1/320 that was factored out of its associated Egyptian fraction term (5R/n), and Daressy had been on the correct historical path in 1906. A table of 29 hekat partitions were compared by Ahmes to hinu partitions, (as included in Gillings Mathematics in the Time of the Pharaohs book). One partition used a ro scaling factor, and the other omitted the use of ro, thereby exposing the common form of remainder arithmetic used by both partitions for those that look closely at the raw data.
B. Rhind Mathematical Papyrus
1. The 'what' side of this text begins with the 2/nth table. Algebra was recorded by scribes using 'false position', a notation that is not easily supported when remainder arithmetic methods of writing vulgar fractions are considered. In addition, Ahmes also wrote hekat partitioning problems, using two forms of shorthand that omitted the details that the Akhmim Wooden Tablet scribe used to explain the method. In total, Ahmes used hekat division, within its 1/10th unit,, to show that a hinu unity was written as 640/64, allowing n to reach 640 following the relationship:
(640/64)/n = Q/64 + R/(n*64) when
n = 10, or
(640/64)/10 = 64/64 = 1 hin.
However, Ahmes in his 15 other RMP 80 uses of hin cited
hin = 10/n
a relationship that avoided the use of ro and the 1/64 remainder term.
Again, let it be stressed that Ahmes listed a table of 29 hekat and hinu partitions, the first using the scaling factor ro, and the second omitting ro. Clearly, there is ample hieratic evidence that Ahmes thought and wrote
in these multiple levels of remainder arithmetic.
2. The 'why' side of the 2/nth table shows that algebraic identities were used by Ahmes, one for the majority of the 2/p conversions to Egyptian fraction series, and another for the majority of the 2/pq series. In addition, the 'false position', short for 'false supposition' method of Egyptian algebra was not used at all, as reported for
over 100 years. The Liber Abaci shows that certain scribes had access to two methods to solve algebra problems. The easiest was the direct method, one that appears to have been used by Ahmes.
The vulgar fraction methods, used in all the Ahmes algebra problems, solved for unknowns by first finding a vulgar fraction. A sorting routine, denoted by red auxiliary numbers, was then used, from time to time, to find the optimal solution to the vulgar fraction Egyptian fraction series, following 2/nth table rules. No supposition or 'false position' thinking has been found using 2/nth table rules that follow remainder arithmetic considerations.
Concerning hekat division by n, the hin unit, 1/10th of a hekat followed a different remainder airthmetic statement, it being:
hin = 10/n
example, n= 64, the largest n in his table, or
(64/64)/64 = 1/64
with its corresponding hinu value being found by:
hin = 10/64 = (8 + 2)/64 = 1/8 + 1/32
Note that the hinu method omitted the use of ro, a special term that had reduced the size of the hekat method by writing exact remainders. Ahmes' table of 16 hekat two-part statements therefore had included the use of (5R/n), an Egyptian fraction series, for reasons best left to the scribes, with the corresponding 16 hinu values writing 1/10th of a hehat hinu values using the shorthand 10/n relationship by stressing the use of divisor n.
C. Egyptian Mathematical Leather Roll (EMLR)
This scroll had gathered dust in the British Museum unrolled from 1864 to 1927, a total of 64 years. When its 26 lines of Egyptian fraction series were translated, only the additive aspects were stressed in 1927. There had been no comparison of EMLR methods to the methods used to create the 2/nth table, or any other hieratic text, in 1927. Myopia had governed the 1920's scholars.
The text cites six methods to convert 1/p and 1/pq rational numbers to Egyptian fraction series. The first five are multiples of 3, 4, 5, 7 and 25. The sixth is an identity form 1 = 1/2 + 1/3 + 1/6, a method found in the RMP 2/nth table to write 2/101. A seventh method introduced the six methods by showing that a miltiple of 2 of four unit fractions only allowed for equal fraction sub-divisions, technically not Egyptian fraction answers.
The multiple of 25 case has been reported in the journal Historia Mathematica. The 1/8 example can be solved by an algebraic identity.
example A = 25
1/8 = 1/25* 25/8
= 1/5 * 25/40
= 1/5 * (3/5 + 1/40)
= 1/5*(1/5 + 2/5 + 1/40)
= 1/5*(1/5 + 1/3 + 1/15 + 1/40)
= 1/25 + 1/15 + 1/75 + 1/200
A multiple of 25 version of this problem follows Occam's Razor, as all 26 EMLR answers can be read.
Note that the out-of-order listing of 1/25 + 1/15 was left as a clue to the scribal methodology. The abstract method was repeared when calculatinng a line that began with 1/16, using A = 25. The abstract method was most likely used to prepare beginning students later to accept the closely related and generalized RMP method:
2/pq = (2/A) * (A/pq),
with A = (p + 1), as used 24 times in the RMP
or by a multiple of (p + 1)/(p +1) such that
2/pq = 2/pq x ((p + 1)/(p+ 1) = 2/(p + 1) x (p + 1)/pq
D. Reisner Papyri
Gillings correctly showed that the Reisner Papyri, housed in Boston, divided worker digging rates by 10. The raw data actually follows a quotient and remainder structure, as all vulgar fractions were written. The quotient appeared as an integer, and the remainder appeared as an exact Egyptian fraction series.
E. Liber Abaci (Wikipedia)
The Liber Abaci (blog) is a 1200AD document that copied the tradition Egyptian fraction methods from Arab sources. Please refer to the linked Wikipedia and blog information on this document.
Translating the seven methods, or distinctions, into the older Egyptian fraction context. The first 126 pages summarize this information. The seven methods describe at least 10 medieval and older methods that Leonardo used to convert vulgar fractions to Egyptian fraction series. Additional discussions on this suhbect can be found at::
http://en.wikipedia.org/wiki/Liber_Abaci
Method one contains three methods, as noted above. The first method may date to Ahmes and his Egyptian scribal style of writing parts of a fraction, in long hand, as the EMLR and RMP used factoring and identities to convert several vulgar fractions. Leonardo converted 1/18 by factoring 1/2 x 1/9, with 1/2 = 1/3 + 1/6 such that
1/18 = 1/27 + 1/64. The EMLR used this method four times, and the RMP used it to convert 2/101.
The second two methods discuss medieval arithmetic and its short hand notation, all useful in finding elegant coversions. Leondard's medieval arithmetic notation, at times, wrote out elegant and not-so-elegant Egyptian fraction answers. These medieval arithmetic notations used by Leonardo have not been thoroughly discussed by math historians related to their use in finding elegant Egyptian fraction answers. Hopefully I will run across analyses of this class information in ways that connect to this discussion, the oldest form of Egyptian fraction arithmetic, and add it to this blog.
Method two wrote 5/6 as (3 + 2)/6 = 1/2 + 1/3 as the EMLR wrote out all of its 1/p and 1/pq answers, and Ahmes used over and over again. The EMLR increased 1/p and 1/pq to multiples of 2,3, 4, 5, 7, and 25, such that conversions were completed as this LA section details in n/p and n/pq problems, and completed the finding of Egyptian fraction series by parsing the numerator in this manner.
Method three details 8/11 = 2/11 + 6/11, a tabular method that 400 AD Coptics used. The Coptic style was reported by David Fowler in Historia Mathematica in 1982 citing answers to problems from n/5 to n/31, and n/4 to n/32, or thereabouts. The RMP 2/nth table can also be seen as the 2/11 aspect of this method. Knowing any 2/n Egyptian fraction series, a second table entry can be found by doubling, such that 2/p + (n- 2)/p = n/p.
Methods four, five and six focus on the Hultsch-Bruins method, as used in all of Ahmes 2/p series. These three methods increasingly define complex definitions of the very old Hultsch-Bruins method, as first reported in the modern era by F. Hultsch in 1895.. The basic H-B method is discussed in method four. It was used by Ahmes to convert 2/p vulgar fractions to unit fraction series by first selecting a first partition with a highly composite denominator, and so forth, as explained elsewhere. Methods five and six show that the first partition need not have been a unit fraction, a style that Ahmes did not adopt. For example, Leonardo method six to convert 20/53 by subtracting 3/8 after raising it to a multiple of 6, 18/48, writing out an answer, 18/48 1/8 0/53. This answer is confirmed by subtracting each fraction, given a little practice, exactly as method four and five were solved.
Method seven includes two methods. The first method is an extension of the Hultsch-Bruins method. Leonardo allowed a second subtraction, when the remainder's vulgar fraction could not be converted by method two, creating a not-so-elegant answer, possibly a form of recreational mathematics.
The second method discussed under method seven covers a factoring method first noted in the RMP 2/nth table, by Fowler and others, where 2/95 was factored as 1/5 x 2/19, with 2/19 taken from the 2/nth table. Adding method one (a) and method seven (b) generally factoring was available to Leonardo, Ahmes and everyone working with Egyptian fractions, any time during its 3,200 year recorded history.
In Leonardo's case Sigler footnotes stressed that an error had been made converting 4/49 by a quasi-greedy algorithm method. However, factoring 4/49 to 1/7 x 4/7 with 4/7 = 1/2 + 1/14 such that an elegant unit fraction answer 4/49 = 1/14 + 1/98 was preferred by Leonardo. Again, the not-so-elegant 'quasi greedy algorithm' version of the 4/49 problem may have been recreational in scope.
There is more to the story. For example the RMP 2/pq method seemingly was not discussed by Leonardo. However, recalling that Ahmes used a (p + 1) multples was an aide in this work, it is easy to see that Leonardo also used the (p + 1) relationship to solve his Hultsch-Bruins and n/pq problems in methods four, five and six. Additional research will be conducted to determine if Leonardo also wrote his 2/pq elegant answers in a manner that Ahmes wrote his 2/pq answers.
That is, these seven suggestions show that the Liber Abaci assists in parsing aspects of the oldest Egyptian mathematical texts, and its traditional Egyptian fraction arithmetic. The 1200AD work of Leonardo, particularly the Liber Abaci, will continue to be parsed in extended ways, allowing additional aspects of the oldest Egyptian mathematical texts, such as the EMLR and the RMP 2/nth table, to read in deeper ways as scholars first began this quest 120 years ago, by solely analyzing the RMP.
REFERENCES
1. Sigler, L,E,, " Fibonacci's Liber Abaci, Leonardo Pisano's Book of Calculations" Springer , New York, 2002, ISBN 0-387-40737-5.
2. Lüneburg, Heinz (1993). Leonardi Pisani Liber Abbaci oder Lesevergnügen eines Mathematikers. Mannheim: B. I. Wissenschaftsverlag.
3. Ore, Oystein (1948). Number Theory and its History. McGraw Hill. Dover version also available, 1988, ISBN 978-0486656205.
F. Summary
Honoring Hana Vymazalova's 2002 discovery of a hekat unity equalling (1 = 64/64) is required because she unknownly opened a major door to scribal remainder arithmetic. Large sections of previously hard to Middle Kingdom mathematical texts can now be easily decoded and fully translated into modern base 10 decimal notation. In the past only fragments of quotients and remainders had been translated, with many fragrment had left unread by scholars (ie Chace noted one of the oversights in his 1927 RMP paper). Therefore, Peet's odd and often complicated 1923 views of Egyptian multiplication, and his related additive issues citing 'false position',
can now be proposed to be revised and dropped as historical in several areas of Egyptian mathematics. A generalized web page attempts to compare Ahmes' remainder arithmetic and Egyptian fractions, and is titled Remainder Arithmetric vs Egyptian Fractions. It includes a few scribal points serious students might like to consider, and debate, thereby replacing Peet's themes with attested .
Scribal remainder arithmetic, when seen in its original EMLR and RMP context, and used in the Liber Abaci in several closely related, ofters several challenges to our academic communities that wish to hold on to the 1920's authors' views.. It is expected that academics will, over the next 5-10 years, enter into serious discussions on this topic, and thereby update their 1920's views. It is also expected, that within the next 10 years, the 130 year old debate may confirm, in unexpected ways, the intellectual origins of Egyptian fractions. At that time Egyptian fraction studies may be revitalized within a well defined form of ancient remainder arithmetic, first reported in the Akhmim Wooden Tablet.. At that time, thanks to L.E. Sigler and others, a wider and deeper context will be presented in academia, describing the work of scribes and their Egyptian fraction arithmetic, algebra and weights and measures. Conclusions have been reached by the leaders of the history of math and the Egyptology communities, replacing the outdated views of the 1920s, once and for all. One example is given by a re-classification of Ahmes 84 problems.
The history of Egyptian fractions is outlined by three documents, two ancient Egyptian and one medieval. The older of the two documents, the EMLR, was written by an unknown student scribe before 1800 BCE.The text describes 2/2, 3/3, 4/4, 5/5, 6/6, 7/7 and 25/25 multiples of 1/p and 1/pq. scaled to vulgar fractions in alternatives ways. For example 1/8 was scaled by multiplication three ways; the most complicated: 1/8(25/25) = 25/200 = (8 + 17)/200 = 1/25 + [(17/200)(6/6) = 102/1200 = (80 + 16 + 6)/1200] = 1/25 + 1/15 + 1/75 + 1/200, one of two out-of-order series cited in the text.
The second document is the Rhind Mathematical Papyrus, written by Ahmes in 1650 BCE, and a hard-to-decode 2/n table. The decoded 2/n table extended two EMLR conversion methods converted 2/p and 2/pq fractions to concise Egyptian fraction series, also in a multiplication context in 2011. Ahmes used third and forth conversion methods. The third was re-discovered by F. Hultsch in 1895 and confirmed by E.M.Bruins in 1944 . A generalized fourth method appears in RMP 36, scaled 53/53 = 2/53(30/30) + 3/53(20/20) + 5/15(12/12) + 15/53(4/4) + 28/53(2/2).
L.E. Sigler in 2002 reported 1202 AD medieval rational number conversion methods recorded in a new subtraction context devised by Arabs in 800 AD . Silger's report includes seven conversion methods, written down by Leonardo de Pisa (Fibonacci) Liber Abaci. in 1202 AD. Sigler's seven methods, or distinctions, converted vulgar fractions to elegant and not-so-elegant Egyptian fraction series. Methods one and two factor and convert vulgar fractions following rules developed in the EMLR and RMP 2/n table. Method three parses 8/11 to 2/11 + 6/11 and converts each vulgar fraction to an Egyptian fraction series. Methods four, five, and six detail Egyptian and medieval versions of the Hultsch-Bruins method. Method six is an exclusive medieval method. Method seven includes two methods, the first being an extension of Ahmes' H-B method. Leonardo allowed a second subtraction, a method that modern algorithms have extended to n-subtractions, creating no-so-elegant Egyptian fraction series. The second half of the seventh method followed the pattern set down by the first of the seven methods. It shows that Leonardo generally factored vulgar fractions before choosing two or more component vulgar fractions, and calculating elegant Egyptian fraction series from the smaller vulgar fractions.
BACKGROUND
Modern scholars have had great difficulties in formally pinning down, and confirming, a clearly defined historical context in which Egyptian fractions were first understood by ancient scribes. Scholars had often assumed that scribal arithmetic had only been been additive, based on aspects of the hieratic Egyptian fraction notation being suspected to have been linked to the Old Kingdom and hieroglyphic binary arithmetic. Modern scholars have often read Egyptian fraction documents, like the Liber Abaci, and skipped over its easy to read 1/p and 1/pq series and closely related quotient and remainder threads, two subjects that connect a subject that dominated Western mathematics for at least 3,6000 years. The EMLR , Reisner Papyri , Akhmim Wooden Tablet, and the Kahun Papyrus, a medical text, document the beginning era of Egyptian fractions, while the Liber Abaci documents its last chapter, as an obituary.
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It should also be noted that the early cursive hieratic form of writing numbers through its binary fraction (Horus-Eye) math statements, has been oddly seen by scholars (ie Neugbauer), as having gone into a form of 'intellectual decline' when reaching the Middle Kingdom and it hieratic numeration and hieratic math.
One aspect of the long proposed, and unconfirmed intellectual decline, tends to stress comparisons of Egyptian astronomy and weights and measures to its Babylonian counter parts. The older Babylonian numeration, astronomical and certain math facts have bee stressed as intellectually superior (by Neuebauer), while at the same time scholars (like Neugebauer, writing in Exact Sciences in Anitiquity) have overlooked and minimized the certain aspects of the body of mathematical knowledge contained in the Egyptian hieratic math texts, beginning with the unified and set of generalized methods for writing rational numbers in the RMP 2/nth table, as decoded by modern arithmetic.
One proposed positive suggestion made by scholars noted that Horus-Eye binary numeration formed a basis of additive views of Egyptian fractions, a view that clouded any potential suggestion of hieratic numeration or math containing any abstract properties.
Another less frequentlly formalized suggestion was that cursive binary numeration formed an algorithmic method utilized by Egyptian scribes, a proposal that may mean that Old Kingdom math contained abstract features at one time, most of which were lost by the time of the Middle Kingdom and hieratic math.
The modern Egypotology community has often stressed a proposed hieroglyphic and Horus-Eye link to such an extent that the hieratic math texts have not been independently read by many Egyptologists and math historians to rigorously test if other intellectual foundations may have existed during the Middle Kingdom. Otto Neugebauer, Exact Sciences in Antiquity, one of the most often quoted experts, has written the definitive minimalist treastise. Neugebauer concluded, without citing specific examples or attested references, that the 2/nth table table method of writing Egyptian fractions followed a mathematical form that appeared in a 'intellectual declined' state.
Thus, a problem of scholars not frequently or not fairly attempting to look for an 'original' first context from which Egyptian fractions may have first been used, beyond the hieroglyphic paradigm, considering only the hieratic texts, has been a major translating, decoding, 'blind- in-one-eye' problem since the RMP was published in 1879. The blind- in-one-eye metaphor is pertinent because scholars have tended to overstress unproven hieroglyphic foundations of hieratic math, while, at the same time, not attempting to independently read the hieratic texts as a potentially separate body of knowledge.
Cleary early 1920's attempts to fairly and fully decode read the Middle Kingdom Egyptian fractions texts, often used 'one additive eye' beginning with the Rhind Mathematical Papyrus (RMP) and its 2/nth table, 8 pages of the RMP's 25 pages. The 2/nth table has proven to be filled with several types of modern logical dead-ends, and conflicting opinions., reported by modern scholars. Conflicting opinions have been offered by members of both of the history of mathematics and the Egyptology communities. Opinions have ranged from the overly 'restrictive' minimalist to several 'far out' pedagogical positions, most often based on very small samples of scribal math ematical texts, and then making overly broad a priori generalizations (i.e. Neugebauer's intellectual decline proposal).
J.J. Sylvester was the first noted mathematician that provided serious comments on the subject, taking a very small slice of Liber Abaci data. Sylvester first wrote on Egyptian fractions in 1883, consulting with Cantor, suggesting a contrived analysis and basis of the 2/nth table. Later, in 1891, Sylvester suggested that an algorithmic pattern may have been used by Leonardo in creating n/p and n/pq Egyptian.fraction series. Sylvester offered his 1891 suggestion by fairly noting that Leonardo (Fibonacci) had reported a recreational use of a proto-greedy algorithm in 1202 AD. Leonardo had revised the traditional Hultsch-Bruin method, a one subtraction method that calculated elegant Egyptian fractions, to a two step subtraction method that calculated not-so-elegant Egyptian fractrions.
Among Sylvester's minor oversights was one where he had unfairly omitted Fibonacci's two parametric methods that converted vulgar fractions into Egyptian fraction series. One Fibonacci method converted n/p vulgar fractions, and the second method converted n/pq vulgar fractions. Heinz Leuneburg detailed this two-part subhject in his 1993 review on the Liber Abaci, as did L.E. Sigler in his 2002 complete translation of the Liber Abaci. Sylvester may have only offered a 'far out' proposal, one that several researchers have refuted by noting that Fiboancci's method was limited to a second subtraction, possibly not using any form of algorithm. Ahmes may have also only used one subtraction to build his 2/p series.
Given that the general class of algorithms, in which greedy algorithms are a member, are confirmed to have been formalized by Islamic mathematicians (around 800 AD), any potential greedy algorithm use by Ahmes should have been given a very small historical chance to have been confirmed, by Sylvester or anyone else, unless a large number of 2/nth table members can be computed by it. Given than only four of 51 fall into this 'possible' category, Ahmes, therefore, most probably had not gained significant insights from algorithmic thinking, The EMLR shows that multiiples of 2, 3, 4, 5, 7, and 25 were generalized in the RMP 2/nth table to convert 2/pq series by selecting the a multiple (p + 1)/(p + 1).
That is, Fibonacci's recreational work that took place 3,000 years after Ahmes' work, most likely had not been based in Islamic suggestions, but a simple extension of the traditional Hultsch-Bruins method. Therefore, Sylvester's algorithmic method should be seen unrelated to the Liber Abaci unit fraction statements, several of which (six of ten) that had been passed down from Middle Kingdom scribes writing 2/nth tables.
Yet, despite the well defined historical 3,200 year out-of-sequence facts, related to the first appearance and use of the greedy algorithm, and other discussions related to its possible use by Ahmes and other scribes, continue to appear on otherwise reputable web pages. One is St. Andrew's University, Scotland. St. Andrews cites David Eppstein's modern use of 10 algorithms, as first published in "Intelligencer" in 1991, rather than Sylvester's 100 year older views. Why does St. Andrews and other reputable universities cite the greedy algorithm when discussing Egyptian fractions?
Are these web pages trying to cloud what form of mathematics that the scribes like Ahmes actually used, reading only the hieratic texts, use to build their 2/nth tables? Or, are the web pages reporting a little of ancient history to justify a modern mathematical subject? Readers of this blog can make up their own minds if the greedy algorithm, and its many clones, had anything to do with scribal unit fraction arithmetic statements. For myself, I say none, or a very small chance (even in the Liber Abaci). But others are free to disagree, provided that hard scribal data is attached to a fair discussion.
Other scholars have differed with Sylvester. In 1895, F. Hultsch, an Egyptologist, suggested that an abstract solution to an aspect of the 2/nth table solved the majority of 2/nth table members by touching on a 'why' side of the scribal arithmetic. Hultsch's suggestion pointed out that divisors of the first partition of 2/nth table members could be further sub-divided, opening a divisor door to create members of the 2/nth table, without using an algorithm. This method has been confirmed several times, the first independent work being completed by E.M. Bruins in 1944 and published in 1952.
E.M. Bruins, working 50 years later in 1944 reached the same 'aliquot part', a non-algorithm, conclusion that Hultsch had discusssed. Today the method is named after both men, or the Hultsch-Bruins' Method. Oddly the Hultsch-Bruins method is not widely discussed in either the Egyptology or history of math communities in 2006. Yet, its main thesis still serves as a central method used by many that directly and easily decode many 2/p members of the 2/nth table and other Egyptian fraction series published elsewhere in other hieratic texts, especially in the subtraction mode, knowing the first partition derived by some previous study. Ahmes left a clue to his previous study of first partitions. It was 'red auxiliary' numbers, an LCM sorting routine.
G. Daressy, another Egyptologist, translated and analyzed the Akhmim (Cairo) Wooden Tablet in 1906. He pointed out a possible abstract link to the oldest form of Egyptian fractions. Daressy had shown that three of the five divisions of a hekat, using the divisors 3, 7, 10, and not 11 and 13 were exact two-part form statements. All five divisions were written such that the first-term used binary fractions and the second term included Egyptian fractions (followed by ro).
Daressy had been unable to directly prove that the division by 11 and 13 problems were also exact. Given no general rule for all five AWT division statements, his translation project gained little traction in the Egyptology or history of math communities. Sadly his raw data and analysis were soon forgotten, as the Hultsch-Bruins Method had also been often forgotten. Possibly by chance, in 2002, Daressy's 1906 work began to resurface, per the work of Hana Vymazalova, a Charles U., Prague, graduate student.
Before discussing Daressy in the 21th century, let's continue in the 20th century. T. Eric Peet, a mathematician, in 1923, attempted to unify the conflicting practical views concerning the contents of Egyptian fraction arithmetic. Peet's writings disputed Daressy and Hultsch's analyses and conclusions. Peet cited Daressy's incomplete logic and inconclusive data as insufficent, while (at the same time) totally ignoring Hultsch's earlier work. Peet's form of Egyptian arithmetic came down on the side of the additive side of the discussion, prematurely concluding, as a minimalist, that scribes had not used any form of abstract mathematics, or had not experimented or developed any abstract definitions of number. Yet it is clear that a hekat unity, 1 = 64/64, was used in hekat division, as well
as several additional examples of abstract definitions of number are included in the Egytptian Mathematical Learther Roll and the RMP and its 2/nth table.
One of Peet's proposals attempted to show that scribal division had been closely linked to scribal multiplication by using an inverse method, all additively based. 'False position', a second century AD East Indian innovation, possibly linked to Diophantus and 100 AD Chinese texts. False position, an algebraic method of finding roots, entered the hieratic Egyptian math literature at the time when Peet and others discussed Egyptian algebra dated to 1650 BCE. How could have 'false position' been found in Egypt, 2,000 years before it was discovered in China, Greece or India? The 1202 AD Liber Abaci covers this topic very well, as related to this history of Egyptian fractions subject.
Formal academic literature, published by the history of mathematics community in the 1920's, shows that Peet's analysis and conclusions only related to additive scribal arithmetic. Even though the proposed use of false position was widely accepted by DE Smith, Chace, Neugebauer any many members of the history of math community, this researcher has not been able to find a published hieratic text that expliciitly defined the method. Later Gillings and the bulk of the Egyptology community also signed-on to Peet's additive and related minimalist points of view. Minimalist thought is a sad situation that continues to cloud the historical record related to the actual scribal methods used by Ahmes, et al.
In 1987 Robins-Shute wrote their view of the RMP, the 2/nth table and Hultsch-Bruins, mentioning products (and not quotients) and other bits of information that they properely called remainders. Yet, in their otherwise
interesting book Robins-Shute explicitly denied the general use of quotients and remainders, as well as the Hultsch-Bruins Method, by Ahmes.
In 1995 Kevin Brown posted a clearer statement, citing a wide ranging view of Egyptian fractions on his web site. Brown suggested that Egyptian fractions, as a notation, may have been birthed from Egyptian gambling
practices. Math Cats, a grammar school based web site out of Maryland may have said it best, " We know they (Egyptians) used this system for over 2,000 years .. but we don't know why!".
That is, confusion related to the minimalist reading of Egyptian fraction arithmetic continued, infrequently challenged, until 2002. In 2002 a clear publication of the bottom half of the Akhmim Wooden Tablet by Hana
Vymazalova changed everything.
THE 'WHAT' AND 'WHY' SIDE OF HIERATIC TEXTS
The 'why' side of Egyptian fraction began to be reported in 2002. A clear picture of the scribal context of Egyptian fraction was published in 2005 on the internet follows these facts:
A. Akhmim Wooden Tablet, 2,000 BCE, began to 'decoded' in 2002
1. The 'what' side of this text shows that a scribe partitioned a hekat by five divisors, (3, 7, 10, 11 and 13). This was done by beginning and ending with a hekat unity 1 = (64/64), as Hana Vymazalova, the Charles U, Prague grad student, clearly detailed in "Wooden Tablets from Cairo ..., Archiv Orientakni, Vol I, pages 27-42, 2002.
The first modern analysis of the AWT text had been published by G. Daressy in 1906. Daressy had not clearly seen all of the 'what' aspects of the text. What Daressy had seen caused Peet, in 1923, to strongly comment in opposition. Peet attempted to negate Daressy's incomplete analysis, one that tended to show say that Daressy's view of Egyptian division was incorrect, with a short comment concerning ro equalling 1/320th of a hekat. Peet's paper on Egyptian arithmetic excluding the methodology used to exactly divide a hekat by 3, 7, 10, 11 , 13 and
by implication, any n upto and including 64.
Well, it turns out that Peet had correctly viewed ro, a numerical word, equal to 1/320th of a hekat. But little else of Peet's analysis was accurate when he discussed the AkhmimWooden Tablet (AWT) in terms of a Egyptian division. For example, Peet had suggested that scribal division followed an inverse relationship to scribal multiplication. The AWT reports scribal division was closely related to scribal subtraction, a method that looked very much like the Hultsch-Bruins' method, and proof its 2-part division answers by multiplying the answers by the initial divisor (3, 7, 10, 11 and 13) calculating 64/64, the starting hekat unity value (as Hana Vymazalova published in 2002).
2. That is, the 'why' side of the AWT details abstract aspects of remainder arithmetic that Peet had not expected, nor reported in any way. The AWT acrtually reported the hekat unity (64/64) divided by n, by the statement:
(64/64)/n = Q/64 + (5R/n)*ro
with n = 3, 7, 10, 11 and 13.
a form that replaced the remainder 1/64th term with 5/320, thereby creating a constant 1/320 that was factored out of its associated Egyptian fraction term (5R/n), and Daressy had been on the correct historical path in 1906. A table of 29 hekat partitions were compared by Ahmes to hinu partitions, (as included in Gillings Mathematics in the Time of the Pharaohs book). One partition used a ro scaling factor, and the other omitted the use of ro, thereby exposing the common form of remainder arithmetic used by both partitions for those that look closely at the raw data.
B. Rhind Mathematical Papyrus
1. The 'what' side of this text begins with the 2/nth table. Algebra was recorded by scribes using 'false position', a notation that is not easily supported when remainder arithmetic methods of writing vulgar fractions are considered. In addition, Ahmes also wrote hekat partitioning problems, using two forms of shorthand that omitted the details that the Akhmim Wooden Tablet scribe used to explain the method. In total, Ahmes used hekat division, within its 1/10th unit,, to show that a hinu unity was written as 640/64, allowing n to reach 640 following the relationship:
(640/64)/n = Q/64 + R/(n*64) when
n = 10, or
(640/64)/10 = 64/64 = 1 hin.
However, Ahmes in his 15 other RMP 80 uses of hin cited
hin = 10/n
a relationship that avoided the use of ro and the 1/64 remainder term.
Again, let it be stressed that Ahmes listed a table of 29 hekat and hinu partitions, the first using the scaling factor ro, and the second omitting ro. Clearly, there is ample hieratic evidence that Ahmes thought and wrote
in these multiple levels of remainder arithmetic.
2. The 'why' side of the 2/nth table shows that algebraic identities were used by Ahmes, one for the majority of the 2/p conversions to Egyptian fraction series, and another for the majority of the 2/pq series. In addition, the 'false position', short for 'false supposition' method of Egyptian algebra was not used at all, as reported for
over 100 years. The Liber Abaci shows that certain scribes had access to two methods to solve algebra problems. The easiest was the direct method, one that appears to have been used by Ahmes.
The vulgar fraction methods, used in all the Ahmes algebra problems, solved for unknowns by first finding a vulgar fraction. A sorting routine, denoted by red auxiliary numbers, was then used, from time to time, to find the optimal solution to the vulgar fraction Egyptian fraction series, following 2/nth table rules. No supposition or 'false position' thinking has been found using 2/nth table rules that follow remainder arithmetic considerations.
Concerning hekat division by n, the hin unit, 1/10th of a hekat followed a different remainder airthmetic statement, it being:
hin = 10/n
example, n= 64, the largest n in his table, or
(64/64)/64 = 1/64
with its corresponding hinu value being found by:
hin = 10/64 = (8 + 2)/64 = 1/8 + 1/32
Note that the hinu method omitted the use of ro, a special term that had reduced the size of the hekat method by writing exact remainders. Ahmes' table of 16 hekat two-part statements therefore had included the use of (5R/n), an Egyptian fraction series, for reasons best left to the scribes, with the corresponding 16 hinu values writing 1/10th of a hehat hinu values using the shorthand 10/n relationship by stressing the use of divisor n.
C. Egyptian Mathematical Leather Roll (EMLR)
This scroll had gathered dust in the British Museum unrolled from 1864 to 1927, a total of 64 years. When its 26 lines of Egyptian fraction series were translated, only the additive aspects were stressed in 1927. There had been no comparison of EMLR methods to the methods used to create the 2/nth table, or any other hieratic text, in 1927. Myopia had governed the 1920's scholars.
The text cites six methods to convert 1/p and 1/pq rational numbers to Egyptian fraction series. The first five are multiples of 3, 4, 5, 7 and 25. The sixth is an identity form 1 = 1/2 + 1/3 + 1/6, a method found in the RMP 2/nth table to write 2/101. A seventh method introduced the six methods by showing that a miltiple of 2 of four unit fractions only allowed for equal fraction sub-divisions, technically not Egyptian fraction answers.
The multiple of 25 case has been reported in the journal Historia Mathematica. The 1/8 example can be solved by an algebraic identity.
example A = 25
1/8 = 1/25* 25/8
= 1/5 * 25/40
= 1/5 * (3/5 + 1/40)
= 1/5*(1/5 + 2/5 + 1/40)
= 1/5*(1/5 + 1/3 + 1/15 + 1/40)
= 1/25 + 1/15 + 1/75 + 1/200
A multiple of 25 version of this problem follows Occam's Razor, as all 26 EMLR answers can be read.
Note that the out-of-order listing of 1/25 + 1/15 was left as a clue to the scribal methodology. The abstract method was repeared when calculatinng a line that began with 1/16, using A = 25. The abstract method was most likely used to prepare beginning students later to accept the closely related and generalized RMP method:
2/pq = (2/A) * (A/pq),
with A = (p + 1), as used 24 times in the RMP
or by a multiple of (p + 1)/(p +1) such that
2/pq = 2/pq x ((p + 1)/(p+ 1) = 2/(p + 1) x (p + 1)/pq
D. Reisner Papyri
Gillings correctly showed that the Reisner Papyri, housed in Boston, divided worker digging rates by 10. The raw data actually follows a quotient and remainder structure, as all vulgar fractions were written. The quotient appeared as an integer, and the remainder appeared as an exact Egyptian fraction series.
E. Liber Abaci (Wikipedia)
The Liber Abaci (blog) is a 1200AD document that copied the tradition Egyptian fraction methods from Arab sources. Please refer to the linked Wikipedia and blog information on this document.
Translating the seven methods, or distinctions, into the older Egyptian fraction context. The first 126 pages summarize this information. The seven methods describe at least 10 medieval and older methods that Leonardo used to convert vulgar fractions to Egyptian fraction series. Additional discussions on this suhbect can be found at::
http://en.wikipedia.org/wiki/Liber_Abaci
Method one contains three methods, as noted above. The first method may date to Ahmes and his Egyptian scribal style of writing parts of a fraction, in long hand, as the EMLR and RMP used factoring and identities to convert several vulgar fractions. Leonardo converted 1/18 by factoring 1/2 x 1/9, with 1/2 = 1/3 + 1/6 such that
1/18 = 1/27 + 1/64. The EMLR used this method four times, and the RMP used it to convert 2/101.
The second two methods discuss medieval arithmetic and its short hand notation, all useful in finding elegant coversions. Leondard's medieval arithmetic notation, at times, wrote out elegant and not-so-elegant Egyptian fraction answers. These medieval arithmetic notations used by Leonardo have not been thoroughly discussed by math historians related to their use in finding elegant Egyptian fraction answers. Hopefully I will run across analyses of this class information in ways that connect to this discussion, the oldest form of Egyptian fraction arithmetic, and add it to this blog.
Method two wrote 5/6 as (3 + 2)/6 = 1/2 + 1/3 as the EMLR wrote out all of its 1/p and 1/pq answers, and Ahmes used over and over again. The EMLR increased 1/p and 1/pq to multiples of 2,3, 4, 5, 7, and 25, such that conversions were completed as this LA section details in n/p and n/pq problems, and completed the finding of Egyptian fraction series by parsing the numerator in this manner.
Method three details 8/11 = 2/11 + 6/11, a tabular method that 400 AD Coptics used. The Coptic style was reported by David Fowler in Historia Mathematica in 1982 citing answers to problems from n/5 to n/31, and n/4 to n/32, or thereabouts. The RMP 2/nth table can also be seen as the 2/11 aspect of this method. Knowing any 2/n Egyptian fraction series, a second table entry can be found by doubling, such that 2/p + (n- 2)/p = n/p.
Methods four, five and six focus on the Hultsch-Bruins method, as used in all of Ahmes 2/p series. These three methods increasingly define complex definitions of the very old Hultsch-Bruins method, as first reported in the modern era by F. Hultsch in 1895.. The basic H-B method is discussed in method four. It was used by Ahmes to convert 2/p vulgar fractions to unit fraction series by first selecting a first partition with a highly composite denominator, and so forth, as explained elsewhere. Methods five and six show that the first partition need not have been a unit fraction, a style that Ahmes did not adopt. For example, Leonardo method six to convert 20/53 by subtracting 3/8 after raising it to a multiple of 6, 18/48, writing out an answer, 18/48 1/8 0/53. This answer is confirmed by subtracting each fraction, given a little practice, exactly as method four and five were solved.
Method seven includes two methods. The first method is an extension of the Hultsch-Bruins method. Leonardo allowed a second subtraction, when the remainder's vulgar fraction could not be converted by method two, creating a not-so-elegant answer, possibly a form of recreational mathematics.
The second method discussed under method seven covers a factoring method first noted in the RMP 2/nth table, by Fowler and others, where 2/95 was factored as 1/5 x 2/19, with 2/19 taken from the 2/nth table. Adding method one (a) and method seven (b) generally factoring was available to Leonardo, Ahmes and everyone working with Egyptian fractions, any time during its 3,200 year recorded history.
In Leonardo's case Sigler footnotes stressed that an error had been made converting 4/49 by a quasi-greedy algorithm method. However, factoring 4/49 to 1/7 x 4/7 with 4/7 = 1/2 + 1/14 such that an elegant unit fraction answer 4/49 = 1/14 + 1/98 was preferred by Leonardo. Again, the not-so-elegant 'quasi greedy algorithm' version of the 4/49 problem may have been recreational in scope.
There is more to the story. For example the RMP 2/pq method seemingly was not discussed by Leonardo. However, recalling that Ahmes used a (p + 1) multples was an aide in this work, it is easy to see that Leonardo also used the (p + 1) relationship to solve his Hultsch-Bruins and n/pq problems in methods four, five and six. Additional research will be conducted to determine if Leonardo also wrote his 2/pq elegant answers in a manner that Ahmes wrote his 2/pq answers.
That is, these seven suggestions show that the Liber Abaci assists in parsing aspects of the oldest Egyptian mathematical texts, and its traditional Egyptian fraction arithmetic. The 1200AD work of Leonardo, particularly the Liber Abaci, will continue to be parsed in extended ways, allowing additional aspects of the oldest Egyptian mathematical texts, such as the EMLR and the RMP 2/nth table, to read in deeper ways as scholars first began this quest 120 years ago, by solely analyzing the RMP.
REFERENCES
1. Sigler, L,E,, " Fibonacci's Liber Abaci, Leonardo Pisano's Book of Calculations" Springer , New York, 2002, ISBN 0-387-40737-5.
2. Lüneburg, Heinz (1993). Leonardi Pisani Liber Abbaci oder Lesevergnügen eines Mathematikers. Mannheim: B. I. Wissenschaftsverlag.
3. Ore, Oystein (1948). Number Theory and its History. McGraw Hill. Dover version also available, 1988, ISBN 978-0486656205.
F. Summary
Honoring Hana Vymazalova's 2002 discovery of a hekat unity equalling (1 = 64/64) is required because she unknownly opened a major door to scribal remainder arithmetic. Large sections of previously hard to Middle Kingdom mathematical texts can now be easily decoded and fully translated into modern base 10 decimal notation. In the past only fragments of quotients and remainders had been translated, with many fragrment had left unread by scholars (ie Chace noted one of the oversights in his 1927 RMP paper). Therefore, Peet's odd and often complicated 1923 views of Egyptian multiplication, and his related additive issues citing 'false position',
can now be proposed to be revised and dropped as historical in several areas of Egyptian mathematics. A generalized web page attempts to compare Ahmes' remainder arithmetic and Egyptian fractions, and is titled Remainder Arithmetric vs Egyptian Fractions. It includes a few scribal points serious students might like to consider, and debate, thereby replacing Peet's themes with attested .
Scribal remainder arithmetic, when seen in its original EMLR and RMP context, and used in the Liber Abaci in several closely related, ofters several challenges to our academic communities that wish to hold on to the 1920's authors' views.. It is expected that academics will, over the next 5-10 years, enter into serious discussions on this topic, and thereby update their 1920's views. It is also expected, that within the next 10 years, the 130 year old debate may confirm, in unexpected ways, the intellectual origins of Egyptian fractions. At that time Egyptian fraction studies may be revitalized within a well defined form of ancient remainder arithmetic, first reported in the Akhmim Wooden Tablet.. At that time, thanks to L.E. Sigler and others, a wider and deeper context will be presented in academia, describing the work of scribes and their Egyptian fraction arithmetic, algebra and weights and measures. Conclusions have been reached by the leaders of the history of math and the Egyptology communities, replacing the outdated views of the 1920s, once and for all. One example is given by a re-classification of Ahmes 84 problems.
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