An Introduction to Egyptian Fractions
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There is a commonly held idea that Egyptian fractions, a Middle Kingdom innovation, was 'useless' or ' a sign of intellectual decline'. This non- utilitarian idea has been stressed for over 100 years. Currently, the majority, if not all, of the Egyptology community considers base 10 hieroglyphic fractions as more important to study than its younger hieratic version. In addition there is a high percentage of modern math historians that also follow a related conclusion, as stated in the work of Otto Neugebauer, and his famous book: " Exact Sciences in Antiquity". Neugebaur cites the RMP 2/nth table, and its apparent awkward construction and concludes that all of the Egyptian fraction notation was 'a sign of intellectual decline'. This may mean that hieroglyphic math was implied to be superior to its younger hieratic version. But was it?
This amateur researcher will confront both of the above conclusions, taking a cross cultural analysis. The analysis may show that Egyptian fractions were a sign of intellectual advancement, confirmed by several useful Old Kingdom and Middle Kingdom applications. Three analytical steps will cover: the Egyptian Old Kingdom, Competition with Babylon, and the Egyptian Middle Kingdom
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The Egyptian Old Kingdom
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Beginning with the Egyptian Old Kingdom an arithmetic form of writing fractions appeared. It as called Horus-Eye, and it came to dominate the culture in all its areas. Horus-Eye was a mixed numeration system, composed of base 2 and maybe base 5, written as a base 10 infinite series.The name Horus-Eye involves a mythic origin, but its arithmetic defined a simple system, one using a doubling operation, may be dating back to its base 2 days, whenever and wherever that was.
A comparison will be made to nearby Babylonian arithmetic. The comparison is intended to outline the simple structure of Horus-Eye, as well as touching upon a few other of its practical properties. Horus-Eye arithmetic was easy of use and it was fairly accurate, since additional digits were easily added, when needed.
Babylonians used another infinite series, written in base 60, within a tradition that was earlier limited to base 10. The base 60 Babylonian arithmetic, wrote number and fractions in base 60. The fractions part is all that will be discussed here, and it looked and acted very much like its Egyptian counter part. That is, Egyptian and Babylonian scribes, around 2500 BC, both worked within an infinite series form of numeration, accepting rounded off fractions, in a manner that scribes understood on levels of accuracy and others basic limitations.
One major difference between Egyptian and Babylonian fraction was that Egyptians later added a finite numeration system, hieratic fractions, around 2200 BC. In my reading of Egyptology literature, hieratic script and its fractions were only a cursive, a shorthand, version of the formal hieroglyphic form of writing. That is, the finite hieratic fractions may be best read today by considering the doubling methodology first set down in the Old Kingdom, and not by any unreported algebraic thinking.
But was hieratic fractions only a condensed version of hieroglyphic fractions, Horus-Eye traditions, or even a step backward as Otto Neugebaur and others have concluded? How and why could have scribes hidden methods that computed hieratic fractions, that were not first computed by a doubling process?
Reintroduction of Babylonian mathematics
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To start again, at the onset of the Egyptian Old Kingdom, it has been reported that it was formed with the merger of Upper and Lower Egypt. Either Upper or Lower Egypt may have used base 2 and the other base 5. Upon the merger Old Kingdom scribes used mixed base 10 Horus-Eye math. This literally meant that numbers smaller than or equal to one were defined by fractions, usually 6-terms or less, by halving or by an inverse doubling operation. The standard definition of one (1) is most often shown in Horus-Eye by:its first 6-terms.
1 = 1/2 + 1/4 + 1/8 + 1/.16 + 1/32 + 1/64
There are of course mythological aspects of Horus-Eye's origins, but they need not be explained at this time. What is important is to go beyond the myth and note that 1/64 was needed to exactly compute one ( 1). The missing 1/64th was not added, but what was done was that another 6-terms as added, when needed. The second 6-terms was called ro,
ro = 1/64 = 1/128 + 1256 + 1/512 + 1/1024 + 1/2048 + 1/4096.
It is easily seen that a second 1/4096 was omitted, and need for ro to reach 1/64.. Thus the last term of any Horus-Eye series is often its level of accuracy. The more terms in a series, the more accurate the calculation.
Were Egyptians pleased with the Horus-Eye system, and its pluses and minuses? To probe both points, let us open the neighbor's door, the Babylonian system.
To recall, Babylonians used a nearly pure base 60 system. To understand its structure let us strip off the whole number aspect, and define the fraction n, as:
n = 1/60^1 + 1/60^2 + 1/60^3 + 1/60^4 + ... + 1/60^m + ...
with ^m being an exponent, meaning repeated multiplication of 60.
Babylonians modified the ideal form of base 60 series, by only using multiples of 2, 3 and 5 when writing the unit fraction form of its denominators. Given this limitation, and noting that not all of the real world numbers could be computed, since numerators of multiples of primes 7, 11, 13, 17, ...., could not be represented in their system, the Babylonian base 60 system was practically very accuracy, compared to Horus-Eye.
Considering pertinent comparative details of Babylonian base 60, only the first decimal fraction term was needed to reach the Horus-Eye's first 6-terms level of accuracy. A second base 60 term yielded 1/3600th accuracy, a level that approximately reached the awkward 12-term Horus-Eye series level. Considering that 3-terms of the base 60 system optimally yielded 1/216000 accuracy, and 4-terms may have yielded an amazingly accurate 1/12960000, it may be clear to most readers that the Babylonians, Chaldeans and related Egyptian neighbors and successors like the Hellene Ptolemy (writing in the Almagest) were happy with base 60. There were two reasons, shortness, using at most 4-terms, and accuracy.
It seemed as though Base 60 handles most calculations, all other things equal, within high levels of accuracy, about 6 times the 'decimal fractions' accuracy rate of Horus-Eye.
Seen on this level, for the ideal calculation, taking out human error as a consideration Babylonian math was much more accurate than Egyptians, prior to 2200 BC. But did Egyptians catch up, or pass Babylonians in the accuracy of their scientific and most importance business and engineering"
So, looking closely at the Egyptian Old Kingdom, it is well known that most of the pyramids, for example, were accurately constructed. Building tolerances were very small, after the initial failure of the first pyramid. So, did Egyptians improve pyramid building methods by improving the accuracy of the Horus-Eye notion, by an improved form of calculation, eliminating all error, except the human one?
Asking this question in another way, were scribal engineers motivated to replace Horus-Eye fractions with a more accurate system (around 2500 BC)? And if so, is it reasonable to conclude that a new method did not earn its own script until 300 years later, or 2200 BC? Egyptologists and the majority of math historians say no.
The Egyptian Middle Kingdom
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The first known hieratic fraction document is often considered to be the 2000 BC Moscow Papyrus. As one of its problems, the scribe computed the exact area of a truncated pyramid, by taking exact slices, much as is done in modern calculus. Its notation was new in several respects. Another new feature being, that unit fractions 1/p were written as p', with p being a letter, This form of number was picked up by Greeks, as they would later copy onto their Ionic and Doric alphabets.
By exclusion, this second interesting aspect of hieratic fractions, marked the none use of the awkward hieroglyphic numeral system within certain Egyptian calculations. The old system had acted like Roman numerals, writing the single numeral 4 as IIII, four symbols.Boyer, the noted math historian, called this new number, ciphered numerals.
Ciphered numerals dominated Western Tradition mathematics until the Hindu-Arabic form of base 10 ciphered numeral appeared in the Middle Ages.
But, what about the other hieratic fraction documents, the EMLR, RMP Hibeh P. and other texts. Were all the problems solved by a new numeration system that contained zero error, 100% accuracy?
One answer is that the EMLR contains no problem solving, except the listing alternative ways to compute 1/p and 1/pq series, the first entry to any n/p or n/pq table, the table aspect often appearing in hieratic script. But what was the intended utility of such tables. Were they actually useless as the Egyptology community has long assumed?
Another answer is that the RMP begins with a 2/nth table and further on contains 84 problems, all solved exactly. No explanation of the 2/nth table was provided by its scribe Ahmes. The only calculation of entries that came from the 2/nth table, that appear in the 84 problems, was a duplation one, brief steps, that often missed a little logic, here and there. These steps, taken together looked and acted like an Old Kingdom form of thinking; however, due to the missing steps, as a proof of another calculation that Ahmes may have done in his head.
Yet, for the modern Egyptologist, the reading of the Old Kingdom hieroglyphic records, and its fractions contained therein, seem to be sufficient. Little is written on the issues that may have existed the birthed hieratic math and its 100% accurate fractions, For practical purposes, the hieroglyphic rounded off form of arithmetic appeared to still be dominate in the Middle Kingdom, New Kingdom and later periods, even if sometimes accompanied by the subtle hieratic form. One subtle aspect was that Egyptian fractions tended to be computed in mental ways, that is not computed for others to confirm in its algebraic form. A proof or a formal confirmation of a particular unit fraction series was only written out using the traditional doubling, or Horus-Eye type of thinking.
So what was the hieratic form of numeration, and how were its series computed?
Hieratic fractions was a 6-term or less unit fraction series that exactly converted rational numbers, numbers less than one. Yes, exactness was a requirement of Egyptian fractions, taking Egyptian measurement to a level beyond Babylonian and other regional weights and measure systems.
There is a great deal more to discuss, that shows Egyptian fractions as containing several important advancements that directly contradict long standing conclusions of Middle Kingdom's reported intellectual decline. One is that ciphered numerals appeared at the same time that hieratic fractions arose, an event that Boyer and others have long discussion as followed by Classical Greeks and Hellene culture, until Arabic-Hindu numerals came on the scene.
Thank you for continuing our little discussion. Deciphering ancient texts should begin and end with respect to the time period of the initial writing. In this case, the Middle Kingdom Egypt should be the primary context for reading Egyptian fractions. However, the literature on this subject has been jumbled, as I'll try to explain on a couple of levels.
As you know, initially, the British Museum gained possession of the RMP and EMLR in the early 1860's, and slowly dribbled out good and bad information related to the texts' contents. The RMP, for example, was released by a 1873 bootleg copy and 50 years later, 1927, the EMLR was first unrolled with little outside influence..
To begin at my beginning, I started playing around with Egyptian fractions in 1988, being provided a paper by Noel Braymer, a neighbor. Noel, a retired electrical engineer, holds several patents (small items like volt meters), loving mathematics since his World War II youth.Noel began working on Egyptian fractions in 1977, based on an IEEE puzzle prepared by a master, such as yourself. Early computers easily overflowed working on this problem, a situation that I had experienced as a FORTRAN systems analyst. Ten years later Noel's completed paper, confirmed with PASCAL, was hand delivered by his daughter, A US naval officer, to the British Museum. Oddly this paper was rejected with a comment, 'to be kept on file'.
As I began assisting Noel, two algebraic identities were teased from the RMP 2/nth table by applying cryptanalytical techniques, that I had learned in the military. William F. Friedman and his cryptanalysis books can be found at
http://www.aegeanparkpress.com/desc.html#c-90 .
My analysis confirmed Noel's basic results, though attempted to be restated to a historical form. Our paper was titled "Middle Kingdom Egyptian Fractions, A Paradigm Shift". The paper and several reviews can be found at:
http://mathforum.org/epigone/math-history-list/roogrimspel/Pine.HPP.3.91.
970201075737.15931A-100000@gaia.ecs.csus.edu
Note that this paper was submitted to a history of math journal, and rejected for being analytically 'too complex'. One of the two 1996 reviewers indicated that this class of paper added little to the historical discussion and therefore could be better discussed or debate on the internet. Disagreeing with the first part of the conclusion, I did take the step to open contextual reviews of Egyptian fractions, mixing in suggestions of Hultsch, Bruins,Gillings, van der Waerden and others that have also reported complex patterns.
My second reply to the 'too complex' review is that the patterns found in the RMP itself were causative, and therefore any rigorous analysis would have required the discussion of at least two 'potential' ancient patterns. It appears that the reviewers had concluded, years earlier as prodded by their peers, that the contents of the 2/nth table were simple minded, containing no significant unifying themes. Neugebauer began writing on this subject in 1926 and later, cited in "Exact Sciences in Antiquity" that the 2/nth table marked a sign of intellectual decline. DE Smith in 1925 concluded that Middle Kingdom Egyptian fractions were only additive in construction. And in 1890 JJ Sylvester introduced the greedy algorithm. Finally, and with a contrived analysis by 1881 Sylvester, consulting with Cantor, cited nothing related to the RMP.
Returning to the British Museum, the 1873 bootlegged copy of the RMP was first published by Eisenlohr in 1877. Eisenlohr's paper sparked an ownership and content debate that continues to this day (much as has taken place recently with respect to the Dead Sea Scrolls - with some historians mixing historical issues with odd personal opinions).
To my eye, the Egyptian fraction literature is jumbled, and exists that unreadable form for two reasons. The first is that professional math historians are not willing to admit that mistakes have been made in their field, and therefore they, as a group, have not acted to retract the early Classical analyses, often built upon a range of personal opinions. The second is that the fields of Egyptology and math history do not formally consult with one another, counter balancing each other's mistakes, in areas that overlap. One day corrective meetings hopefully will be chartered.
Clearly, what exists today, in my opinion, is that the professional journals minimize unifying patterns in the 2/nth table, and the related patterns in other Middle Kingdom texts. However, analytically minded people can decide for themselves what type of blinders, if any, were put on by a particular historian, by either reviewing the 2/nth table, and working backwards, or using programs like Mathematica to generally compute n/p and n/pq conversions forwards.
Mathematica user David Eppstein has offerred 10 interesting algorithms, publicized in Intelligencer, but oddly his analysis provides no discussion of the RMP, or any historical situation (sound familiar?). These are only a few of the clues that show that Egyptian fractions, as an ancient system of thought/numeration system, used by Greeks and their mentioning Egyptians, is officially an unsolved problem, and may remain so for some years to come..
My second paper, with no consulting co-author, is titled "The Egyptian Mathematical Leather Roll, Attested Short Term and Long Term" published in 2003, as you have already scanned. As you may recall, I like to introduce this subject by listing a translation of the EMLR.
http://www.mathorigins.com/image%20grid/BRUCE%20OLD_005.htm
The paper itself can be found at
http://www.math.buffalo.edu/mad/Ancient-Africa/gardner.EMLRL.160202.doc
or
http://mathorigins.com/B_emlr050602.htm
This often ignored text shows that one of the two RMP unifying themes is found in a closely related form. Again, how did Ahmes make his calculations? Was a general rule available to him, as plausibly was available in the Greek two-term 2/pq and n/pq conversions during Hellenistic and later time periods?
To discuss the later time period, I like Howard Eves' explanation of a general Egyptian fraction rule. Eves reported that the 400 AD Akhmim P. contained such a general rule. I first came across this rule in 1962, in a history of math class, as Eves stated in the simple form:
n/pq = 1/pr + 1/qr
where r = (p + q)/n
The rule seems to have been used in the RMP for the unusual 2/35 and 2/91 series, as given by:
2/35, r = (12/2) = 6, or 2/35 = 1/30 + 1/42
2/91, r = (20/2) = 10, or 2/91 = 1/70 + 1/130
For myself, I doubt that Ahmes thought in this manner. I suspect two other algebraic identity ideas as being more likely to have computed 2/35 and 2/91. One is the product of the arithmetic mean = (p + q)/2 and the geometric mean = pq^1/2 , as Howard Eves discusses (in the same paragraph that he reviews the Moscow Mathematical Papyrus and the apparent use of the general formula V = h(a^2 + ab+ b^2)/3). The second algebraic identity may have been a trail and error form of thinking (as Ahmes knew by his use of the method of false position).
Finally the 400 AD Akhmim P. offers other clues to reading Egyptian fractions in historical ways. The AP n/3 - n/32 tables were published by Knorr in 1982. I have condensed the AP to its n/17 and n/19 tables since its. conversions of 4/17, 8/17, 3/19, 7/19 and 9/19 are interesting, using methods that improve upon the RMP optimal series. Kevin Brown, a number theorist, was a major contributor
http://www.ecst.csuchico.edu/~atman/Misc/horus-eye.html
Thank you again for considering these modern recreational math subjects, stated in a proposed historical context (as I know Mr. Wolfram encourages readers of his insightful books to consider). My examples are three Middle Kingdom mathematical texts and one 400 AD Coptic/Greek mathematical text, restated as should have been placed on the same analyst's agenda, beginning 150 years ago.
1. From page = 176:
The fraction 4/23 was discussed in terms of the greedy algorithm, or
4/23 = 1/7 + 1/33 + 1/1329 + 1/2353658
finding a shorter series than The RMP cites a 5-term series
2/23 = 1/12 + (12 + 6 + 4 + 3)/(12*23) = 1/12 + 1/23 + 1/46+ 1/68 +
1/92
meaning that a 6-term series may have been required, at least in Stan Wagon's analysis.
However, upon closer review, 1/6 can be used as the first partition meaning that 4/23 can be calculated directly, without an intermediate 2/23 step, by:
4/23 - 1/6 = (24 - 23)/138
or,
4/23 = 1/6 + 1/138
as Ahmes would have found.
2. From page 206, the 1202 AD Fibonacci greedy algorithm was applied
once again, working on partitioning 5/121, Wagon reporting
5/121 = 1/25 + 1/757 + three more terms in awkwardly growing
denominators.
Wagon went on to show a major improvement by listing Bleicher's improved algorithm, finding
5/121 = 1/25 + 1/225 + 1/3477 + 1/7081 + 1/11737
So how would have the historical Ahmes done?
Ahmes would have first factored 5/121 = 1/11 x 5/11 and went on to find a 3-term series, with smaller denomonators than listed by Bleicher and published by Wagon, by first taking
5/11 - 1/3 = (15 - 11)/33,
or,
5/11 = 1/3 + (3 + 1)/33 = 1/3 + 1/11 + 1/33
such that,
5/121 = 1/33 + 1/121 + 1/363
Given that short class, the historical Ahmes, knowing no algorithms at all, may have given his scribal students a rest, by providing his form of recreational mathematics, much as you as have ably developed a version of the famous St. Ives Riddle.