Talk:Egyptian fraction

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Ancient Egypt

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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)[reply]

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)[reply]

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)[reply]

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

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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

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

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)[reply]

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)[reply]