Egyptian Fractions

2/n roll

The Akhmim Wooden Tablet

Egyptian Math Papyri










by Milo Gardner

Thomas E. Peet, 1923, partially repeated an analysis of the AWT by showing several connections between Egyptian math and the practical experiences of an ancient Egyptian scribe that used two numeration systems, Horus-Eye and the Egyptian fractions (cited in the RMP). Peet muddled where one system ended and where the other system began by only detailing additive aspects of the two numeration systems, missing the exact Egyptian division features using 5ro as a partitioning idea, using numerators and denominators = 320 in an interesting way. Peet also prematurely concluded that the Egyptian division was only an inverse of the Egyptian multiplication operation.

Peet did not directly discuss Egyptian division, as a general operation, as confirmed by the AWT examples. However, contrary to Gillings and Robins-Shute, Peet did seem to compute with 5ro, 4ro, 3ro, 2 ro and ro, but only from a limited view of the AWT student. Peet was slightly myopic, never asking basic questions, such as: were all of the student's divisions required to be exact? More importantly, no comparisons of Peet's view of the AWT were made to the RMP and its 84 problems. At least ten RMP problems, 36-43, and 81-82, have been misread with respect to ro, suggesting it was a weights and measures unit. Ro was actually connected to a generalized partitioning role, as closely related to other exact partitioning methods cited in the RMP, and other Middle Kingdom mathematical texts.

Scholars are, of course, free to explore these issues on their own, commenting on the actual mix of Egyptian mathematics that meets a few of the standards that are deduced from the AWT, and its very interesting set of division methodologies.

Since Peet, two scholars will be mentioned, Gillings, 1972, and Robin-Shute, 1987, that arguably prematurely concluded their analyses of ro data (from the RMP). Neither scholar seemed to grasp the fairly easy to read fact that ro was never a weights and measures unit. Both authors published that ro was a weights and measures unit.

Returning to Peet, his analysis and conclusions, will be shown in the next few paragaphs. He apparently made serious errors with respect to ro and its relationship to Egyptian division, as was vividly declared in the AWT hekat and 1/64 divided by 1/n context. Peet did not see ro's actual association with remainders, though he mentioned remainders from time to time. It is clear that 64 times 5, an early form of mod 5 in the R/3 term, served both as a numerator and a denominator. The numerators

(1) Q + R/3
appeared in two shorthand forms, the first being
(2) 64/n = Q/64 + R/(n*64).
The second form created 320*n denominators, by subsitituting 5/320 for 1/64. The purpose of this modification created
(3) 5*Q*n + 5*R
totaled 320, the numerator of the divisions (for all n < 64?)

Oddly, Gillings repeatedly mentioned ro, 2ro, 3ro, 4ro, 5ro, as a table, though he never directly computed with them as historical examples, as Ahmes suggested following a range of suggestions, such as 5ro= 5/320 = 1/64.

The AWT generally partitioned its remainders by a numerator idea such that the LCM 320 was used as a common numerator and divisor for 1/64th division problems, as mentioned. A major aspect of this shorthand (Q + R/3) statement specificially wrote Q as a Horus-Eye series and R/3, a quasi- mod 3 in this case, was written as 5/(320*3)= 5/3ro, or 1 + 2/3 ro as a Hieratic series. The AWT, therefore completed all five of its 64/n problems by proving that an exact hekat was found in each case, by multipling its (Q + R/n) answer by 3, 7, 10, 11 and 13, with the R/n term being a remainder.

Gillings did show, however, that when quasi-mod 53 data was used in the RMP, a 106 value was substituted (without mentioning ro), RMP 36, and again quasi-mod 22 data was used solving RMP 37, so it should be noted that Gillings thought ro was a powerful tool, leaving its details unspecified.

Robins-Shute cited a two part division process associated with division, partial products and remainders, to use their terms, a mental idea that Gillings seemed not to see. Oddly, Robins-Shute seeing a two part 'product and remainder' did not attempt to probe or generalize the pattern to the Egyptian division operation, while discussing the 'curious' 1/70 x 100 hekat example:
1 + 1/4 + 1/8 + 1/32 + 1/64 and (2 + 1/14 + 1/21 + 1/42 )ro.

Clearly, (6400/64)/70 was being divided, or 91/64 + 30/70*64) was an intermediate result. The Horus-Eye portion of the 'curious' statement should have been seen as asking what 91 added up to, which is, of course, or 1/2 1/4 1/8 1/32 1/64, in a pure Horus-Eye series. Since the 1/2 term was written as 1, may mean that the scribe was thinking of, 91/64 + 150/(70*320) = 1 + 27/64 + (15/7)ro possibly implying that even the Q term was thought of as a remainder, after a certain point in time.

For a broader discussion of divisors greater than 64, problems that may have generally selected 10 hekat or 100 hekat as the numerator, with specific suggestions on this topic appearing in the appendix of this blog.

Robins-Shute did not follow through with his 'curious' problem, by not using any of the decoding approaches suggested by Ahmes. Like Gillings, Robins-Shute proceeded to guess at a method to correct the data, especially the bits of ro remainder information, by interjecting his personal views, thereby missing the actual Ahmes conversation of ro. Ro, and the partitioning numerators and denominator innovations were available to find a 100% accurate hekat, by adding back the missing 1/64th, as first proven by dividing (64/64) by 3, and then dividi ng (64/64) by 7, 10, 11 and 13. In all five divisions, 320 became the numerator, with (n*320) as the modern base 10 denominator. (For those with access to the AWT text, the beginning of the 7, 10, 11 and 13 division, it appears that 320 is being divided. It might be best for Egyptologists and/or other historians to explain these fragmented facts, once the central method of exact division remainder writing is understood.)

This proof of a 100% accurate hekat was achieved by seeing a missing 1/64th unit was required to added back, much as any remainder of any division calculation must not be rounded off. Remainders, were therefore required to be exactly stated, using hieratc unit fractions.

It is clear, even based on the AWT data that both RMP authors (Gillings and Robins-Shute) were attempting to report details from the same standarized Egyptian division operation, but each from different points of view. Each author left out critical details of the ro calculation, thereby, mostly likely, throwing the baby out with the bath water.

Clearly hekat = 320 should not have been cited as a weights and measures unit by historians. Even the RMP data mandates a form of partitioning connected 64 division problems, using denominators, that cite a 53 division, using 106 as a denominator; a 22 division, using 22 as a numerator and so forth.

Almost as interesting is that the RMP cites 1280 as a denominator for solving another 64/n division problem (RMP 47), again, briefly, and inaccurately reported by Robins-Shute. To probe the issue of the scope of Egyptian division, as reported by five 64/n problems, a little time needs to be spent with the AWT.

It appears that all of the AWT's scribal assumptions need to be probed, by closely analyzing all of the AWT's contents, and all of the data reported by modern researchers on the Egyptian division subject, asking, why for this, and why for that?

Hana Vymazalova, repeated an excellent translation of the AWT data and a very good analysis the hekat data, though she only took the AWT student point of view of the hekat, defering to Peet on the Horus-Eye relationships. She did suggest exactness, from time to time, as seen by approaching exactness but only for the grain discussion, as garbled by the student for some unknown reason.

Vymazalova choose not to discuss the higher math implications of the AWT, and other Middle Kingdom mathematical texts, as poorly analyzed by Peet, Gillings, Robins-Shute and others, defining a serious gap that cries out to be formally filled, by rigorous professional studies.

Vymazalova, however, did show that the student was required to prove the division result by multiplying back by 3, 7, 10, 11, and 13, retaining inexact calculations, by chance or by intention? She suggested that the AWT's central process (left unspecified, related to Egyptian proportions and division) worked by the student was more important than the final calculations. Hence the sloppy initial, intermediate and final Egyptian fraction calculations were meaningless (in her eyes) in certain respects, to the main topic of the AWT, detailing a method to find a 100% accurate grain unit.

This blog will cite Peet's analysis, stressing the arithmetic details of his view of the second numeration system, Egyptian fractions, that were available during Peet's lifetime, and facts that have been published since his death. This analysis will also include other pertinent authors like Gillings and Robins-Shute, discussing their views on Egyptian division, along the same lines, updating Egyptian fractions to 2004.

It is expected that the conclusions of this analysis will shed new light on all four of the Egyptian arithmetic operations as known and widely used during the Middle Kingdom, most of which was diffused to Classical Greece. This analysis therefore will stress a goal of being comprehensive, touching as many of the applications of ro that are known, as well as citing other related partitions of rational numbers.

The purpose of this analysis will, therefore, be to report the new and old aspects of ancient scribal thinking, as discussed in the AWT, and as confirmed by all Middle Kingdom texts (that I know of), not stressing one document over another. To do that, a general discussion of Egyptian addition, subtraction, multiplication and division needs to take place, probably placed on another blog.

Therefore another goal is to confirm or refute that all five AWT division calculations were intended to be exact, as required by Egyptian fraction rules set down in the RMP, the EMLR, and other Middle Kingdom texts.

But what were those rules, and why did they develop as they did? The ro discussion provides a strong hint, concerning the division operation, related to the hekat unit.

But can this one fact be a source of origin for Egyptian fractions, as a numeration system, as well as a solution to the Horus-Eye round off problem?

Almost as important, why have so many scholars missed one or more of the minor aspects of Egyptian arithmetic, thereby clouding the broader context of Egyptian arithmetic history, dating back to the Old Kingdom period? In other words, can these missing Middle Kingdom details, if confirmed to have been generally accepted in the Middle Kingdom, shed light on a major subject, here and there?

To follow this discussion to greater depths, read on.

The five AWT division problems:

1. Vymazalova states that the intended hekat division and remainder raw data was worked (exclusively) within the hekat context, by dividing 64/64 by 3, 7, 10, 11 and 13. Her data includes the following:

a. divide 64/64 by 3 = 21/64 = (1/4 + 1/16 + 1/64) = 315/960 (Q), with remainder = (1 + 2/3)* ro = 5/(3*320) = 5/960 (R) and adding (Q + R) = 320/960 = 1/3

b. divide 64/64 by 7 = (1/8 + 1/64) = 9/64 = 315/2240 (Q) remainder = (1/2 + 1/7 + 1/14)*ro = 10/(14*320) = 5/2240 (R) and adding (Q + R) = 320/2240 = 1/7

c. divide 64/64 by 10 = (1/16 + 1/32) = 6/64 = 300/320 (Q) with remainder = 2*ro = 20/320 (R) and (Q + R) = 320/3200 = 1/10

d. divide 64/64 by 11 = (1/16 + 1/64) = 5/64 = 275/3520 (Q) with remainder = (4 + 1/11)*ro = 45/(11*320) = 45/3520 (R) and (Q + R) = 320/3520 = 1/11

e. divide 64/64 by 13 = 1/16 = 4/64 = 260/4160 (Q) with remainder = (4+1/2+1/13+1/26) = 120/(26*320) = 60/4160 (R) and (Q + R) = 320/4160 = 1/13

The overall AWT discusion required the proof of the value of 1/64th, by multiplying by 3, 7, 10, 11, and 13, creating needed steps of showing duplation elements of n/3, n/7, n/10, n/11 and n/13 tables.

The errors that were found in the AWT were the student's arithmetic ones, much as a beginner, or first time user of the method, might experience. While a majority of this data can be pulled into the 21th century by converting to our modern number system, the beginning of the second set of Egyptian fraction statements, division remainder data, containing 5ro, or 5/320, seemed a little muddled, to Daressy, Peet, Gillings, Robins-Shute and other historians, so they ignored the AWT data, replacing it with their unconfirmed personal views on the subject.

To attempt tp clear up confusion from many scholars, additional evidence needs to be considered. The AWT contains bits of fragmented thinking, dating back to 4,000 BC, showing that rational numbers were the center of its mathematics, with exactness being following where ever possible.

The main point division point of the AWT is that all division remainders were always converted to exact Egyptian fraction series, thereby eliminating all 'round off errors'. This point can not be over stressed.

Thomas Peet's 1923 Analysis

T. Eric Peet's reading of Daressy's analysis suggested that Daressy was 'completely erronous' with his multipliers of whole numbers and fractions using powers of 1/2, 1/4 and so on. Peet showed that Moeller was the first to see a hekat (or bushel, per Peet). Sethe also commented, "but even he did not understand Egyptian mathematics, a subject worthy of study", per Peet. Using Peet's words against him, even Peet did not understand Egyptian Middle Kingdom mathematics, as will be made evident, in the next few paragraphs. Peet noticed that hekats were not discussed in terms of 1/7th unit, or any unit except the Horus-Eye version, modified by a remainder statement that included ro, as 320 in a hekat.

The AWT conclusions "are nothing more than the expressions of various fraction forms of (1/2, 1/7, 1/10, 1/11, 1/13) of the bushel of the bushel, in terms of the divisors 1/2, 1/4, 18, etc. and the ro. For example, 1/11 of a bushel was shown as

(1/18 + 1/84 + 4 1/14) ro

and this was the only correct way of expressing 1/11 of a bushel in Egyptian", per Peet.

Continuing with Peet's analysis," The Egyptians, however, were more methodological than we are, for each of his units was half of the next, except the ro, which was 1/5th of the 1/64th of a bushel."

Conclusion: Peet never did see the numerator and remainder roles of ro, as a subject that clearly connects to the way that Egypytians saw division, always converting remainders to unit fraction series.

The AWT is a special case, that shows that remainders may have firstbeen generally converted in the search for exact hekat units, adding back any missing parts of the missing 1/64th unit, as first created in the cursive Horus-Eye numeration system. Therefore to Peet, he never saw where the Horus-Eye system ended, with the statement of the missing 1/64th unit, and where Egyptian fractions as a numeration system began, when Horus-Eye numeration was made 100% accurate, by finding the missing 1/64th portion, 1/192 when dividing 64/3.

The AWT is therefore a critical document in understanding Egyptian mathematics, a subject that Peet pronounced that he had mastered. Clearly Peet had not grasped ro and its common numerator and denominator relationship to remainders and other points, all glaring oversights that continues to be a problem in completely reading the RMP itself.

Origination Issues, as suggested by the AWT

1. Daressy : He worked within the Horus-Eye system exclusively, not noticing the exact aspect connected to Egyptian division itself (though my French is not very good, hence others will have to read his analysis on this point). What seems to be true, Peet and Gillings' used a term 'ordinary fractions', to mean Egyptian fractions, as if there was nothing unordinary about them. That is, Egyptian fractions have long been seen as additive in scope, and thereby Egyptian fractions, as a notation, was not generally constructed to be exact in its smallest (rational number) calculations. Daressy may have also accepted this view, one that may now be disproven by reading the AWT, and working its data in a broad manner, including the four Middle Kingdom arithmetic operations.

2. Peet, Gillings and Robins-Shute

They worked on different aspects of the RMP, adding in other Middle Kingdom texts, usually within an additive limitation, as suggested by the majority of ANE scholars. But was the common conclusion that Egyptian duplation was the highest form of Middle Kingdom arithmetic correct, or off just slightly?

Duplation surely was present in all the Middle Kingdom texts, associated with addition and multiplication. But what about subtraction and division? Was duplation a required concept in those operations?

The AWT's two part division process including the ro, begins to answer a few of these problems. However, since so few scholars have explicitly discussed ro as connected to division. It appears that for over 100 years, scholars have backed off critical division details, each in their own personal manner.

Peet is one of those often cited scholars that did skip over an important division aspect of the AWT, by not asking, were all the calculations intended to be exact? The AWT shows that all calculations were intended to be exact, using a very interesting partitioning method.

3. Vymazalova

She exclusvely worked within the grain unit, noting the intertwined 'ordinary fractions'. She did not comment on the Egyptian fractions subject in any depth. Her concern was with the process under taken by the student, to find a full hekat, by pointing out the student's many minor arithmetical mistakes, all leading to the final hekat computation.

4. Gardner

It has been noticed that the AWT documented a method to generally partition the missing Horus-Eye 1/64th unit, by exactly dividing unity (64/64) by 3, 7, 10, 11 , 13 and by implication 1/n, as written in three Egyptian shorthands 64/n verions, via a two-part motiff.

The first (1) Q + R/n was easily seen, yet difficult to prove. The first part(Q) listed a Horus-Eye series, as easily seen from the (2) Q/64 + R/(3*64), with the second part, R/(3*64), being listed a Hieratic series, a subtle but basic decoding pattern. Anyone reading the blog must work through the data to see these relationships, before proceding to the final pattern, the one explaining ro. The second part may suggest a root of the actual origin of the Egyptian fraction system itself.



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