Talk:Egyptian fraction

Untitled

This:

Assume 7. 1 and ¹/₇ of 7 is 8.

was recently changed to this:

Assume 7. 7 and ¹/₇ of 7 is 8.

with description "A hard-to-spot mistake, I haven't read the papyrus but this is the only thing which makes sense)".

This is not necessarily so, for another interpretation of "1 and ¹/₇ of 7" could be "1¹/₇ of 7". So I think we should stick with the old version. --Romanm 07:47, 23 Jul 2004 (UTC)

The greedy algorithms for representing a number using egyptian fractions is attributed to Fibbonaci:

"In 1202, Fibonacci published an algorithm for constructing unit fraction representations, and this algorithm was subsequently rediscovered by Sylvester (Hoffman 1998, p. 154; Martin 1999)."

The last quote is taken from MathWorld, precisely:

http://mathworld.wolfram.com/EgyptianFraction.html

However, the above modern view does not probe the minds of ancient scribes. To do that, math world also cites the EMLR, a 26 line table of 1/p and 1/pq conversions to Egyptian fraction series, per

http://mathworld.wolfram.com/EgyptianMathematicalLeatherRoll.html

and other texts, as mathworld cites the Akhmim Wooden Tablet and the Rhind (Ahmes) Mathematical Papyrus.

Puzzling example

The article says:

Ahmes converted 19/20 by simply testing the first partition 1/2 within a well known ancient method, known as Hultsch-Bruins, or

19/20 = 1/2 + 18/40,

and solved 18/40 by looking for the divisors of 40, or 20, 10, 8, 5, 4, 2, 1 that add up to 18. Note that by replacing 18 by (10 + 8), or

19/20 = 1/2 + (10 + 8)/40

= 1/2 + 1/4 + 1/5

The choice of 18/40 here puzzles me, and is given without explanation. Why wouldn't Ahmes have started with 9/20 and tried to find a subset of {10, 5, 4, 2, 1} that added to 9? This gives:

19/20 = 1/2 + (5 + 4)/20

= 1/2 + 1/4 + 1/5

more simply. What am I missing here? -- Dominus 00:29, 8 May 2006 (UTC)

I have no idea. Maybe Ahmes had a penchant for never writing fractions in lowest terms? --King Bee 15:15, 3 December 2006 (UTC)

Fragment out of place?

The section at the end of "Fractions in Egypt" dealing with 5/7 seems to be out of place or incomplete. It was in "Notation" and seemed out of place there so I moved it to "Fractions in Egypt." There still seems to be some loss on continuity. --Jbergquist 23:08, 17 June 2006 (UTC)

Unhappy

I feel too close to this material (note my page on the subject is one of only three external references) and too uninterested in its non-mathematical aspects to be a good editor for it, but I am quite unhappy with the present treatment. I feel that it fails both WP:NPOV and WP:OR, that it contains outright falsehoods, and that by paying attention only to analysis of ancient methods it slights some very notable mathematical aspects of the problem.

The ancient scribes said what they said. To not report the scribal view of Egyptian fractions as the ancient texts report its methods seems unfair, and inappropriate. (Milo)

Some specific complaints with what's included here:

• This quote:

But first, it should be noted that modern Harmonic series can be used to prove every positive rational number can be written in the form of an Egyption fraction. However, modern methodologies produce awkward and relatively longer unit fraction series compared to the shorter and more concise ancient methods. (Milo)

is false. Modern computer algorithms are easily capable of finding optimally short series for numbers of the magnitude used in the examples here, and good mathematical bounds on the size of the expansion are proven by G. Tenenbaum and H. Yokota. Length and denominators of Egyptian fractions. J. Number Th. 35, 1990, pp. 150–­156, and by M. Vose. Egyptian fractions. Bull. Lond. Math. Soc. 17, 1985, p. 21. Also the attitude that the ancients knew better than we know now fails WP:NPOV.

• Too much off-topic discussion of the origins of algebra. It's notable, just not on-topic for this page.

• No citations for Hultsch-Bruins. Is this original research?

No this is not original research, F. Hultsch first published it

in 1895. Its 2/p form is documented in the RMP 2/nth table was independently confirmed by E.M. Bruins in 1944, hence the name modern Hultsch-Bruins, as generalized by

n/p - 1/A = (nA -p)/aP

with A being chosen in the range p/n < A< p and the divisors of A being used to optimally find the vale (nA -p).(Milo, 11/10/07)

• Long sequences of calculations that add verbiage but not a lot of understanding.

...and what's not included:

Egyptian fraction notation changed during Egyptian eras. To be brief Middle Kingdom Egyptians were first to use the fully developed method of writing n/p and n/pq by exact and concise Egyptian fraction series, methods that included a proof that each statement was correct. By the time of the Greeks and Archimedes the exact way of writing Egyptian fraction series were stressed in proofs for infinite series situtations (for example Archimedes' Lemma 4A/3 = A + A/4 + A/16, ... became A + A/4 + A/12). Later, Byzantine and Middle Ages mathematicians tended to drop the last step of writing its algebra and every day math, the conversion of vulgar fractions to Egyptian fraction series. This time period tended to write its final algebra and higher math answers as vulgar fractions, or as we call today improper fractions, steps that had long been used in earlier algebra. As a user of several notations, Fibonacci 1202 AD was the one of the last Middle Ages writer to fully use the Egyptian fraction notation, while at the same time experimenting with modern greedy algorithms and other ideas form a wider region ( such as math that came from Islamic and Vedic mathematicians, as Diophantus had earlier used Chinese math and the Chinese Remainder Theorem to solve indeterminate equations). By 1600 AD modern base 10 decimals came onto the scene and attempted to remove Egyptian fraction notation from our math history memories. Sadly, the 3,600 year history of finding optimal partitioning values A, and its aliquot parts, needed to solve n/p and n/pq conversions to exact unit fraction series remain only as faint ways in most math history books. The death of the Egyptian fraction notation, and its many versions and uses, has sadly not been properly noted. Egyptian fractions had survived for over 3,600 years, longer than any notation in the Western Tradition. Yet, many modern obituaries on the topic have often been negative, such as Otto Neugebaur's "Exact Science in Antiquity" - where Neugebaur falsely suggests, without proof, that Egyptian fractions marked intellectual decline in the RMP, rather than its actual historical role, intellecutal advancement. (Milo 11/11/06).

• Sufficient historical context. Who, exactly, used this notation for what time period, and why did they stop?

The improvement of Huygens' telescope was one of the major reasons while Egyptian fractions are not used today. Napier developed logs scales, needed to find planetary bodies within base 10 decimals, thereby creating more accurate and smaller units of measure. The new notation freshly used zero as a place-holder. The new base 10 decimal system notation had not been used or anticipated in any way by Egyptian, Greek, Roman and Middle Age mathematicians in their practical and theoretical mathematics. Yet, all the mathematics of the ancients are worth studying, since they developed both practical and theoretical arithmetic, alegbra, geometry and calculus within its Egyptian fraction notation that intellectually empowered it.(Milo 11/11/06).

• More references. The subject has a big literature which is barely touched on here. See http://www.ics.uci.edu/~eppstein/numth/egypt/refs.html for starters. As another instance of a reference that might make for an interesting cross-link with another WP entry, Ron Graham's Ph.D. thesis was on Egyptian fractions.

• The existence of modern methods other than greedy. The greedy algorithm is interesting mostly for its pathological behavior. Modern number theorists have developed much more effective methods. In particular the connection with continued fractions seems important enough to mention: the sequence of continued fraction convergents to a number gives something almost exactly like an Egyptian fraction series, but with both positive and negative signs, and it can be converted to a true Egyptian fraction using the secondary series of the continued fraction. (M. N. Bleicher. A new algorithm for the expansion of continued fractions. J. Number Th. 4, 1972, pp. 342–­382.)

Modern number methods do not commmonly factor the rational number being converted to a unit fraction series. Hence the greedy algorithm, as Stan Wagon "Old and New Unsolved Problems in Plane Geometry and Number Theory mangling easy problems like 5/121, writing out an improved 1/25 + 1/1225 + 1/3477 + 1/7081 + 1/11737, from the very awkward greedy algorithm with a 25 digit last term denominator. Egyptian scribes simply factored 5/121 into 1/11 x 5/11 and wrote 5/11 = 1/3 + 1/11 + 1/33, and 5/121 = 1/33 + 1/121 + 1/363. (Milo. 11/10/06)

The greedy method is not modern, and some modern methods do in fact depend on factorization.

[1]. On the other hand, it would not surprise me at all if the scribes did exactly as you say. But do you have a source for that specific expansion? —David Eppstein 20:57, 8 November 2006 (UTC)

David, an EMLR paper, based on a 2002 publication, and condensed into the "Encylopedia of the History of Science, Technology and Medicine of Non-European countries", edited by Helaine Seline, available in an updated version since 2004 provides adequate information on this topic. I have dropped it into your email inbox, so let me know how you read the EMLR and its connections to the RMP and its 2/nth table. Milo, 11/10/06.

• The 4/n problem. That is, is every number 4/n (n a positive integer) expressable as 1/x+1/y+1/z (again positive integers)? Now that Fermat's Last Theorem has been proven, one of the most notable remaining open problems involving specific Diophantine equations.

• The odd greedy problem. Does repeatedly subtracting the largest possible unused odd fraction eventually terminate, when starting from a rational with an odd denominator? There are other ways of generating all-odd expansions (e.g. R. Breusch. A special case of Egyptian fractions, solution to advanced problem 4512. Amer. Math. Monthly 61, 1954, pp. 200–­201) but this is notable anyway, partly because it's a well-known and easily understood open problem in number theory, and also in part because it's one of a few cases like the Collatz conjecture where it seems very difficult to tell whether an algorithm always terminates.

—David Eppstein 04:14, 8 September 2006 (UTC)

Too right. Article started out as more an essay on egyptain mathematics, see for example this version. It still needs a lot of work. Edit at will. --Salix alba (talk) 07:35, 8 September 2006 (UTC)

I can help with some of this, but much is beyond my level of expertise. I understand your reticence, but I hope you will edit as much as you feel comfortable doing. Other than avoiding linking to your own material, I know of no other restrictions. Tom Harrison ^Talk 14:21, 8 September 2006 (UTC)

Ok, I'll think about it for a few more days but probably do some editing. For future reference I note that the 4/n problem has its own article: Erdős-Strauss conjecture (which also needs work; I'll probably deal with that earlier), and therefore doesn't need a lot of detail here. —David Eppstein 23:29, 8 September 2006 (UTC)

Good stuff. It finally looks like a decient article. --Salix alba (talk) 01:10, 1 October 2006 (UTC)

Thanks! There's a little more left to do (the greedy algorithm redlinks), but I'm pretty pleased with how this came out. —David Eppstein 05:02, 1 October 2006 (UTC)

Confusing

it is very confusing!!!!!!!!!!!!!!!!! —The preceding unsigned comment was added by 195.147.207.170 (talk • contribs) 12:20, 14 November 2006 (UTC).

Could you be more specific? Which sections do you find confusing, and what parts of them do you have trouble understanding? —David Eppstein 15:55, 14 November 2006 (UTC)

Struik

Dear David: Your use of Struik, 1967, as a guide to unfairly degrades ancient Egyptian fractions. Struik's views looks and acts a great deal like Otto Neugebauer's improper reporting of the 2/th table in Exact Sciences in Antiquity. There is no question that base 10 decimals replaced the Egyptian fracion numeration system, the later having domimated Western math for 3,600 years. Yet, Struik and Neugebauer like to declare that only the Greeks developed abstract forms of arithmetic, and that ancient Egyptian fractions showed intellectual decline in the RMP, rather than reporting the fact that every noted Greek mathematician studied in Egypt, learning its mathematics beginning with its numeration system. That is, a broader view of the history of math and the history of science needs to be offered before your 'time warp' jump to modern greedy algorithm studies like your own, as I first read in Intelligencer in 1991. Best Regards, Milo

• • • Dear David, I understand that you are using Struik in a way that is not consistent with the above. However, Struik's point of view only cites a late Hellene text, and not the 2,200 year older Egyptian fraction system, say as Brown discussed. The Babylonian base 60 system rounded off every n/p and n/pq conversion in its awkward and incomplete unit fraction system. Whenever p and pq were not multiples of 2,3 and 5 Babylonian methods, that Struik oddly prized highly, offered poor approximations. Egyptians beginning in 2,000 BCE always wrote exact unit fraction series for its n/p and n/pq conversion. Hence, Struik was incorrect in his evaluation, and his name and conclusion should be removed from Wikpedia. Best Regards, Milo 11/24/06.

This differs how from the use of rounding in modern decimal notation? —David Eppstein 17:19, 23 November 2006 (UTC)

I am using Struik only as a reference for the assertion that these fractions continued to be used practically into medieval times, and for Ptolemy's complaints. I don't think there is anything in the article about Greek supremacy or intellectual decline, though perhaps there is content like that elsewhere in Struik. In what sense is the current treatment unfair, or missing any important history of this specific notation? This is not the place for a broader history of mathematics or of the Egyptians' contributions to abstract thought. —David Eppstein 15:50, 16 November 2006 (UTC)

Dear David, Struik was incorrect in his practical assertion as well. Ptolemy's use of base 60 fractions is another matter. Base 10 decimals and Napier's log tables solved Ptolemy's issues, and removed the 3,600 year history of Egyptian fractions from our 'immediate' memories. Struik's analysis was muddled, and not to the theoretical or practical point of Hellene or older times. Hellenes like Ptolemy needed the use of modern statements of latitude and longitudes, a subject that Ptolemy applied base 60 fractions replacing simple base 10 fractions. Best Regardd, Milo 11/24/06.

David,

I had shown that the greedy algorithm was birthed after 800 AD, hence it should not be connected to historical term Egyptian fraction. Mahavira in India also applied the Islamic algorithm to the area of recreational math, the proper title of the arena that your discussion of the greedy algorithm and Egyptian fractions belonngs.

That is, mixing ancient abstract arithmetic, be it Egyptian, Greek or medieval, sometimes using unit fraction notation for the final step (common in Egyptian and Greek eras) and sometimes not (as was common in medieval times) as is being done in the Egyptian fraction section of Wikipedia is confusing, historically and mathematically. Need I detail the areas of confusion in more detail? Milo

Like it or not, modern number theory is an important aspect of this subject and much of the number theory involved is far from recreational. The use of "Egyptian fraction" to refer to all aspects of mathematics involving sums of distinct unit fractions, not just ancient methods, is not and should not be in question. It is standard mathematical usage, and anything else would be a violation of WP:NPOV, as I already discussed much earlier under the "unhappy" heading. Similarly, I personally am not especially interested in the ancient history, but I have included it prominently in the article (in the longest and first section) because it is also an important part of the subject, and omitting it would be a violation of WP:NPOV. I am not prepared to seriously consider any suggestion of scaling back this article to discuss only one side of the subject. If you want an article that reflects only the historical aspects, unadulterated with modernism, go to Egyptian mathematics or go to some other site than Wikipedia. —David Eppstein 00:55, 17 November 2006 (UTC)

Dear David,

Your continued evaluation of ancient Egyptian fractions as only practical in scope does not consider our discussions of 10 + years ago with Kevin Brown, also a noted modern number theory mathematician. I'll look for Brown's web page where he rigorously shows that the Akhmim Papyrus, written around 400 AD, and its n/p and n/pq table going up to p = 32, pr thereabouts, were more optimal that any version of awkward modern versions of the greedy algorith, the first 10 of yours published in 1991. Clearly modern number theory methods, be they Erdos, Wagon, Bleicher, et al, have not been able to come close the to optimal unit fraction series created by Copts, Greeks or Egyptians. Your introduction of NPOV seem odd. As the moderator has indicated, please provide reference(s) that Greeks, well known to have used abstract arithmetic, as later used by Copts writing in the Akhmim Papyrus, was only practical in scope. Best Regards for an interesting and lively discussion. Milo 11/18/06.

Dear David, you are free, as is anyone to disscuss any subject in a modern context. The term Egyptian fraction is a modern phrase, one that the ancients may not have used. Surely unit fraction series was a common way of writing final answers for 3,600 years. However, when the historical records are reviewed closely, the greedy algorithm was birthed in India and Medieval Europe betweenm 800 AD to 1200 AD, it being a 'brute force method'. In contrast, the historical Egyptian fracion method that dates back to 2000 BCE and continuously used in several forms untel 1600 AD was not 'brute force'. The oldest method found exact, concise and optimal unit fraction series for n/p and n/pq in ways that modern brute force methods have not been able to duplicate. I'd be happy to begin with the EMLR and its 1/p and 1/pq methods, and proceed to the RMP and its closely related 2/p, 2/pq and several n/p and n/pq method in Akhmim Wooden Tablet, methods that Greeks and medievals used until base 10 decimals removed these abstract arithmetic methods from our immediate memories. Yet, the ancient records exist. And following C.F. Gauss' rule, to understand an ancient mathematician, the ancient records must be read, advise that I have been following since I first read in it Oystein Ore's History of Number Theory textbook in 1964. St. Andrews' web page oddly links your modern brute force work to ancient non-brute force methods, a misleading link if I every saw one. This point of view that oddly is also appearing on Wikipedia. Modern brute force 'greedy algorithm's' have nothing to do with historical Egyptian fractions methods used prior to 800 AD, as the ancietn texts reports them. Best Regards (Milo)

Why do you continue to claim that the greedy method is modern? Your insistence on using the greedy algorithm as the paragon of modernism, when so much better algorithms are known now, comes off as intellectually dishonest. If the article were claiming that the Egyptians used the greedy method, or that the greedy method was somehow superior to what the Egyptians did, it would be mistaken and need fixing, but I don't believe that the article as written supports either of those statements. —David Eppstein 23:31, 17 November 2006 (UTC)

David, the greedy algorithm is a modern looking 'brute force' method, a technique that developed after 800 AD, as noted by Leonardi Pisani in 1202 AD. The greedy algorithm belongs in its own recreational math section, thereby not continuing to confuse the historical discussion of 1/p, 1/pq, 2/p, 2/p, ..., n/p and n/pq non-brute force methods used by Egyptians, Greeks and others, even Fibonacci in 1202 AD. Your suggestion that my views may be intellectually dishonest is odd. All of my views were directly parsed from the ancient Greek and Egyptian texts. You have properly concluded that the greedy algorithm has nothing to do with the ancient history of Egyptian fraction methods. That is, do you not agree that the main threads of the greedy algorithm need to be placed in a mutually exclusive category that do not 'rub up against' the 3,600 year history of writing n/p - 1/A = (nA -p)/Ap, with A chosen in the range p/n < A < p, with the divisors of A used to optimally find (nA -p), a method that F. Hultsch found in 1895, always finding a shorter and smaller last term than even the most modern of the 'greedy algorithms', such as Syvlester's 1891 discussions on the subject?

That last sentence epitomizes what I find objectionable about the writing of the article before I made such major revisions to it, writing that I believe you were largely responsible for: (1) It is run-on, badly written, and difficult to comprehend. (2) It tries to pass off the entire history of Egyptian fractions as a conflict between Hultsch and Fibonacci, ignoring any recent contributions to the subject. (3) Its claim that a method described in 1200 is the epitomy of modernism while a method described in 1895 is ancient comes off as strange and ahistorical. (4) Its claim that modern methods are incapable of finding optimally short expansions is false.

• • • David, show me any modern number theory person that have duplicated the RMP 2/n table's optimal series, as David Brown discussed. Please also note that the Coptic Akhmim Papyrus wrote n/p conversion tables listing n upto and including. 32. Brown published the Coptic n= 17 and n= 19 tables, available somewhere on his web page. Clearly, by inspection, the

ancient scribes found relatively more optimal series than published by any modern mathematician used the lastest 'brute force' algorithms, as implied by Brown and stated by myself. Milo 11/18/06

(5) Its claim that modern methods are incapable of finding expansions with the minimum possible denominator is false.

Claiming that some idea is false is easy, in words, but very hard to achieve by modern algorithms/proofs. Ancients used algebraic identities, as F. Hultsch showed in 1895, per the rule, n/p = 1/A + (nA-p)/Ap as we previously discussed 10 years ago (even cititng your following example, as I recall.) Yes, we have already discussed the 31/311 case. It can be found by H-B by taking a second differential, 1/A and 1/B. Our current questions deal with the published 2/n table from 1650 BCE, where n ranged upto and including 101, and n = 32 with often more optimal series in the 400 AD Akhmim Papyrus. Where are the modern algorithms that duplicate and or improve on the ancients? Since there are none, to my knowledge, the ball is in your court. Best Wishes, Milo 11/18/06. ****

(6) Its implication that there always exists a single expansion that is both optimally short and having the minimum possible denominator is false. Example: 31/311 = 1/12 + 1/90 + 1/311 + 1/1244 + 1/1866 + 1/2799 + 1/3110 = 1/12 + 1/63 + 1/2799 + 1/8708; there is no four-term expansion having maximum denominator 3110. —David Eppstein 17:03, 18 November 2006 (UTC) 

• • • I have never claimed that a 4-term series was required, in the

RMP or the Akhmim Papyrus related to a specific denominator. Scibes seemed to find 5-term term or small series, consistent with the length of the traditional 5-term Horus-eye series. Yet, scribes have even recorded using 6-term series. This the ancients place no 4-term, 5-term or even 6-term limit. The 31/311 case is an odd-ball case as you know. Its main 2/p structure can be pierced by the EMLR and RMP rule 31/331 = 31/331 *(1/2 + 1/3 + 1/6). More directly, using Hutlsch-Bruins' 4-term or 5-term series can obtained. I'll try finding a more optimally series than your suggestion, by NOT soley using A = 12 as teh first partition, where the divisors of A can not be optimally found to solve (nA -p) = 61. I send you a solution to this problem over 10 years ago, as you have noted in part (by citing the answer without mentioning the author). A second subtraction step is needed in this case, an ancient form of a second differential. In the 31/331 case, combining the divisors of the first two partitions, divisors of A = 12 and B= 63, is sufficient, as I'll discuss off-line, if you desire. Best Regards, Milo ****

By the way, regarding Hultsch's method — it certainly has some explanatory power, as not all expansions have the form he describes. For instance the two expansions I list above for 31/311 do not include 1/A as terms in the expansion for A=180 and A=252 respectively. So I suppose that 1/12 + 1/63 + 1/2799 + 1/8708 would not be generated by this method,

This looks like the solution that I sent you 10 years ago, and a second subtaction step, using the second step, 1/B = 1/63 and its combined divisors with 1/A = 1/12, solves the problem in the H-B style. Best Regards, Milo 11/19/06.

and in fact Hultsch's method seems unable to find a four-term expansion for 31/331 no matter which A is chosen. But more generally the method does not specify how to decide which A to use, nor how to find the best subset of divisors, so I would not call it an algorithm.

• • • Agreed, on the point concerning the rule for selecting A or B. There are no known rules other than by inspection, thereby showing the degree of work put in by the student/scribe(a form of recreational math?). Here I agree with you, on one point. The ancient H-B method and its clones tended not to follow algorithmic rules. Your methods certainly are algorithms. Hence we have been discussing apples and oranges QED. Milo 11/18/06.

The latter part of the problem, selecting an appropriate subset of divisors, is itself nontrivial: see subset sum problem. —David Eppstein 18:34, 18 November 2006 (UTC)

Dear David, Ron Knott's web pape, as cited in Wikipedia's references discusses finding the best Egyptian fraction series. Greedy algorithms were no help to him, or to you in finding optimal series, I suspect. Knott referenced a web page by Scott Williams, who in turn cited my MAD work on the subject. So what is the difficulty of either Knott or yourself in directly citing my work? Kevin Brown and my work both fairly and accurate read the ancient texts in their optimal series, as modern algorthms (the greedy and its clones have not calculated or published) do not. Milo 11/25/06.

Dear David, Ron Knott does discuss the 'best Egyptian fraction' series cited in one ancient text, the RMP 2/nth table. Kevin Brown's analysis on this subject is to the point, and does not wander. Knott's incomplete review, the later citing you and myself, does not logically or completely discuss the 'best' Egyptian fraction series, a problem that faces all modern greedy algorithms.

For example, Eric W. made not one attempt to make sense of the 4,000 year old ancient data, using only the incomplete modern form of Egyptian fractions created from 'greedy algorithms'. Hence, Erik's otherwise excellent book needs to be removed as a citation from Wikpedia and replaced by Kevin Brown's clear and historically interesting analysis. Best Regards, Milo Gardner. updated 11/25/06.

I am not citing Weisstein for any historical information, but as a relevant web page on related material. I am aware that his site contains mistakes. Brown is not an adequate citation for historical information for reasons I expand on here, but I welcome the addition of a link to his site. —David Eppstein 16:45, 20 November 2006 (UTC)

Dear David, I'll look into your reasons for seeing Brown as you do, and get back to you later. My view appears to be very different. Best Regards, Milo 11/21/06.

Thank you for adding Brown to the reference list. The paper opens up a much needed discussion. Modern Egyptian fraction discussions limited to modern number theory have long left a large hole. The modern hole never attempted to use Hultsch-Bruins and related ancient methods to attempt explain the RMP 2/nth table and other optimal ancient conversions of rational numbers. H-B and its algebraic identity clones work, while modern algorithms do not! Best Regards, Milo Gardner 11/25/06.

Churn

Milo, I confess not understanding where you are trying to go with these recent changes. And my lack of understanding is causing the article to go back and forth without much change, because I don't see your changes as improvements but you keep doggedly reinserting them.

• You want to insert something about Fibonacci studying n/pq expansions. But you are inserting it into the wrong place, a paragraph about the greedy method. I don't understand from your insertion what you are trying to say about Fibonacci's studies, but if it is different from the Engel expansion and the greedy method then it deserves a separate paragraph and a more clear explanation, because it's relevant material not covered elsewhere in the article. Your summaries say to check Lueneberg's book but I haven't had time to track that down yet.
• The fact tag is there because of the specificity of the 1600 date. I cited Struik precisely for the statement that Egyptian fractions were used into medieval times, nothing more. A specific date like 1600 for the death of the notation needs a more specific citation to actual literature where that date may be verified. This citation should go into the references section with a proper Harvard-style citation in the main text. Please do not remove the fact tag unless you add a citation to the article or remove the 1600 date.
• Many of your other changes do not change the content significantly, and I don't find them objectionable in principle, but I don't see why you think the added verbosity improves on the text as it is.
• If you think there is some significant topic not appropriately covered by this article, please say so. Your sequences of smaller changes don't leave me with the impression that you have a plan for what you want to see different in the article.

P.S. please leave this block of text in one piece when replying. Your habit of breaking things up into pieces and not indenting the parts you've added makes it very difficult to follow these conversations. —David Eppstein 02:39, 27 November 2006 (UTC)

OK, since you obviously don't understand requests like "leave this block of text in one piece", here is your response reformatted according to that request. The problem with your interleaved responses is that it makes it impossible to figure out who wrote what when. I also removed the line breaks in the middle of the paragraphs since they break the formatting.

• David, we do have a communication problem. Your view is to maintain

your incomplete and confusing flow of the history of Egyptian fractions, (per 'leave this block in one piece) and not associate my comments after your corresponding paragraph. My citations generally follow your apt comment that our discussion 'war' will be resolved by the rules of mathematics (as we both may agree do not follow the rules of English grammar?). Our discussions seem to have followed the rules of one hand clapping, not allowing both ends of the history of Egyptian fractions to touch, except in your mind. My reviews of the history of Egyptian fractions, prior to and including 1600 AD, have been much clearer and mathematically comprehensive than yours. So why all of fuss on the transition from Egyptian, Greek and medievel forms of Egyptian fractions to the modern view?

• Clearly, your reliance and love of algorithms, per its recent 800 AD

birth, is involved in our communication problem. Leonardi Pisani was able to convert n/p and n/pq into the traditional Egyptian fraction series, facts that you seem unwilling to read up on, described in 1993 by Heinz Leuneberg, or anyone else, at any other time. Your view continues to improperly cite one small aspect of the Liber Abbaci, and Leonardi's recreational use of one algorithm as Sylvester reminded everyone in 1891. Yet, F. Hultsch came along in 1895 and proved that the 2/nth table dating to 1650 BC was built upon a 2/p conversion method (an algebraic identity) that excluded the innovative greedy algorithm, or any algorithm. Hultsch's view was idependently confirmed by Bruins in 1944, so the ancient n/p method has been known as Hultsch-Bruins for over 50 years. Hence, medievels used algebric identities, written as indeterminate equations by the time of Pisani, the dominate method used in 1202 AD. The greedy algorithm was a footnote in Pisni's book, a historical fact that you may feel that I am overly stressing.

• That is, we are conducting a verbal war over algorithms and algebraic

identities. I look forward to a peace treaty, and a proper statement of the medieval uses of both algebraic identities and (greedy) algorithms to create exact Egyptian fraction series in 1202 AD, and at other times prior to the death of Egyptian fractions in 1585 AD (with the rise of base 10 decimals.)

Best Regards, Milo Gardner 11/28/06.

• David, I am properly discussing the contents of the 1202 AD Liber Abbaci, as fully translated and reported by Heinz Lueneburg in 1993. Omitting Leonardi Pisani's understanding and methods of writing n/p and n/pq rational numbers into Egyptian fraction series is odd. Leonardi's method for writing n/p conversions extended to search for A to 2p, a point that should be of interest to many, if it was allowecd to to re-published in Wikipedia. Milo 11/27
• David, in context of the total Wikipedia article, the citation of Struik says that Egyptian fraction notation was a clumsy notation. This statement is incomplete and confusing since it does not specify the source of the clumsiness. Was clumsiness based in rounding off rational numbers to unit fraction series, as noted in the Hibeh P? Milo
• David, my plan is to always shows that the term Egyptian fraction references to two historical and two modern facts. First, the observation of Egyptian fraction series, in ancient or modern times, means that a rational number had been expressed. Skipping over the mentioning of the referenced rational number, at any time, needs to stop (even if the ancient scribe did not cite it specifically). Second, given a rational number and an Egyptian fracion series, facts are present to work backwards or forwards to determine the method or methods that were used by the ancients, or by moderns like yourself. Leonardi was kind to summarize his n/p and n/pq method in abstract (arithmetic). Yet, Fibonacci' work in this area continues to be skipped over, as you and others have skipped over 43/48, and the primary definition of a an Egyptian fraction (series) when mentioning 1/2 + 1/3 + 1/16. Milo 11/28
• David, citing 1/2 + 1/3 + 1/16 as an Egyptian fraction is an incomplete and confusing sentence. Somewhere the fact that 43/48 was converted to 1/2 + 1/3 + 1/16 to an Egyptian fraction series needs to be stated, be the situation ancient or modern. That is, the current generalized a/b rational number to a modern Egyptian fraction (series) statement is not properly introduced and therefore not defined as it was used by ancients or by moderns like yourself.. Milo 11/28

David, given that you have now written easy to reply paragraphs, hopefully, we can progress to discuss the fundamental issues. Previously, you had tended to throw multiple points into one hard to reply paragraph, not leaving a logical spot to reply in context. Milo 11/28.

And now some replies, numbered for your convenient reference.

1. Who is "Leanardi Pisani"? That is not a standard name for Fibonacci in English.
2. If there is a third topic on Egyptian fractions studied by Fibonacci in Liber Abaci, beyond greedy expansion and Engel expansion, it would be reasonable to include it as a third topic paragraph in the "medieval mathematics" section. But "method for writing n/p conversions extended to search for A to 2/p" doesn't really tell me what this third topic consists of. When you use the letter A, are you suggesting that he used the same method of summing divisors of Ap that you had previously been attributing to Hultch? Did he write out tables of expansions according to some unknown rule that we must infer ourselves, as Ahmes did?
3. Re clumsiness see Ptolemy, not Struik. My expectation is that it had to do with the difficulty of performing arithmetic calculations and comparisons using this notation. Since the context was astronomical observations, the issue of rounding vs exact representations of rational numbers is irrelevant, as all observations are rounded.
4. I don't see what value the word "series" adds to the first sentence. It just means that it's a sum, but that's obvious enough from the plus signs in it, and we already use the synonymous "sum" in the same sentence. It's redundant and therefore confusing. And why do you insist on stating that 1/2 + 1/3 + 1/16 is a representation of a number rather than the number itself when you don't make the same insistance that 43/48 is a representation of a number and not the number itself? In any case these sums have other uses than as representations, so if this talk about representations belongs in the article at all it belongs, not in the first sentence where you have been trying to insert it, but in the third sentence where the application of these sums as a representation (or, as the sentence uses, "notation") is introduced.
5. Your response still talks about individual elements that you have repeatedly attempted to insert into the article. Do they form part of a larger plan? Is there a pattern to them? Will there come a time when you say "enough is enough, the article can be left alone" or do you foresee adding these quibbling changes continually for a long time to come? Because any one of these changes may not be so bad on its own but the reason I keep reverting them is that the pattern I see is that they are causing the article to slide, very slowly, back towards the total disaster it was, organizationally, mathematically, and grammatically, before I reorganized it.

—David Eppstein 17:33, 27 November 2006 (UTC)

Dear David, I'd very much like to respond to each and every point mentioned below, in context. My goal is to discuss Egyptian fractions, in its three part structure. The three parts are the rational number, the method of conversion, and the historical Egyptian fraction series. That is, the second point covers the range of methods that have been used to convert the rational number to unit fraction series, a subject seems not to have grabbed your attention in a serious way. Generally, for all of its history, Egyptian fraction series were created by two algebraic identities, one for n/p rational numbers and one for n/pq rational numbers, as I'll happily detail, given the chance.

It appears that your discussions often cite only two of the required (logical and historical) three points that define Egyptian fractions. Example, your definition of an Egyptian fraction as a unit fraction does not go on and include any reference to the beginning rational number, 43/48 case, when you cite the series 1/2 + 1/3 + 1/16. Skipping over one of the three required elements has been a confusing issue to myself, reading your work, and to historians that have begun with the historical Egyptian fraction series, say of Ahmes, and not gone on and cited the correct rational number.

The beauty of Egyptian fractions is that given an Egyptian fraction series, and an ancient method, a historical rational number can easily be found. Given that historians have often omitted the mention of scribal rational numbers, and stressed Egyptian fractions as a separate subject (which it really is not), texts like the Akhmim Wooden Tablet were not properly read until 2002 AD. Yes, I am discussing Egyptian and Greek rational number arithmetic and its related algebra, defined by algebraic identities, a notation that was first defined in the EMLR and its 1/p and 1/pq examples. Note that the RMP's 2/nth table generalizes 2/p and 2/pq methods, as Hultsch, Boyer, Brown and others have pointed out. Yet, a large group of classical scholars, i.e Neugebauer, Peet, et al, mostly trained or working in the 1920's have not appreciated the 2/nth table discussion as proto-number theory. Is not time to discuss Hultsch's 1895 discovery? I think that 111 years of waiting is long enough.

Finally, may I respond to your following comments, paragraph by paragraph so that Winston Churchill's apt comment, "the US and Britain are separated by a common language", does not further apply to our discussions? Best Regards, Milo Gardner 11/28/06

David, I have tried to patch up a flawed article, paragraph by paragraph. Let me stop this practice, and begin at the top by freshly defining an Egyptian fraction as containing three elements: the beginning rational number, the method used to convert it to a unit fraction series, and the actual unit fraction series (the so-called Egyptian fraction. Do we have a deal? Best Regards, Milo 11/28/06.

I don't believe we know what method was used by the ancients to produce their expansions. I am willing to assume that Hultsch was correct that most of the 2/p fractions are formed by summing divisors of Ap for some sufficiently small and highly composite A, but that pattern is not itself a method — there are multiple different methods that could have led to the same pattern. One is that they explicity looked for factors of this type by systematically trying possible values of A and all possible combinations of divisors. Another (unlikely to have been used by the ancients) is that they used a dynamic programming technique to find appropriate combinations of divisors without trying all combinations. Another is to subtract small round fractions such as 1/2, 1/6, etc until the remainder can be expressed using only its own divisors. These would all end up generating expansions that fit the same pattern but are very different as methods. I also think "the beginning rational number" presupposes that Egyptians thought of their numbers as ratios primarily, and unit fraction series only secondarily, which seems an assumption unlikely to be strongly supported by the actual evidence. For instance, when adding two Egyptian fractions, did they convert them both back into ratios, add them as ratios, and then expand the resulting ratio as a sum, or did they simply combine the fractions from the two summands using their 2/p table to eliminate duplicates? The former would support your insistance that the ratio come first, but the latter would not. —David Eppstein 00:31, 29 November 2006 (UTC)

David, thank you for accepting H-B as a possible way to read the 2/nth table 2/p series. No other method has been offered for 111 years, so the jury should be in on this subject. One interesting implication of H-B is that it shows and confirms that scribal subtraction consisted of:

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

with A being larger than p/2, and less than p (before medievals extended the range to 2p), with the divisors of A being used to 'best' find the solution to (2A -p). H-B was also generalized by Ahmes and the AWT scribe to mean,

n/p - 1/A = (nA -p)/Ap,

with A being larger than p/n, and so forth.

Your citation of a ratio seems odd. Ahmes and all other scribes used a method of division that looked like this, when dividing a fraction by a fraction:

(a/b)/(c/d) = ad/bc

as used in all of the RMP's algebra problems. For example, the one that has been shown can not be solved by 'false (sup)position', is RMP 33. Note the above rule for division solves this problem, as written by Ahmes and his final Egyptian fraction series by:

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

(97/42)x = 37

x = 14 + 1/4 + 1/56 + 1/97 + 1/194 + 1/338 + 1/679 + 1/776

as I'll happily show you came from

x = 14 + 28/97 = 14 + 2/97 + 26/97

as the 2/nth table calculated for 2/97 and H-B computed for

26/97 - 1/4 = (104 - 97)/(4*97) = (4 + 2 + 1)/(4*97).

Yes, I can barely see how you may have misread this situation as a ratio. However, note that the algrebraic form of this division is really out modern 'invert and mutiply' rule, one that sadly not often has not been stated for our youth in its algebraic form. Egyptian algebra as shown above did NOT use false (sup)position, as long extrapolated from 1920's scholary stabs in the dark, since Occam's Razor says that the most direct and simplest method was the historical method.

Again, to summarize, Egyptian fraction was used by scribes to write exact remainders created from remainder arithmetic rational numbers, as 28/97 was parsed to 2/97 + 26/97 in RMP 33. To fairly translate any ancient document to modern terminology the ancient Egyptian fraction series MUST be fist replaced by the appropriate rational number that create it (sometimes by hard to decipher scribal methods, with one being H-B). When the correct rational numbers are written in, the arithmetic found in the RMP, AWT and all the other texts report remainder arithmetic. For a second example, from the AWT, the division of a hekat unity, 64/64, by n was written as, and fairly translated as:

(64/64)/n = Q/64 + (5R/n)*1/320

when n was less than equal to 64. When n was greater tham 64, a hin unit was written as 10/n, and a ro unit was written as 320/n, as 2,000 medical text data points report.

Thank you in advance for the willingness to read about Egyptian fractions and the scribal exact weights and measures system, with (5R/n) being an Egyptian fraction. To ancient scibes Egyptian fractions were remainders within a remainder arithmetic form of arithmetic. Best Regards, Milo Gardner, 11/29/06

I was reacting to your insistance that 1/2 + 1/3 + 1/16 should be first written using a ratio-based notation, 43/48, rather than being left to stand on its own without conversion to a ratio. —David Eppstein 16:34, 29 November 2006 (UTC)

43/48 is a rational number, and not a ratio. Scribes often ran into rational numbers, as RMP 33 cited 28/97, and then finalized their answers by unit fracton series conversions. Milo 11/29/06.

43/48 is the representation of a rational number as a ratio of two decimal integers. It is not the number itself. The same number has other representations, for instance in other bases, or as a Dedekind cut, or as an egyptian fraction 1/2 + 1/3 + 1/16. —David Eppstein 18:37, 29 November 2006 (UTC)

David, discussing Egyptian numeration: all of Egyptian numbers were ciphered onto their hieratic alpabet, including 43/48, or 28/97, as Greeks ciphered its numerals onto their Ionian and Doric alphabets. All numbers at any time, counting numbers or rational numbers, are only representations of its base ideas, sometimes written in clear ancient or clear modern notations, and sometimes not. Egyptians drew a line over the its hieratic ciphered numerals to denote a fraction, while Greeks wrote its fractions like 1/p as p'. Both ancient notations are clear to me. Best Regards, Milo Gardner, 11/30/06.

Style

I think something like this is likely to be incomprehensible to most of our readers: "So what we know about ancient methods for calculating with Egyptian fractions has been based on reading the EMLR, and its 1/p and 1/pq methods, and the RMP 2/nth table and its 2/p and 2/pq methods, that taken together extrapolate into n/p and n/pq conversion methods used in the AWT, MMP and other texts." This might be appropriate for a specialized journal, but does not go well here. Tom Harrison ^Talk 14:52, 30 November 2006 (UTC)

Tom, please feel free to reprhase the historical facts into a more readable modern sentence, or a modern paragraph. To simply observe the ancient Egyptian fraction series and not attempt to show that a specific rational number, and a specific method of conversion, was used by a scribe is not to fairly consider F. Hultsch. E.M. Bruins and many other scholars over the last 111 years.

My suggestion to resolve my sometimes 'specialized journal' language is to simply say: all Egyptian fraction series, ancient or modern are based in a rational number, and a range of conversion methods. Ancient scribes used algebraic identities like 1/p = 1/p*(1/2 + 1/3 + 1/6), as used in the EMLR and RMP. Modern conversion methods have stressed the use of algorithms, beginning with Fibonacci's 1202 AD greedy algorithm. Best Regards, Milo Gardner 12/1/06.

Your obsession with only the ancient aspects of the problem, and in particular with 2/n tables, has led you astray. The identity 1/p = 1/p*(1/2 + 1/3 + 1/6) will not ever terminate if you try to use it to expand 3/n or bigger. E.g.

3/4 => 1/4 + 1/4 + 1/4 => 1/4 + 1/8 + 1/8 + 1/12 + 1/12 + 1/24 + 1/24 = 1/4 + 1/8 + 1/12 + 1/12 + 1/16 + 1/24 + 1/24 + 1/24 + 1/48

and now you have three 1/24's, not any better than the three 1/4's you started with. In fact no identity like this, with denominators linear in the input, can work in general [2]. However if you want identity based methods the article describes two that do work: the one in the bullet with the citation to Takenouchi (1921), which is useful for showing that the requirement of distinct denominators does not make any Egyptian fraction longer than it would be without that requirement, and the one in the next bullet by Graham and Jewett, which produces bad expansions but is interesting as an example of an algorithm that's nontrivial to prove termination of. Also your edit comment repeated the two lies that ancient methods were optimal and that modern methods are not; I wish you would read what we have written about modern methods and learn otherwise. In fact I explicitly included in the article the sentence beginning "It is possible to use brute-force search algorithms" to counter the false statement in MathWorld that no optimal algorithm exists. —David Eppstein 02:14, 2 December 2006 (UTC)

Or, if you don't like that example, here's another: try adding 1/11 to 1/11 + 1/22 + 1/66, using only the identity 1/p = 1/p*(1/2 + 1/3 + 1/6) to simplify any repetitions you find. The existence of the non-identity 2/p expansions in the RMP masks this problem for the simpler fractions but doesn't really get rid of it. —David Eppstein 02:36, 2 December 2006 (UTC)

David, your use of modern 'strawmen', and thereby not reading/using the ancient texts as your primary guide is amazing. Graham and Jewett never discussed the 2/nth table and its Egyptian fraction patterns, as cited per your link.

The ancient scribal kit bag was filled with tools, or several algebraic identities. David continues to overlook the vivid EMLR to RMP link of 1/p = 1/p*1, with 1 = 1/2 + 1/3 + 1/6. Ahmes only used it ONCE to show that any 2/p conversion could be made by 2/p = (1/1 + 1/2 + 1/3 + 1/6). So why have you gone on and discussed the 3/p case, a subject that I have not seen discussed by any scribe by the 1 = 1/2 + 1/3 + 1/6 method, as first developed in the EMLR (I'll send you a paper on the EMLR, as published in another encyclopedia, based on a longer journal article published in 2002, if you are upto it). That is, if you want me to guess on scribal thought, I would use Hultsch-Bruins for 3/p, ..., n/p, as I would use on the silly 4/p or n/4 cases, often limited to three terms in modern (recreational) number theory. The ancient scribes usually found 5-term or shorter solutiuons - but even that rule was not always followed, as ancient texts confirm.

Oh yes, David, my offer to fairly define any ancient or modern Egyptian Fraction (series) as containing three elements: (1) the initial rational number, (2) the method used (be it an algebraic identity or algorithm, and (3) the final Egyptian fraction series, is still open. I look forward to your 'obvious' comments on the topic. Best Regards, Milo Gardner 12/02/06.

I still don't see why it is necessary to start with a ratio, use some method to convert it to a series, and only then talk about the series itself. It seems a very indirect method of approaching the problem, and one that is focused on issues of using these series as representations of rationals, issues that are really only relevant for the ancient section of this article. For the modern parts, the series itself is primary, not the value found by summing it. —David Eppstein 18:52, 2 December 2006 (UTC)

David, Tom, et al, all medieval, Greek and Egyptian math was rational in their arithmetic forms. Medievals made no bones about it, often dropping the convention of unit fractions for its final arithmetic statements, citing only the vulgar fraction. The Liber Abbaci, written in 1202 AD was an exception. It cited two updated n/p and n/pq ancient conversion methods, as well as the modern greedy algorithm. However, Greeks having learned their math and arithmetic in Egypt, followed the Egyptian practice of converting all rational numbers and vulgar fractions to short and concise unit fraction series, as a final notation. Egyptians fraction conversion methods used by scribes have confused scholars in Egyptology since the RMP began to be read in 1880's by Sylvester(someone that grossly failed in two papers, one oddly adding greedy algorithms to our menu) and others like Peet, Neugebauer and the 1920's classical scholars.) Why else all the fuss when Pythagoras showed that irrational numbers were real, and that the square root of 2, and so forth, had to be included in the scribal kit bag? Clearly, all of Egyptian math was rational number based, with no irrational or higher oder numbers being allowed. Greeks slowly added the irrationals, but, at the same time, maintained the neat conventions of using Hultsch-Bruins and other algebraic identities as methods to 'encode' their vulgar fractions into short and concise unit fraction series.

That is, Egyptians did not use a ratio, at any time that I or you have seen in an ancient text, in any sense of the term. Hence anyone's continued use of ratio in terms of Egyptians needs to stop, unless an ancient text has been shown to contain it! Best Regards, Milo Gardner 12/5/06.

2/p expansion for primes

In connection with the 2/n expansions for primes, it may be of interest (though I think too much WP:OR for the article itself) that the 1/n + 1/2n + 1/3n + 1/6n expansion used in the RMP for 2/101 is optimal (in the sense of minimizing the denominator) for all but finitely many primes. The only exceptions are n = 3, 5, 7, 11, 13, 17, 29, 31, 43, or 73. Note that many of the RMP expansions are nonoptimal in this sense. For proof and a 2/n table of exceptional cases see here. —David Eppstein 16:53, 5 December 2006 (UTC)

David, using Hultsch-Bruins, a small number of 2/nth table series are not optimal. I would not categorize the total of non-optimal series as 'many'. I'd be happy to discuss cases with you, to show you what I mean. I'll begin by looking up the web site cited above, and get back with you. Of course, Ahmes' kit bag of algebraic identities was large, the number and exact definitions I'd also like to discuss. Milo Gardner 12/5/06.

Ok, based on the list of 2/prime RMP expansions in Brown's page, and with "optimal" meaning that the expansion minimizes the max denominator (which is what I meant by "in this sense" above):

• Optimal: 3, 5, 7, 13, 19, 31, 73, 83, 101
• Not optimal: 11, 17, 23, 29, 37, 41, 43, 47, 53, 59, 61, 67, 71, 79, 89, 97

Dear David, the list that I read from your web site listed 2/7, 2/13, 2/17, 2/19, 2/29 2/43 and 2/73 as not being optimal. No where did you cite the method or methods for calculating your optimal series. Clearly your method or methods (maybe brute force) since a neat algebraic idenity was used by Ahmes. Whatever method that you did use, in each case, several exceeded the bounds of p used by Ahmes. For example your optimal citation of 2/29 and 2/43 cited A = 30, and A = 60 respectively was out of bounds. Fibonacci's 1202AD method accepted such a 2p limit, and Fibonacci may have thought that your values for 2/29 and 2/43 were optimal. But surely Ahmes or any Egyptian scribe would have disagreed with you. Ahmes' method, whatever it was, did not exceed p for the first partition, as you can determine for yourself. Milo 12/6/06.

The list 2/7, 2/13, 2/17, 2/19, 2/29, 2/43 and 2/73 are the fractions for which the 1/n + 1/2n + 1/3n + 1/6n expansion is not optimal. The list above are the fractions for which the RMP expansion is not optimal. They are very different lists because the RMP used the 1/n + 1/2n + 1/3n + 1/6n expansion only once. —David Eppstein 16:21, 6 December 2006 (UTC)

I think the word "many" is justified. Whether one should conclude from this that the Egyptians had some other criterion for preferring the fractions they listed, or whether one should conclude that they did want to optimize the maximum denominator but lacked the mathematical knowledge to do so consistently, is a different question that I take no position on. The RMP expansion that most baffles me, though, is one of the composite ones mentioned in the article: 2/95 = 1/60 + 1/380 + 1/570. Was it so important to match the 2/19 expansion that they preferred this to 1/60 + 1/228? Or was this just an oversight? —David Eppstein 01:26, 6 December 2006 (UTC)

David, my position is that the EMLR set down non-optimal solutions related to the partition any 1/p and 1/pq rational number. The methods used by the EMLR student, as I have sent to you by email, include the 1/pq = 1/A times A/pq method used for 1/8 and 1/16, a method that works very well in the RMP for its 2/pq cases, allowed A = (p + 1). Clearly the EMLR contained recreational data, and therefore, at times, was not a search for the 'best' series. The EMLR showed that a small kit bag of algebraic identities worked, and that students should work with them before moving on to the more serious 2/n methods.

The same may be be said for the RMP, but only a small number of examples were not the best. To my eye, Ahmes was not trying to find the best, or the optimal series for all 2/p and 2/pq conversions. Ahmes showed that a short and concise conversion was always available, based on using only four methods, a great step forward in mathematics, a position that I have taken on this subject. Clearly, Ahmes methods included H-B for most 2/p series, with the first three term series 2/13 not being optimal using anyone's definition. Ahmes showed that taking the first available composite number in the range p/2 < A < p worked nicely, and was possibly cited for historical purposes, since the system of Egyptian fractions may have been very old, and took years to develop.

While you panned Ahmes' 2/43 conversion, looking at Ahmes actual series, I appreciate his hard work, given that Ahmes looked through all the possible A's, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40 and 42, and selected 42. I find that the following work as neat, compared to the several non-opimal series that were chosen by Ahmes:

2/43 - 1/42 = (84 -43)/(42*43), with divisors of 42 = 21, 14, 7, 6, 3, 2 allowing, 2/43 = 1/42 + (21 + 14 + 6)/(42*43) = 42' 86' 139' 308' (using Greek notation,since I find it easier to read).

Clearly Ahmes and all Egyptian, Greek and medieval scribes used the aliquot parts of the denominator of the first partition to find solutions of n/p and n/pq conversions, a great innovation that is used in modern number theory. I acknowledge the birth of the idea of aliquot parts, in the work of Ahmes and the earlier scribes. Best Regards, Milo 12/6/06.

Concerning the 2/95 case, discussed by many historians, Ahmes kit bag of algebraic identities including factoring. Note that 2/19, taken from the RMP, A = 12 using H-B, times 1/5 produces Ahmes 2/95 series. This factoring fact is critical in understanding scribal arithmetic and its range of methods to generally find short and often the best conversions of n/p and n/pq. But in the end, Egyptian fractions was only a convention, used for a purpose that I have not fully determined, other than as a final notation. Clearly scribes never computed with Egyptian fractions related its its four arithmetic operations, as long falsely associated with division (false position), and subtraction, by scholars not showing H-B as the scribal method. Milo 12/06/06.

My position remains unconvinced that H-B was used as a method. It is an accurate description of the expansions but the same expansions may be found in other ways as well. The 2/43 expansion in the RMP is hard to fault, given that the optimal expansion 1/60 + 1/86 + 1/129 + 1/172 + 1/215 requires five terms and does not fit the H-B pattern, but I did not mean "nonoptimal" as a criticism but as a rigorous mathematical statement. As for 2/95, I recognize that the expansion they used can be explained as coming from factoring 95 as 5×19, but that doesn't answer whether they recognized that 1/380+1/570=1/228 but didn't want to simplify it because it would break the parallelism from 2/19, or wether they failed to recognize that such a simplification was possible. —David Eppstein 16:21, 6 December 2006 (UTC)

David, thank you for the discussion. We are closing the gap between ancient algebraic identities, a term that sems to escape the keys of your keyboard, be it H-B or another, and your modern view of the subject. Ahmes actually used five algebraic identities to create his 2/nth table (1) a method for 2/pq, one that many have offer solutions, though I like the one suggested in the EMLR -THAT BEING (2/A)*(A/pq), with A = (p + 1). (2) a method for 2/p, one that Hultsch-Bruins has offered since 1985 (and not refuted by anyone that I know of. Saying a method may not have been used bu Ahmes does not mean it was not used), (3) a method for 2/35 and 2/91, one that may have simply used an odd number in H-B, and not a necessarily a version of n/pq = 1/pr + 1/qr, where r= (p +q)/n used in 400 AD, or the one suggested by Kevin Brown, an inverse arithmetic mean and inverse geometric mean; (4) a factoring method for 2/95, allowing 1/n to be multiplied by any 2/p series, in this five times 2/19 solved by H-B, and finally, (5) the simple EMLR method for 2/101, stated as 2/p =1/p*(1/1 + 1/2 + 1/3 + 1/6), as any 2/p < 2/101 can be created, as you have acknowledged.

In summary, all that the EMLR student scribe and Ahmes, the professional scribe, were doing was to easily convert any n/p and n/pq rational number to a short and concise unit fraction series, as a final notation attached to several classes of problems. They did this by beginning with 1/p and 1/pq problems, and progressed to 2/p and 2/pq problems, and finally as Ahmes used several times the n/p and n/pq cases. All the RMP algebra problems were solved in this manner, with the final rational number being written out in the traditional 'best' and not necessarily optimal Egyptian fraction series (using modern glasses). Thus, optimal Egyptian fraction series termed by Gillings and others seems to be a misleading statement. Best Regards, Milo Gardner 12/06/06.

David, and Tom: It appears that our Egyptian, and Greek discussion has progressed very well in fairly reaching out to the medieval method(s) of writing its rational number arithmetic. All three periods of time used Egyptian fractions as a final notation, and never- to my knowledge- as a context to solve any algebra or any technical problem that Babylonians or Ptolemy would would worked on. That is, I would like to fairly and directly describe and report the Egyptian fraction context of Fibonacci's Liber Abbaci, as written up in 1993 by Heinz Leuneburg, or anyone of your choosing (taken from old Theory of Equation textbooks, one of which I studied as an undergrad, or where ever) that discusses more that the greedy algorithm aspect of this 1202AD text. A brief summary will do, of course, other than Struik's incomplete citation.

My position continues to be that Ptolemy's work in the Almagest was looking for a better way to compute and write final latitude and longitude problems and answers. Only the 'final' aspect of Ptolemy's work was implied by Struik, thereby unfairly concluding that Greeks dropped computing with Egyptian fractions since Babylonian numeration was better.

Egyptians, Greeks and Hellenes never computed with Egyptian fractions, other than as proofs, as noted by Archimedes, per his famous lemma. The lemma shows that a 'method of exhaustion' related to our modern calculus limit theorem was not used. What was used is noted by:

1. 1/4th geometric Infinite series statement, as noted by Dijksterhuis in his biography of Archimedes:

4A/3 = A + A/4 + A/16 + ... + A/4n + ...

2. Proof (Egyptian fraction finite arithmetic)

4A/3 = A + A/4 + A/12,

which means

1/3 = 1/4 + 1/12 using Egyptian fractions, as almost any ancient infinite series was proven by finite Egyptian fraction series.

I have not thought or read a great deal about the 1/4th geometric series, since the ancient subject has not been well reported in the modern media. What has been reported is that Archimedes and his mentor Eudoxus, and others, used it to find the area and volume of a wide range of shapes, opening up modern differential calculus, applied to slices of a parabola, as noted above.

Thank you both for discussing and moderating an important aspect of the history of Western Tradition mathematics. Best Regards, Milo Gardner 12/8/06

David, your suggestion that 2/pq = 1/pr + 1/qr, where r = (p + q)/2 is the same as 2/pq = 1/A x A/pq is wrong. The first splitting method defines the special case 2/35 and its sibling 2/91. The second method is found in the EMLR, and it only finds the regular cases, excluding 2/35, 2/91, 2/95 and 2/101.

Best Regards, Milo Gardner 12/18/06

That's not the pair that I meant were the same. The two same ones are (1) "2/pq = 2/pq = 1/aq + 1/apq, where a = (p+1)/2" (as described in the article as it stands now) and (2) "2/pq = 2/A x A/pq//, where A = (p + 1)" (and then using the representation of A as the sum p+1 to split the fraction into two unit fractions), in the reverted version of the same sentence. Obviously you disagree, but I find the first way of writing this identity to be easier to understand, even ignoring the extra slashes, because it more clearly describes the two unit fractions that result. —David Eppstein 23:41, 18 December 2006 (UTC)

David, My 2002 paper only shows one method, 2/pq = 2/A x A/pq where A = (p + 1) as directly parsed from the EMLR. The singular method that you have chosen had been noted by Boyer, Gillings and others, parsed from 'who knows where'. Given that you have chosen to cite my 2002 paper, the EMLR method, as used 4 + times, this method is not optional. It must be cited. Of course, I do not mind Boyer, Gillings and others work cited in this context, the more the merrier.

To further state alternatives to the 2/35 and 2/91 case, I see two clear alternatives: (1) 2/pq = (1/p + 1/q)*(2/(p + q), and (2) 2/pq = 2/AH, where A = (p + q)/2, and H = (1/p + 1/q)/2 = 2/pq/(p + q). Yes, A is the arithmetic mean, and H is the harmonic mean. Takin A and H together, this proposal may report an inverse 'golden proportion', a classic phrase mentioned by van der Waerden in 'Science Awakening'.

That is, ancient scribes were arguably nimble in selecting their wide range of methods to generally convert n/pq to Egyptian fraction series. Scholars just have not explored this nimble issue very well. The EMLR reports five basic methods, excluding Hultsch-Bruins. Four of the methods that have been agreed upon since 1927. The fifth is my 2/pq = 2/A x A/pq method, one that connects to the one error in the EMLR. My view is that the RMP accepts all of the EMLR methods, plus it adds the Hultsch-Bruins, and 2/95 methods. Gee, at least four methods have been confirmed in two texts, a fact that should not go un-noticed ( possibly even in the Liber Abaci, as we'll discuss in January?).

Taking a broad encyclopedic view of scribal n/pq method requires the discussion of alternatives, if for no other reason, alternatives gives each text and class of 2/n series an opportunity to 'call-out' or specify its most likely conversion method. This form of discussion is not often entered into by scholars. Yet, beyond the standard 1920's scholars, and their sometimes 'rush to judgment' the ancient texts can be seen as speaking for themselves, loudly in some cases. Read me, and look at my work, and see how I thought. Best Regards, Milo Gardner 12/21/06.

There is no difference. They are not two different methods, they are merely two different ways of writing the same algebraic identity. I don't see why we should use the same exact sequence of symbols in the same order as any particular source; we should be free to write algebraic identities in an order that is most understandable. And frankly, I find your preferred syntax, "2/pq = 2/A x A/pq where A = (p + 1)" difficult to understand. It gives an expression that does not look like an Egyptian fraction but rather a product of two vulgar fractions; one has to know that the second A (but not the first!) in this product causes the fraction it is part of to be replaced by the sum of two smaller fractions, apply the distributive law to the product, and then simplify each term, none of which is written out or obvious. In contrast, the form I wrote it in, "2/pq = 1/aq + 1/apq where a=(p+1)/2", while mathematically equivalent, has already been written as an Egyptian fraction; the reader need do no additional algebra to interpret it. If you think this version of the identity needs a different citation than the one given, but I feel strongly that the easier to understand form should be the one used in the article and that "2/pq = 2/A x A/pq where A = (p + 1)" should not be described as a different method than it, or even included, as this alternative way of writing the algebra is more confusing than helpful. —David Eppstein 17:38, 21 December 2006 (UTC)

David, your insistence on Boyer's form for converting RMP 2/pq series, reminds me of Gillings' EMLR analysis where he only used the well known identity 1= 1/2 + 1/3 + 1/6 to correct 1/13 = 1/28 + 1/49 + 1/98, by suggesting that the student may have meant 1/13 = 1/26 + 1/39 + 1/78.

Clearly the student was thinking mod 7, since 1/7 x(1/4 + 1/7 + 1/28) = (1/7) x (12/28) = (1/7)= (3/7 = 3/49, rather than the correct 3/39 beginning fraction. So given that the student may have been asked to use 7 to convert 1/13, how can 7 be used? One method that discusses an A, the Egyptian 'think of the number' clue that I prefer, or A = 7 in an identity 1/n = (1/A) x (A/n) such that:

1/13 = 1/7 x 7/13, allowing 7/13 = 1/2 + 1/26, or 1/13 = 1/7 x (1/2 + 1/26) = 1/14 + 1/182,

as the student did not find.

The student may have muddled his version of the 3/39 problem, writing 3/49, because he/she did not know how to write 7/13 using Hultsch-Bruins or by a related method, since that type of method had not been taught. The EMLR was only stressing methods to convert 1/p and 1/pq.

I'll not go on and discuss the EMLR cases that used A = 3, 5, and 25 as a proving ground to show that the RMP A = (p + 1) method was superior that did appear in the EMLR. Again, my view is that scribes were taught to "think of a number", as Gillings shows was used in other areas. In the cases of 1/p, 1/pq, and 2/pq conversions an ' A' may be the 'best' way to describe the ancient scribe's thinking in this area. Our limited views of the ancient scribal methods needs confirmation by at least one additional text, a standard that seems to to missing in several of your conclusions. May we soon resolve our rhetorical differences by discussing the related mathematics, modern and ancient, that relates to the 'word' problem. Best Regards, Milo Gardner 12/22/06.

The latest batch of revisions

Some comments on the revisions of the last couple days (now headed "final draft?" so I think it's safe to comment and edit). If you (Milo) wish to respond in the middle of these, please obey the Wikipedia convention of indenting your comments by putting two colons (::) in front of them; your usual habit of interspersing unindented comments makes things very difficult to read.

• "with the final answers" expressed as Egyptian fractions — I have no issue with this clarification

• "These expansions do not match those produced by any algorithm; rather, several different algebraic identity methods were used to produce the expansions in this table:" — you removed "single" before algorithm. Frankly I find your distinction between algorithms and methods bizarre and difficult to understand. An algebraic identity is an algorithm, if a very simple one. I am editing the word "single" back in. In addition, I think "algebraic identity methods" is redundant, and I don't care for overuse of this grammatical style of piling up noun phrases together; I think "algebraic identities" would convey as much information and be more understandable, if I agreed that these methods were in fact algebraic identities. However, the part of Hultsch-Bruins in which we find sets of divisors summing to 2A-p is not really algebraic in nature. So I think the vaguer "methods" is a better choice here. Instead I added the word "algebraic" before the next use of the word "identity" (in the first rule that I think really is an algebraic identity) so that that phrase is not lost from the article.

• "Curiously, to those that use algorithms," — this phrase comes across to me as an attempt to insult the intelligence of me specifically, a very unencyclopedic sentiment. I don't see how it adds any useful information to the article. And I don't see what algorithms has to do with curiosity over a missed number-theoretic simplification. Deleted.

• "clumsiness of the base 10 vulgar fraction notation from which Egyptian fractions were written" — you have completely inverted the meaning of this citation, which clearly in context is a complaint about Egyptian fractions, and turned it instead into something about decimal notation. I consider this a deliberate falsehood. Reverted.

• "The primary subject of the Liber Abaci is calculations involving proto-decimal and vulgar fraction notation, both being eventually used, along with zero as a place-holder, to replace Egyptian fractions with the modern basde 10 decimal system" — in fact the notation in Liber Abaci itself was decimal (even including some decimal fractions, in a different notation to ours but decimal nonetheless), and included zeros. Nothing eventual about it. Compared to the previous sentence "The primary subject of the Liber Abaci is calculations involving decimal and vulgar fraction notation, which eventually replaced Egyptian fractions" I see no new information but a lot more words making the sentence more difficult to understand. Reverted. However I have no issue with your choice to replace the semicolon here with a period and split the sentence in two; the two parts are sufficiently unrelated that making them separate sentences works well enough.

• "Fibonacci himself used a complex notation for fractions involving a combination of a mixed radix notation with sums of fractions, that dated to 2,000 BCE." I would be very surprised if Fibonacci's sum-of-mixed-radix-fraction notation itself dated to 2000BCE. The sum of fractions part does, of course, but your addition to the sentence is unhelpful. Readers will already have seen the ancient Egyptian part of the article, so repeating the date here is redundant and serves only to confuse.

• "a second Hulstch-Bruins subtraction" in the 8/11 example: It's incomprehensible (what is a second subtraction supposed to mean), unhistorical (Hultsch and Bruins were long after Fibonacci), and mathematically wrong (Fibonacci describes in detail the method he uses to find this expansion and it is very different from Hultsch-Bruins). Not to mention poorly spelled. And removing the description of how he solved each of the two resulting parts doesn't seem to be well motivated. Reverted.

• "Fibonacci describes similar methods for denominators that are two or three less than a number with many factors, using only two subtractions, a method that modern mathematicians like Sylvester recognized as the greedy algorithm and its n-subtractions." Your addition here is grossly inaccurate. This sentence (in its original form without the part about subtractions and greedy) describes the methods Fibonacci numbers as his fifth and sixth, which are purely algebraic identities. The greedy method, which has nothing to do with proximity to highly composite numbers, is Fibonacci's seventh method, and is not described in this article until later. And Sylvester did no such thing; he reinvented the method himself, rather than recognizing anything about what Fibonacci did.

• "If all these other methods fail, which is rare" — I don't feel this flows very smoothly; reworded slightly to "In the rare case that these other methods all fail".

• "Fibonacci suggests a Hultsch-Bruins second subtraction that looks and acts like the modern greedy algorithm". No no no! You give me the impression that you have a copy of Liber Abaci; have you actually tried to read it? Subtracting the largest possible unit fraction is not something that can plausibly be credited to Hultsch and Bruins. And as this sequence of paragraphs of the article clearly stated before you messed with it, Fibonacci goes through examples in which he repeats this greedy subtraction multiple times until a complete expansion is produced. That is not something that "looks and acts like the modern greedy algorithm", it is the greedy algorithm, and since this algorithm was described so clearly in the middle ages it is very unhistorical to call it modern.

• "Fibonacci did this because there are rational numbers that may require another method" is extremely misleading, placed where it is in the middle of a discussion of the greedy algorithm. There are no rational numbers that the greedy algorithm fails to expand. Perhaps you meant some other method than the greedy method, but why are you confusing the discussion of the greedy method by talking about other methods in the middle of it?

• Discussion of Coptic mathematics: might be an interesting addition if placed in an appropriate section of the article, not in the middle of the material about Fibonacci. However, it cannot be added to the article without a citation to external literature clearly describing the Coptic contributions to the problem.

• Clarification that termination of the greedy algorithm was not addressed by Fibonacci but by later mathematicians: agreed and kept.

• Replacement of "greedy algorithm" by "second subtraction proto-algorithm": confusing, wrong, and a violation of WP:NEO.

• "Hultsch-Bruins explanation used in the ancient RMP 2/p expansions" makes no sense. The ancients did not use the H-B explanation for anything, as it didn't exist until H and B published it. What they may have used was the method described in the H-B explanation. In fairness, the original version of this sentence was only a little better. Rewritten.

• "as Fibonaccis solved by factoring 5/121 = 1/11 x (5/11), and solving for 5/11- 1/3 = (3 + 1)/33 = 1/11 + 1/33 or 5/121 = 1/11 x (1/3 + 1/11 + 1/33) = 1/33 + 1/121 + 1/363." In fact the 5/121 example is not in Liber Abaci at all. I don't think the current wording implies that it is. And I don't see how describing a different method used to find the better expansion is relevant to a paragraph describing the faults of the greedy method. Deleted.

—David Eppstein 02:00, 16 December 2006 (UTC)

David,

• Thank you for the extended comments. You are correct, I am not fully aware of certain Wikipedia practices, such as :. I'll look it up,

and try to write more readable mathematics and proofs that almost anyone can read.

• Related to algebraic identities and algorithms, these are not redundant terms in the historical sense, as you yourself have noted with respect to the first subtraction 2/p - 1/A = (2A -p)/Ap. The remainder

(2A -p)/Ap and the general case n/p = (na -p)/np often do not act as an algorithms, hence my name algebraic identity seem more appropriate than your later algorithm name.

C*oncerning algorithms, as a separate idea, as you may be aware Islamic mathematicians developed n-iteration algorithms after 800 AD. Prior to 800 AD scribes used one or two subtractions that may look like iterations, hence not an algorithm in the pure modern sense. Of course, you may argue that 3,000 BCE recursive numeration of Babylonian base 60 and Egyptian base 10 were both essentially algorithmic. However, on the Egyptian side of the discussion the algorithmic aspect was modified in 2000 BC with the formalization of Egyptian fractions, updating the traditional binary Horus-Eye notation in important ways. After 2,000 BC Egyptian fraction math was not algorithmic as the first iteration used in the 2/p conversions clearly denote, even though its finite remainder arithmetic, did at times, act as one or two algorithmic iterations, at later times.

(Your Euclidean algorism discussion has always been seen as a special case, and never as Euclid or any Greek using algorithms as defined by Islamic mathematicians. Again, I have continued to cite Islamics as the inventoris of the algorithms that we have been discussing, hence my citations of iterations is only used as you (and Sylvester) have been trying to associate the modern greedy algorithm with Fibonacci and others. Milo 12/16/06)

• That is, your point of view consistently takes a modern point of view, skipping over the subtle and important innovations of the ancient scribes, be the subject aliquot parts, or its connection to the first and second subtraction methods, all necessary for Islamics to develop

the modern algorithm, is an issue to keeps arising in our 'debate'. (You may yell out modern definitions of algorithms all that you wish. The definition that I am using resides in the ancient texts, no more, or no less).

• Concerning Hultsch and Sylvester, I am offering a compromise to include both scholars that re-discovered these important historical methods. Both methods are necessary to read Egyptian fractions as scribes wrote their texts over 3,200 years, 2000 BCE to 1202 AD. Note that the first and second subtraction methods, as noted in my writings, specificially explains, in the simplest possible number of steps (as required by the history of science Occam's Razor Rule, as I show implied that Ahmes, Greeks, Coptics and Fibonacci used this method in its simplest form).

• Muddling Fibonacci's method and reasons, into the more complicated and less optimal modern algorithm is unfair, and therefore must be justified on your part - well beyond your current descriptions. Fibonacci computed 17/29 as he did, using only two subtractions and not n-iterations as you have the strong modern tendency to cite, over and over again, without explicit justification. All ancient scribes were required to find a second method to convert 5/58 when using the second subtraction method, since it can not be computed by the first subtraction method, named Hultsch-Bruins, or named as you may feel is appropriate. However, as you known, 17/29 was computed by the two step 10/29 + 17/29 method, as you apply cited with respect to 8/11 = 2/11 + 6/11.

• Concerning, Fibonacci's used of zero, you have not cited it once, in mathematical form. Clearly zero had reached Europe by 1202 AD, so Fibonnaci was aware of it, but he did not use it as a place-holder as you seem to be implying by a few muddled words. Stevins has been given credit for writing up zero as a place-holder in 1585 AD in two books, one for science and one for business, as approved by the Paris Academy, a point that Oystein Ore and many others cover, one that I'd be pleased for you to consider when discussing Fibonacci. That is, all of my work connects to very old scholarly works, where several areas of your writings tend to take only a modern point of view, leaving out an important historical fact, as you interconnect your version of history. Again, your writing style tends to throw the ancient algebraic identity (ancient Egyptian fraction) baby out with the modern algorithmic (Egyptian fraction) bath water, as I have been pointing our for some time now.

Concerning the issue of radix, and so forth, I have not wished to send you a copy of my latest Akhmim Wooden Tablet paper, one that may confirm that remainder arithmetic, that you are seeing in Fibonacci and the Liber Abaci dates to 2,000 BCE, was in full use by 2,000 BCE. Against my better judgment, I will be sending a draft paper by email later today that may tend to close a few gaps that now exist between our two points of view. The proposed ancient remainder arithmetic fact, that cites quotients and remainders, as you may be meaning by radix and so forth, has been confirmed over 29 times in the RMP, in a newly published paper that was due to be released in 12/1/06. I can not cite the book's ISBN number and other details until I receive my author's copy of the book. Hana Vymazalova implied aspects of these early proto-decimal facts in 2002. However, her work did not explicitly cite and clarify the general use of scribal remainder arithmetic. Hana's work only showed that an hekat unity, 64/64, was generally divided by n, with 0 < n < 64, such that (64/64)/n = 64/n + (5R/n)*(1/320), facts that you have already seen, and fairly reported on Wikipedia.

• Let me stop at this point, and go over a few of your other comments, later today. Best Regards, Milo Gardner 12/16/06.

I have no idea what you mean when you use the word "algorithm", but it doesn't seem to be what I and everybody else means. An algorithm does not have to involve iteration; it is merely a sequence of steps that if followed consistently will produce a desired calculation. And even iterative algorithms were invented long before you say: generally, Euclid's algorithm for greatest common divisors is called the first algorithm. Your restrictive redefinition of words is I think unencyclopedic and wrong. But I have no time now to respond in depth to this, so more later.

Re the article itself, I see you've made more changes. Expect 90% of them to be reverted, as usual, when I or someone else has time to do so. In particular, I am ok with the first change I see, noting that most of the RMP methods were algebraic in nature, but extremely unhappy with your iterated attempt to twist Ptolemy's complaint into being about decimal rather than being about Egyptian fractions. —David Eppstein 14:39, 16 December 2006 (UTC)

David,

Ancient astronomy and Ptolemy's reasons for choosing base 60 to express his latitudes and longitudes is a subject that actually has little or nothing to do with Egyptian fractions. Your insistance on using Struik, someone I respect with respect to his Minoan work, citing Horus-Eye binary fractions before 1500 BC on Crete, continues to puzzle me. Let's try sticking to Egyptian fraction texts, and read those, and see where the discussion, guided by documented scholars, leads up.

I'll try to answer two of your questions, the first: what does the second subtraction mean? The Egyptian scribal subtraction methd, first solved for 8/11 - 1/2 = 5/22, reports a rational number that mimics 5/58 problem. Note that 5/22 and 5/58 are fractions that are not easily written into Egyptian fractions by solely the first subtraction method. Yet, 5/22 and 5/58 can be solved by a second subtraction method, one that you continue to prematurely associate with the greedy algorithm, when the proper fraction is selected. In the 5/22 case, subtract 1/5 or 5/22 - 1/5 = 3/110 = (2 + 1)/110, or 8/11 = 1/2 + 1/5 + 1/55 + 1/110. In the 5/58 case, subtract 1/12 or 5/58 - 1/12 = 2/696 = 1/348, meaning 17/29 = 1/2 + 1/12 + 1/348.

Of course, Fibonacci was not required to use second subtractions, or Sylvester's version of a proto-greedy algorithm. The 8/11 case was easily created by solving 2/11 + 6/11, and 17/29 was solved by finding the Egyptian fraction series related to 7/27 + 10/27, as I have shown several times. That is, one point of view is that the second subtraction step (or your greedy algorithm) may have been shown for recreational purposes (as I have been mentioning for some time now) more than as a primary computation method, as you have been suggesting.

Your second question: have I read your Liber Abaci reference book?

No, I am planning on doing so, at the CSU-Sacramento or the UC-Davis library. For now, my Heinz Leuneburg Liber Abbaci (German spelling) is a reference in my personal library, a book that I highly recommend. Given that we have often offered dueling references, some of mine being German or French, and many of yours British or 1920's USA, we may gain instruction from the fact that every translation of the RMP or any specific ancient Egyptian text limits its selection of the contents of each document, often in subtle and important ways. Chaces' RMP transation is very different from Robin-Shute's translation. The same subtle oversights, or omissions, may be taking place in our discussion of the Liber Abaci. I'll look for, and read, the original Latin, if available, and or see if our English and German translations are complete. Best Regards, Milo Gardner 12/16/06.

Zero as placeholder, and the distinction between two subtractions and n subtractions

The distinction between a method that involves two subtractions and a method that involves n subtractions is subtle to nonexistent, as far as I am concerned. Fibonacci, in the translation I have, takes six lines to perform the first step for the expansion of 17/29, into 1/2 + 5/58 or as it is written in my translation $\scriptstyle {\frac {5}{58}}\,{\frac {1}{2}}$. He then states "$\scriptstyle {\frac {5}{58}}$ must be made into unit fraction parts, namely by the same distinction" (distinction is the word used for the seven methods he lists, of which the greedy algorithm is the seventh). That is, it is very obvious that he is going through his list of methods and applying the first one that fits. Writing down a list of methods, applying the first one that fits, and (if the method happens to be the seventh) going through the same procedure for the remainder is certainly an algorithm, though it is not quite the same as the greedy algorithm because of the possibility that one of the other methods might be applied in one iteration. He says nothing about applying this method only if the number of subtractions can be limited to two, he merely applies the same sequence of choices to the remainder 5/58. That to me implies possibly unbounded iteration. I have seen modern algorithms expressed less clearly, that are still clearly intended as algorithms.

His other example of 4/13 is possibly more interesting for the history of the greedy algorithm specifically, though, because he expands it two ways. First, he expands it as I have described above: using the first applicable distinction (the seventh greedy one) and then using the first applicable distinction on the remainder. This greedy step produces the unit fraction 1/4 and a remaining fraction which he writes in his mixed radix notation as $\scriptstyle {\frac {3\ 0}{4\ 13}}$ (note the zero as placeholder contradicting your claim that this usage did not occur until much later), meaning the vulgar fraction 3/52. Keeping it in mixed notation he expands it as $\scriptstyle {\frac {1\ 0}{2\ 13}}+\scriptstyle {\frac {1\ 0}{4\ 13}}$ using his second distinction, from which he finds the non-greedy expansion $\scriptstyle {\frac {1}{52}}\,{\frac {1}{26}}\,{\frac {1}{4}}$. So far, applying the same method as the 17/29 example, which is not quite the same as the greedy method because of the possibility of switching to a different distinction at later iterations. But the part that convinces me that Fibonacci had in mind the pure greedy algorithm, unbounded iteration and all, occurs when he, after this, mentions that the expansion of 4/13 can be found in another way, in which one applies the seventh distinction again to the remainder 3/52, and after several lines of calculation he reaches the pure greedy expansion $\scriptstyle {\frac {1}{468}}\,{\frac {1}{18}}\,{\frac {1}{4}}$.

You can cavil all you want that he only ever applies two subtractions. But he is presenting finite examples: of course they do not continue indefinitely. If he had presented expansions with at most five terms, would you then call them "the fourth subtraction pseudo-greedy proto-algorithm algebraic identity method"? Well, given what I've seen of your writing so far, you probably would, but it would be wrong. To me it is clear: he describes a method (his seventh distinction, in the version described for the second 4/13 example and in the 17/29 example) which implicitly requires applying the same method to whatever remainder is left, if it not be a unit. That is all that is needed to described the greedy algorithm as we would call it modernly.

Re your idiosyncratic attempt at using the word "algorithm" in some imagined historic sense: we are writing in modern English, for readers of modern English. Therefore "algorithm" must be taken in its modern meaning.

As for whether "first subtraction" and "second subtraction" are acceptable terms to use in an article for a general audience without some definition of what those terms mean: no. And WP:NEO to me implies that you should avoid such definitions unless absolutely necessary. I see no such necessity here.

Until I can be persuaded that you are willing to be reasoned with, and I find your recent additions to the talk page very unpersuasive in this respect, I am going to eschew such long responses as a waste of my time, and return to a policy of reverting your mistakes without the courtesy of an explanation. And I am going to continue to be frustrated at having to waste my time policing what would remain a perfectly satisfactory article if you could only be persuaded to leave it alone.

—David Eppstein 01:01, 17 December 2006 (UTC)

• David, thank you for the clear use of zero, as a remainder arithmetic place-holder, and not a zero place-holder in our modern base 10 decimal system, where 10^ 0 = 1. Stevins used a broader definition of an arithmetic algorithm, that we both readily recognize, to structure our base 10 decimal system.

• Fibonacci's zero was interesting, writing out medieval proto-decimal information within none algorithmic remainders, a form of arithmetic that would soon (280 years) be fully written into an algorithm. Prior to Stevins, algorithms were not generally used in the ancient arithmetic that we are discussing. (you may continue to disagree by citing the Eucidean algorithm, or the Eudoxus or Archimedes 1/4th geometric series, where Archimedes Lemma stated: 4A/3 = A + A/4 + A/16 + ... A/4n + ...

(not an intended algorithm?) as proven by Dijksterhuis/per the 1906 reading by Heiberg by Egyptian fractions: 4A/3 = A + A/4 + A/12 (also not an intended algorithm?)

• Several of the above facts have been included in my overall position for some time now. Can you begin to re-state your position(s) on Fibonacci's zero being brought into our base 10 decimal system, since we tend to disagree in important ways? Again, beyond our respective pedagogical positions (positions that neither of us have fully written out for the other's full consideration) we are very close to an overall agreement on the details of how ancients split their rational numbers into Egyptian fractions, mathematically.

• Once the facts, of the Liber Abaci or whatever text, are placed before both of us our disagreements tend to disappear. One fact that is before us is: what was the finite arithmetic that was used by Fibonacci? Were central aspects of Fibonacci's right to left integer and unit fractions used 3,200 years earlier? Was Fibonacci's arithmetic the remainder arithmetic that I have been discussing for some time now, dating back to 2,000 BCE. Or, was some other form of arithmetic used, an arithmetic style that we can properly name, or identify in other ways, so that modern readers of Wikipedia can easily grasp the details of the Egyptian fractions subject?

• My positiion: Fibonacci wrote out quotient integers and remainder unit fractions (with both the quotient and remainders being defined as the Egyptian fractions) as Egyptians, Greeks and Coptics had previously taught, from right to left, with few strucutural changes to the remainder arithmetic (the one that you have seen that partitioned the hekat by n = 3, 5, 7, 10 and 11 in 2000 BCE). What type of red flags are needed, for both of us to agree on a few of the subtle connections of Egyptian fraction (remainder) methods covering the 3,200 year period preceeding Fibonacci's Liber Abaci?

• I'll have to take another look at the seven methods cited by Fibonacci, and double check the transations that are available from the original Latin. The Liber Abaci is an important text for two reasons that both of us may not have fully considered.

• (1) It explicitly states finite methods that converted rational numbers to unit fraction series, written in a language (Latin and mathematics) that we should be able to fully read - without an intermediate translator. We tend to disagree concerning certain data presented in the text (i.e. which data was abstract and which data was practical -fuzzy wording on my part - trying to pin down a critical issue that dates to Hellene and Egyptian eras).

• (2) it provides a 'dirty window' context, a potential clear window to the past, maybe 2850 years that almost anyone may soon be able to use to read medieval and ancient Egyptian fraction methods. Are we not gaining important clues from the 1202 AD text that sheds light on Ahmes and the intermediate Hellene and Coptic scribal methods?

• Oddly, to my view, 19th and 20th century translations have included the use of the term algorithm when discussing the Liber Abaci (Sylvster et al), an evaluation that clearly is not pertinent to the RMP and Fibonacci 'first subtractions' (my preferred term, a term that does not sit well with you).

• That is to say, the first RMP 2/p operation, whatever the name is connected to the RMP 2/p - 1/A = (2A -p)/Ap splitting method, a form that you wish to only write and consider as 2/n = 1/A + (2A -n)/An. Clearly this method was implied by Ahmes and used by Fibonacci, both 'scribes' using closely related aliquot parts of A, an important fact in the history of number theory (per O. Ore, and others stated in generalized ways without naming Ahmes). Yet, Ahmes, and Fibonacci only used the H-B method, clearly not an algorithm - as you have fairly concluded several times, for the prime cases. So why advocate the writing of the method only using n, when it was only used when n equalled p, and so forth?

• I could go on, but I'd like to stop for a few hours or days. I'll be considering the status of our 'nearly complete' debate, hoping to complete it is ASAP (by Jan. 2007?). As you have aptly advised, I need to go to the library to read your Liber Abaci reference(s), and do a few other quick studies on our base 10 decimal system, and its earlier Egyptian fraction and base 10 roots. Best Regards, Milo Gardner 12/17/06.

• I have ordered a copy of Sigler's 2002 Liber Abaci, so I may be off-topic for awhile, awaiting its arrival. The book's table of contents stresses subtractions as I have been mentioning, so the seven methods as outlined in one chapter may appear in several other chapters of the book, as we can discuss later. Best Regards, Milo Gardner 12/17/06.

• If you find any discussion of methods of conversion in other chapters I'd be interested to find out where. As I wrote, Egyptian fractions are used often throughout the book, but the only discussion I saw of conversion methods is in chapter 7. Much of the book is rather boring, consisting of very detailed working out of arithmetic and algebra problems, so it takes some effort to find the more theoretical parts. —David Eppstein 17:58, 17 December 2006 (UTC)

The basic three-in one Egyptian fraction numeration system, cited in the Introduction of Sigler's book, shows more than 1/2 2/3 4/5 = 29/30, as we written as 4/5 + 2/15 + 1/30. It shows that when only one was allowed in the numerator, the notation exactly followed the oldest Egyptian fraction system. More importantly, the notation was used for parsing prime number numerators and denominator of vulgar fraction, following a version of the FTA, a fact picked up and used by Egyptians, Greeks and Medievals. Finally, Euclid, Greeks and others expanded the notation to Euclid's Arc, a method of writing fractions that may be beyond this encyclopedia. Fiboancci subtracted everyday Egyptian fractions from Euclid's Arc, a notation that I have not fullu digested - but it looks very interesting, right? Best Regards, Milogardner 00:47, 10 January 2007 (UTC)

Definition

My mathematics dictionary (Dictionary of Mathematics, Borowski and Borwein, Collins Reference, 1989, ISBN 0-00-434347-6) says that "Egyptian fraction" is a synonym for "unit fraction", and does not mention the "sum of unit fractions" definition given here. I believe there may be two definitions in existence, or am I mistaken?

I ask because this is being hotly debated over at Wiktionary. — Paul G 13:02, 4 March 2007 (UTC)

The two subjects are obviously related, but all of the several books and dozens of research papers that I've seen that use the phrase "Egyptian fraction" use it to refer to the sum of unit fractions definition here. Do you have any published references for the other usage other than that dictionary? —David Eppstein 18:15, 4 March 2007 (UTC)

No. I own three mathematics dictionaries/encyclopedias, including the extremely thorough and comprehensive "Encyclopedic Dictionary of Mathematics" (second edition, 1993, ISBN-10 0262590204, ISBN-13 978-0262590204) and two English dictionaries (the full OED, 1989, and Chambers, 1998) but only Borowski and Borwein has an entry for "Egyptian fraction". I would not expect to see one in EDM as that covers higher mathematics, but I would have thought the OED would include it (at least, I haven't been able to find it under either "Egyptian" or "fraction"). B & B does not give a bibliography or references for its content.

I notice that Mathworld agrees with your definition, and in its bibliography for the entry does not include B & B (which Mathworld does cite in some of its other entries). Not having seen any other reference before reading Wikipedia's entry, it initially looked to me that Wikipedia had it wrong. As Wikipedia and Mathworld have a large number of references, it looks very likely that B & B made a mistake. — Paul G 17:53, 14 March 2007 (UTC)

David Eppstein and I have been conducting an informal year long debate related to the definition of the term: Egyptian fraction, a discussion that impacts Wikipedia, Planetmath and other modern definitions of the term.

My position is: given that Middle Kingdom Egyptian scribes invented the term's use, at some point, the 'ancient' texts, and their innovation applications (such as remainder arithmetic) must be factored into any modern definition of the term - as used in ancient mathematical studies, or any modern recreational math project (the context of David Eppstein's Wikipedia posts).

That is, Wikipedia, Planetmath, and other modern definitions of the term follows a common, and trite, modern definition, one that only recognizes the modern existence of Egyptian fractions, and not the origin of 'Egyptian fractions, as an ancient notation that was continuously used for over 3,400 years (ending with the rise of modern base 10 decimals, built upon algorithms).

It should be noted that David Eppstein is a modern algorithmic adherent of Egyptian fractions, first squeezing ancient Egyptian fractions into (his) 10 modern algorithms, while noting the 800 AD origin of algorithms, as used during the final 800 years of Egyptian faction notation's life, but not the origin of Egyptian fractions, itself, as a mathematical notation, that was used 2,800 years before Arab algorithms arrived into our mathematical literature. Hence, the origins of the term Egyptian should not be closely associated with later algorithms (as Eppstein had indirectly implied, over and over again).

I'll not go on, other than to say, an Egyptian fraction, by anyone's modern or ancient definition, allows a concise representation of a rational number, such that its (practical and theoretical) unit fraction components can be used for weights and measures, and other classes of problems (another being arithmetic progressions noted in the Kahun Papyrus, the RMP and the Liber Abaci).

Best Regards, Milo Gardner Milogardner (talk) 11/28/07

Further reading

LeadSongDog added the following references as part of a new "further reading" section:

1. Mahmoud Ezzamel (2002). "Accounting for Private Estates and the Household in the 20th Century BC Middle Kingdom". Abacus. 38: 235–263. doi:10.1111/1467-6281.00107.
2. Milo Gardner (2002). The Egyptian Mathematical Leather Roll, Attested Short Term and Long Term (DOC file). Hindustan Book. {{cite book}}: Unknown parameter |seriestitle= ignored (help)
3. Milo Gardner (2006). An Ancient Egyptian Problem and its Innovative Solution, Ganita Bharati. MD Publications.
4. Richard Gillings (1982). Mathematics in the Time of the Pharaohs. Dover Books. ISBN 048624315X. OCLC 363921.
5. T. Eric Peet (1923). "Arithmetic in the Middle Kingdom". Journal Egyptian Archeology.
6. Tanja Pommerening. "Altagyptische Holmasse Metrologish neu Interpretiert" and relevant pharmaceutical and medical knowledge (in German). Vol. 26. doi:10.1002/bewi.200390001. {{cite book}}: |journal= ignored (help) taken from "Die Altagyptschen Hohlmass, Buske-Verlag, 2005. ISBN 3875484118
7. Leonardo Fibonacci, Laurence E. Sigler (2002). Fibonacci's Liber Abaci: Leonardo Pisano's Book of Calculation. Springer. ISBN 0387407375.
8. Hana Vymazalova (2002). "The Wooden Tablets from Cairo:The Use of the Grain Unit HK3T in Ancient Egypt". Archiv orientální. 70 (1). ISSN 0044-8699.

My impression is that this is heavily weighted towards the history of Egyptian mathematics (not the primary subject of this article) and away from the mathematics of Egyptian fractions specifically. Additionally, several of these references duplicate ones already present in the references section. Could I see some explanation, please, of what specifically these references have to add to the subject of Egyptian fractions (as distinct from Egyptian mathematics more generally) and why they should be listed as further reading rather than cited as references within the article? In the meantime, I have undone the changes. —David Eppstein (talk) 19:36, 7 October 2008 (UTC)

It's an attempt to put an end to the blog add/delete cycle by providing the underlying information to support direct referencing. If you prefer them on the talk page, thats cool with me. I stumbled in to what appeared to be a revert war and tried to find a way out. Do as you like.LeadSongDog (talk) 19:49, 7 October 2008 (UTC)

You mean the edit war from last July? —David Eppstein (talk) 03:17, 8 October 2008 (UTC)

Gads! Don't I look the dolt now... Sorry for the fuss. LeadSongDog (talk) 03:38, 8 October 2008 (UTC)

Sum of exact two fractions

Question: Is it possible to decide, whether a fraction n/p can be represented as a sum of exact two fractions? If so, how could these fractions be computed? There a known special cases, e.g. 2/(2k-1) = 1/k + 1/(k(2k-1)), but what about the general case? —Preceding unsigned comment added by 141.20.50.147 (talk) 11:50, 5 December 2008 (UTC)

From http://www.ics.uci.edu/~eppstein/numth/egypt/force.html : to solve the equation x/y=1/a + 1/b; rewrite it as (ax-y)(bx-y)=y^2, and letting the two factors of y^2 be r and s we can solve a=(r+y)/x, b=(s+y)/x. Simply try all factors r of y^2 for which r<y and see which ones work. It's also possible to use a slower but simpler brute force search in which you try all values of a between y/x and 2y/x. —David Eppstein (talk) 16:08, 5 December 2008 (UTC)

Applications

The article does not mention any applications of Egyptian fractions, but applications do presumably exist. I am not aware of what they are and would be glad to know. If anyone can include mention of applications, that would be appreciated. Thanks. 74.195.16.39 (talk) 16:32, 20 April 2009 (UTC)

What are the applications of arithmetic? Egyptian fractions are primarily a system of arithmetic, used long ago and now replaced by different notation. The modern number-theoretic study of them is largely motivated by concerns of pure mathematics rather than applications. —David Eppstein (talk) 16:42, 20 April 2009 (UTC)

Dyadic rational definition

Are dyadic rationals the set of numbers of the form 1/2^n as this page says or m/2^n as the dyadic rational page says? Dakane2 (talk) 04:36, 10 May 2011 (UTC)

The m/2^n definition is the correct one. But the numbers 1/2^n are examples of dyadic rationals (just not all possible dyadic rationals) and using sums of them any dyadic rational can be formed. —David Eppstein (talk) 04:46, 10 May 2011 (UTC)

Incorrect use of the symbol Omega

(To David Eppstein). Look now. I have been doing and publishing research in analytic number theory for forty years, and one of the main topics I studied is Omega results. I know exactly what the symbol means. It is never used to replace a number. You can write that the number of terms needed = Omega(loglogx), meaning that there is a relation between the number of terms and the function loglogx, or alternately that the function counting the number of terms belongs to a certain set of functions. But you cannot use Omega(loglogx) in order to denote a number of objects. It is just never done in the literature. Also, there is no need to be insulting in your comments. Sapphorain (talk) 07:02, 19 April 2016 (UTC)

How could "the number of terms needed = Omega(loglogx)" be any different than "the number of terms needed is Omega(loglogx)"? Especially given that "=Omega" is in my experience usually pronounced out loud as "is Omega" as It's not actually Intended as an equalIty? But I admIt that my experIence wIth Omega notatIon Is In theoretIcal computer scIence and combInatorIcs, where the conventIons may be dIfferent (certaInly the defInItIon of Omega Is somewhat dIfferent) and Its usual use Is Indeed to lower bound (not denote) a number of objects, In exactly the same way that O-notatIon Is used In the same paragraph to upper bound the same number of objects. ThIs artIcle Is not partIcularly analytIc, but It Is number theory, so I suppose that takes precedence. —David Eppstein (talk) 07:21, 19 April 2016 (UTC)

Alternative modern notation?

I don't know whether this sort of notation is outdated, but I have often come across unit fractions being represented as the number with a dot over it. For example, 1⁄2 is written ${\dot {2}}$, 1⁄10 is written ${\dot {10}}$, and so on. (Though I am not sure whether there is a way to express a 2⁄n fraction using this sort of notation.) I am just pointing this out because I don't see any mention of it in the article yet it seems very common. 98.115.103.26 (talk) 16:44, 19 June 2017 (UTC)

Can you point me to a published source (like a book or journal article) that uses this notation for Egyptian fractions? Because I don't recall seeing it, but that doesn't mean it's not out there somewhere. —David Eppstein (talk) 16:53, 19 June 2017 (UTC)

Sorry, I think I remember seeing it in some translation or transliteration of one of the Egyptian mathematical texts, but I can't seem to find it. Guess it wasn't as common as I thought. Perhaps it was just one author doing it his own way rather than following the standard convention. 98.115.103.26 (talk) 19:34, 19 June 2017 (UTC)

All right, I still haven't been able to find a source, but I did come across a few images saved on my hard drive and was able to locate them online using Google's image search:

The first is a transcription and transliteration of problem 56 from the Rhind Papyrus: https://i0.wp.com/farm9.staticflickr.com/8081/8286659490_1f53a86564_o_d.jpg That was on some guy's blog, but supposedly it's originally from August Eisenlohr. If so, I'm guessing it's outdated? He was from the 19th century...

The second is a facsimile of problem 50 from the Rhind Papyrus, with transcription and transliteration: https://i.warosu.org/data/sci/img/0080/27/1461531495325.jpg This one has been posted and reposted on several blogs. No idea where it's originally from.

I notice on both of these the transliteration (letter-for-letter) goes from right to left. Seems kind of bizarre... I've never seen anyone do it that way before. But then again, I'm not an egyptologist.

But anyway, you can see in both of those that there's a dot above the numeral. So apparently at least some people transliterated it that way at some point. But I guess it's not that common... 98.115.103.26 (talk) 02:36, 23 June 2017 (UTC)

Erdős–Graham Conjecture

Is the Erdős–Graham Conjecture stated correctly? If the sets contain unit fractions, then their reciprocals are natural numbers greater than 1. It would be pretty hard to find a set of those that sum to 1. I think we must either define the sets as containing natural numbers greater than 1, or take out the "reciprocals". Sicherman (talk) 03:53, 30 October 2017 (UTC)

Yes. Changed to integers instead of unit fractions. Thanks for catching this. —David Eppstein (talk) 05:02, 30 October 2017 (UTC)

Glad to help! But you omitted the critical phrase "greater than 1." I have supplied it. Sicherman (talk) 02:07, 31 October 2017 (UTC)

History sections seem substantially inadequate

From what I can tell "Egyptian fractions" were the main representation used throughout the Mediterranean and Greek- and Arabic-writing world throughout the medieval period (coexisting with Mesopotamian sexagesimal arithmetic), only displaced late by "common fraction" techniques using Hindu–Arabic numerals. This article hand-waves thousands of years of history with not even a sentence as "continued to be used in Greek times and into the Middle Ages", with organization that forestalls future expansion about it. The substantial focus on Fibonacci per se, without any mention of e.g. medieval Arabic sources, seems like a "neutral point of view" problem. More generally, the focus on algorithms expressed in modern notation pulls the subject from context and gives an anachronistic impression of its substance. The approach seems appropriate for a modern math book, but not really sufficient for a general encyclopedia. –jacobolus (t) 13:51, 26 October 2023 (UTC)

Example source: Saidan, A. S. (1974), "The Arithmetic of Abū'l-Wafā'", Isis, 65 (3): 367–375, JSTOR 228959. –jacobolus (t) 15:55, 26 October 2023 (UTC)