Talk:E (mathematical constant)/Archive 8

Known digits edit -- first million digits of E
I edited the table "Number of known decimal digits of e" to add the calculation for the first million digits of e; this information had been on this page some time ago (not added by me, BTW), and I noticed today it was gone so added it back.

This was undone with the reason "entries more recent than 1978" are "rather ridiculous." (Why are they "rather ridiculous"?) I added the information again, and noted that "This is a significant increase over the previous calculation and 1,000,000 is a notable number."

Undone again, with the (partial) comment "Already linked in external links. Secondary source needed for mention here." The external link referred to doesn't assign credit to those who computed it nor when this was done. I added the entry back with another reference as a secondary source.

Undone again, with the comment "the added source isn't particularly reliable; it's just a listing in a table on some web page, and it disagreed about the number of digits by a factor of 10 -- also, 1 million isn't a "notable number", it's just a round number."

The added source is a website maintained by two French mathematicians.

The number one million is notable enough to have its own Wikipedia page, and the one million digits of e that were calculated were used in research. Obviously, I think this is a worthwhile entry, so I added the entry again, removing the reference to the French maths site and adding references to two research articles.

Undone again, with the comment "Nothing about this entry makes it notable given the current state of affairs. Use the talk page to make your case if you must."

The current state of affairs is not relevant -- the state of affairs in 1994 is, and the calculation of e to one million digits (by two PhD astrophysicists at NASA) at that time is indeed notable; it's a significant increase over the previous result. Further, these are results that have been used in research (more recently, however, the 2-million digits of e have been used).

I hope the most recent removal of the info I attempted to add (rather, restore) is undone; the information is useful, assigns credit, and the only thing controversial about it is that a number of editors appear to want the table to end with Wozniak. Nice to be reminded (again) of why I hate editing Wikipedia pages and do it so rarely. Ciao. Owlice1 (talk) 05:26, 5 July 2018 (UTC)


 * A few things. First of all, the fact there's an article for 1 million is completely irrelevant.  That has no bearing whatsoever on whether or not this entry should be in the list.  Also, you're neglecting to mention that this entry that you want to restore is just one of a whole mess that got removed (the list has expanded and shrunk at various points).  Presumably we want to draw the line somewhere.  I honestly think this wouldn't be terrible if it were added back in, but these become of increasingly less historical significance as we go on.  Finally, why are you so eager to restore this entry but not any of the others?  –Deacon Vorbis (carbon &bull; videos) 14:45, 5 July 2018 (UTC)


 * The fact that there is an article for 1 million demonstrates the number itself is indeed notable, and I posted that in response to the complaint that the number isn't notable. (If it's not a notable number, delete the page for it.) Yes, you want to draw the line somewhere, and that somewhere is at Wozniak. That's been made very clear. Nevertheless, this one additional entry is useful for the reasons I've already mentioned: it's a significant increase over the previous result, it is a notable number, it is used in research. This result came 16 years after the previous one. Other results of two million, etc., followed closely on the heels of this one, too closely, I would say, to be noted, as with the ever increasing capability in computing power and the speed at which the increases were (and are) being made, that will then always be the case: greater numbers will always be found. I've answered every criticism of the edit. As to why I didn't try to restore any others, well, we've seen how well it worked when I tried to restore just one with so much going for it! Why on earth would I bother with any others, here or anywhere else? I'm done. Owlice1 (talk) 16:06, 5 July 2018 (UTC)


 * It's been asserted that the 1994 calculation is notable. Notability is established by secondary sources.  If this calculation is indeed noted in reliable sources, it can be restored.   Sławomir Biały  (talk) 16:44, 5 July 2018 (UTC)


 * I provided several sources, as one can see from looking at my edits. If you find none of them reliable, then please let me know what is a reliable source, and let me know, too, if you would, why this addition needs more reliable secondary sources than others listed in the table, each of which has one source, at least one of which, which is a link to a website (deemed unacceptable for my addition), doesn't work. Thanks. Owlice1 (talk) 17:21, 5 July 2018 (UTC)


 * While I do not feel particularly strongly on this and I am not disagreeing with Sławomir, I do believe that Purgie is making a valid point, even if a bit too flippantly. Entries in a table like this need to be more than just notable, they need to be interesting, specifically historically interesting. The table's function is to illustrate the growth of the number of known digits. It can not be complete, nor would we want it to be. It has to stop growing at some point and I think that it should stop when the next entry is no longer interesting. For years I did a corresponding lecture on the digits of $\pi$. As each new record was set I was forced to remove some items, even though they were notable at the time I started to talk about them, since they had stopped being interesting (what I could say about them I could easily have said the same about some newer entries). At this point, with over 500 trillion (I didn't really count the zeroes in Ye's table, but the number is up there) digits known, the fleeting record of the millionth digit calculation has lost all interest, at least for me. --Bill Cherowitzo (talk) 17:32, 5 July 2018 (UTC)


 * Yes, it has to stop growing at some point -- I do not disagree with that. I would ask, then, that those who have undone my edit defend stopping the table at 116,000 rather than 1,000,000, a number not achieved until 16 years after Wozniak's. It was after 1 million digits was reached that new significant records (2 million, 5 million, and so on) were set only months, or maybe even weeks, apart, not years. Owlice1 (talk) 17:52, 5 July 2018 (UTC)

The latest revision, with the added sources, looks reasonable to me. I say we let the addition stand. Sławomir Biały (talk) 18:06, 5 July 2018 (UTC)


 * I disagree with Sławomir Biały's and Owlice1's opinion that the entry under reversion should be included in the addressed table. I try to give answers to questions raised by Owlice1 and to explicate my reasons for objecting and also for my suggestion of an expanded table.
 * I admit calling new records in rote computing "rather ridiculous" is quite harsh. I can only mention the "rather" as mitigating: sorry. I do believe, however, that the achieved numbers have no profound importance.
 * I do not doubt the factuality of the intended entry and it being reliably sourced, but the notability of $$10^6$$ in math topics is to me just as big as any decade, perhaps slightly bigger as a multiple of $$10^3$$, favoured in technical contexts. However, I do not see any notability wrt e itself, and not even wrt a number of digits in its representation in positional number systems. There being a WP article on the number $$10^6$$ is largely irrelevant in any other article. Therefore, there is no reason, stemming from $$10^6$$ digits themselves, to appear in the article about e.
 * The v. Neumann entry is relevant for the reason I tried to give: first automatically computed value. I think, v. Neumann is irrelevant in this context, the ENIAC is the relevant information, one of the first floor-sized computers, unavailable to the public. The 1961-entry might be skipped, its importance being perhaps only the increase of available digits in orders of magnitude. The 1978-entry is not important for S. Wozniak, but for the fact that then a publicly available device empowered the almost average Joe to calculate digits of almost any desired math constant to a degree, for which I have no tolerated verbiage. "Trillions" is not to my liking, because of Moore's Law and other reasons I consider obvious to most in good faith. The number of calculated digits is limited just by boredom.
 * I do not deny the existence of research values in the ongoing calculations, but their nexus to e are, at least to my knowledge, confined to the application of specific algorithms, possibly exchangeable to those for other constants, which are often considered barely as useful test samples. I conjecture that even a newly discovered quantum algorithm for calculating digits of e would not justify a new entry, but only an article in WP on its own.
 * I still believe that adding to the table a reason-for-notability column, containing good reasons, enhances the article. Since I was aware that my specific knowledge and active fluency in English would not be sufficient to supply really good entries there, I just did a sketchy draft, and explicitly asked for kind improvement in the edit summary.
 * I did not expect the qualification "flippant" (I am fully respectful!), and other reactions I consider not de rigeur. Purgy (talk) 10:15, 6 July 2018 (UTC)


 * The achieved result of 1,000,000 digits of e is notable. I've already pointed out a number of reasons why. I think you do not grasp this one, however: these digits were (are) available for download; this is actually useful. (Where are the 116,000 digits of e from the previous result? Published in BYTE. How useful were they? What could anyone do with them?) Generating the million digits and then publishing the result online, where the digits can be downloaded, makes them available for research. Here are three research articles that use this particular achieved result:
 * Ginsburg, N. and Lesner, C. (1999) "Some Conjectures about Random Numbers"
 * Shimojo, M., et al (2007) “A Note on Searching Digits of Circular Ratio and Napier's Number for Numerically Expressed Information on Ruminant Agriculture"
 * Lai, Dejian & Danca, Marius-F. (2008) "Fractal and statistical analysis on digits of irrational numbers"


 * Notice the last two articles mentioned were written more than a decade after these results were made available online for others to use, indicating that particular digit-set had some endurance for research (and still may, though I haven't looked for more papers using it; I have run across other papers using even the larger sets generated by Nemiroff & Bonnell, such as 2 million and 5 million digits of e, too).


 * Such research might not be something you'd want to do, but others clearly do. Calculating and then making these digits available online for anyone to use had not been done before. (Editing to add: at the million digit level.) Owlice1 (talk) 11:31, 6 July 2018 (UTC)


 * Yes, I do not grasp how often I have to ruminate that facts, typical of any irrational number, are very well notable at appropriate places, but are not notable within an article about e, which just happens to be irrational. Purgy (talk) 07:47, 7 July 2018 (UTC)

For some reason, the following comes to mind:
 * {| class="wikitable"

! Who || Where || What
 * Colonel Mustard ||align=right| Library || Wrench
 * }
 * }

I've restored Nemiroff & Bonnell to the table, with what I hope are enough references to satisfy all. Thank you for your patience. Owlice1 (talk) 11:47, 7 July 2018 (UTC)
 * It’s just not an interesting or remarkable result. As by 1978 it was already possible to generate over 100,000 digits on a 8-bit CPU, someone could have generated a million digits a few years later, and probably did long before 1994. There would be many such firsts that were not published as they are simply not interesting, no-one has noticed them. It really is not that interesting now anyone can download and run a program to generate digits.-- JohnBlackburne wordsdeeds 12:26, 7 July 2018 (UTC)
 * "not reliably sourced." I provided primary and secondary sources. Which of these did you find not reliable? It is certainly not true that others did not notice these results. They were used in research, the Gutenberg project published them, and the results are even available through Amazon! (The reviews are rather amusing.) The research articles using them are not about algorithms for generating e, but about how to use the generated digits. Can they, for example, be used as a random number generator, or in cryptology? That is what some of this research is, and it's clear that others must not have wanted to generate these digits themselves, as they used (and cite) these results. Speaking of "not reliably sourced," I note this from your post: someone could have generated a million digits a few years later, and probably did long before 1994. (Emphasis mine.) That's not sourced at all. I'm restoring the entry. Owlice1 (talk) 17:32, 7 July 2018 (UTC)
 * The computation of one million digit has not been the object of a regularly published paper. This shows that, even at the time of this computation, this was not considered by the mathematic community as a significant result. Nevertheless, the list of these digits is useful and has been used for other research. This could be mentioned elsewhere in the article, but does not belongs to this section. This list of digits is also listed in the "See also" section, which is its natural place. I have reverted your edit for these reasons.
 * On the other hand, when you disagree with other editors, edit-warring is the worst way for dispute resolution, as you may be blocked for editing because of the WP:3RR rule. D.Lazard (talk) 18:01, 7 July 2018 (UTC)
 * I rarely edit Wikipedia pages; it wasn't until this discussion that I learned the phrase "edit-warring." I brought the discussion to the talk page when it was suggested I do so, and I've answered every criticism of the edit. There was some agreement/acquiescence that the addition could stand, which is why I put it back.
 * BTW, I missed the "regularly published paper" for the Wozniak result that shows it is considered significant by the maths community. Where is that, please? What I see given as a source for that is a BYTE magazine article (which I probably have in the stash of old mags in the basement; I may have to go look). At least one other entry in the table has a link to a website as the source, rather than to a journal article. I have asked before why my addition, which has multiple reliable refereed secondary sources that indicate the value of this result/addition, is unacceptable while others with only one source not as robust still stand; I never got an answer to that. The goal of some of the editors appears to be to end the table at Wozniak, no matter what. Owlice1 (talk) 18:29, 7 July 2018 (UTC)
 * May I point you to the fact that you keep refusing to recognize the statements about the disconnectedness to this article of the sources you mention, backing the non-notability wrt this article of the sourced fact you want included. The research work of these papers is in no way specifically connected to e, but to an assumed structure of randomness in its digits. Any other irrational number would do the same trick. You also seem to ignore in your comparison the given reason for the Apple II entry. I repeat: It is not primarily about Wozniak, but about the public availability of equipment, capable of calculating more or less arbitrarily many digits of any computable number, rendering any new "records" as of no relevance. Please, do not strive to make WP a Guinness Book of Records. Purgy (talk) 19:22, 7 July 2018 (UTC)
 * I'm not disputing the Wozniak entry, at all, although the different standards for sources for its inclusion wrt my addition are noted. You cannot assume randomness in the digits of e; indeed, that's one point of research, one that requires a million or more digits to accomplish. The Nemiroff/Bonnell results are notable for a number of reasons, including enabling that and other research (in computer science, math, and even apparently in ruminant agriculture) where other results at that time did not. I'm not "striv[ing] to make WP a Guinness Book of Records." I'm trying to add a useful/notable addition to that table. That's it.
 * Two additional papers for my own reference (and possibly future others'):
 * BiEntropy - The Approximate Entropy of a Finite Binary String, http://adsabs.harvard.edu/abs/2013arXiv1305.0954C)
 * A New Method for Symbolic Sequences Analysis. An Application to Long Sequences, http://cmst.eu/articles/a-new-method-for-symbolic-sequences-analysis-an-application-to-long-sequences/
 * Owlice1 (talk) 20:01, 7 July 2018 (UTC)
 * I (Rick Nungester, age 63, retired electrical/software engineer, Spokane WA USA) approximated e to 1 million places 12 February 1992. See my attempts starting 21 June 2015 in both the article and Talk pages to get that information posted here. It has all been removed. See my published algorithm and results. I argued with Wikipedia editors regarding both the fact that I did this, and that it was significant. The current page says "Since that time [1978], the proliferation of modern high-speed desktop computers has made it possible for amateurs, with the right hardware, to compute trillions of digits of e." The arrogance of that remark is offensive. Amateurs? I have masters degrees in computer science and electrical engineering (from Stanford). What are the credentials of the person that wrote that? And that statement ignores the significance of advancing the algorithms used to do the approximation. They have changed dramatically since 1978 and contribute much more to the number of approximation digits than advancements in computer hardware have. The usenet post I linked to details 4 improvements to Wozniak's algorithm. See this similar subject in the article for pi. In that article it is a major section with sub-parts. Yet here it is minimized to "the proliferation of modern high-speed computers". Allowing the expansion of the "number of digits" table for e and the algorithms that led to the improvements should be encouraged not squashed. Rick314 (talk) 20:16, 1 April 2019 (UTC)
 * Please read WP:What Wikipedia is not. Only result regularly published may be mentioned in Wikipedia. Moreover these results must be notable, which means that one may cite only results that are mentioned in some reliable secondary sources. In other words, mentioning your computation would be against all Wikipedia policies. By the way, don't be offended to be considered as an amateur, as you clearly state that you are not a professional mathematician nor a professional computer scientist. D.Lazard (talk) 20:53, 1 April 2019 (UTC)
 * Daniel: The dictionary I am using says "professional" is "engaged in a specified activity as one's main paid occupation rather than as a pastime." In addition to a 30-year software engineering career, I currently teach engineering and computer science at local universities. Since you retired in 2008, you are apparently no longer a professional mathematician or computer scientist, whereas I am.
 * You didn't address the obvious differences in the Wikipedia pi article "Modern quest for more digits" section (number of digits history to year 2000, history of algorithms, 5 sub-sections) and your censure of similar content here for e. Why aren't you either censuring the pi article or allowing similar content here? Rick314 (talk) 01:03, 2 April 2019 (UTC)
 * On a side note, I think you need to look up the difference between censure and censor, but that's just for your own benefit. What are you hoping to accomplish exactly?  If you want to have your claim added that you computed a million digits of $e$ based upon a   post with no corroborating evidence, then you're wasting your time because claims like that need to be backed up by reliable, independent sources, and a post on a newsgroup from yourself doesn't qualify.  Not only that, but there's apparently been a lot of sentiment that these computations become less interesting, so aren't really worth including in the article, no matter how well-verified such a claim is.  I strongly suspect that you're not going to get whatever you're wanting here.  –Deacon Vorbis (carbon &bull; videos) 01:48, 2 April 2019 (UTC)
 * I meant censure ("express severe disapproval of (someone or something), especially in a formal statement") regarding the many disapproving responses in the discussion above. I think you should be more careful giving advise regarding grammar, but that's just for your own benefit. Regarding what I want, no more regarding my 1992 work but addressing this whole "Known digits edit -- first million digits of E" topic and the number of digits results after 1978 being "rather ridiculous", not notable, etc. I am trying to site a Wikipedia precedent in a similar article that appears to disagree with what is being done to this article. To repeat: "You didn't address the obvious differences in the Wikipedia pi article "Modern quest for more digits" section (number of digits history to year 2000, history of algorithms, 5 sub-sections) and your censure [disapproval] of similar content here for e. Why aren't you either censuring [disapproving the content of] the pi article or allowing similar content here?" You say "there's apparently been a lot of sentiment that these computations become less interesting...". That is assuming your conclusion, then using it as justification. Read the section I refer to in the pi article -- There is and has been lots of interest in extended precision approximations throughout history. Rick314 (talk) 23:50, 2 April 2019 (UTC)

I would like to add my voice to the consensus here not to extend the table unless there is a clear reason to think an addition is interesting or significant (a la Bill Cherowitzo's comments above). --JBL (talk) 00:29, 3 April 2019 (UTC)
 * Same question to you then -- Why the different standards for pi and e? To add to my argument see the article reference to "e: the Story of a Number" by Eli Maor (227 pages and I have a copy). This and the other sources cited in the references show it is "interesting and significant", just as it is for pi. Rick314 (talk) 01:02, 3 April 2019 (UTC)
 * Since I have not stated a position about π, I see no basis for your supposition that I feel differently about it.
 * Also, in my comment, the referent of "interesting or significant" is "an addition [to the table]", i.e., it is my position that each entry on the table should be independently justifiable as interesting or significant. Your comment about a source that does not mention the event you are trying to add to the table doesn't seem responsive. --JBL (talk) 01:06, 3 April 2019 (UTC)

Thank you to those that replied to me and I am sorry for switching topics in an unclear way among publishing my own work in the table, publishing Nemiroff & Bonnell in the table, and the general topic of additions being interesting or significant or notable. On the last subject I do want to re-state the differences between this article and the existing precedent of the pi article "Modern quest for more digits" section. It has a graph of number of digits to year 2000 and not every point on the graph is independently referenced or justified as interesting and notable and significant. More importantly, it explains the algorithmic advancements that led to the improvements instead of attributing progress to "the proliferation of modern high-speed desktop computers". I understand such algorithm advancements must be properly documented to match Wikipedia standards, unlike my own. On a related topic, I recently came across two articles that you more experienced editors might want to add to this article. They are "Unexpected Occurrences of the Number e" (1989, Mathematics Magazine) and "New Closed-Form Approximations to the Logarithmic Constant e" (1998, The Mathematical Intelligencer). I enjoyed reading both of them. Rick314 (talk) 01:45, 10 April 2019 (UTC)

two comments
The phrase

Because this series keeps >>> many <<< important properties for e^x even when x is complex, it is commonly used to extend the definition of e^x to the complex numbers.

is intriguing. I would like to read a property that is not kept.

According to A. Piccato, Dizionario dei Termini Matematici, Rizzoli 1987 [I realize that an englishman is unhappy of this hard-to-find italian reference], Napier published a table of log-sines in the basis 2.71828_285 (the digits after _ are different from the e's one)

pietro151.29.188.231 (talk) 08:10, 28 June 2019 (UTC)
 * In any case, this "because" clause is definitively wrong. I have fixed it.
 * About Napier approximation, it would be useful to know whether Napier intended to use this base or if this results from the computational approximations. D.Lazard (talk) 08:28, 28 June 2019 (UTC)


 * unfortunately the source Piccato is only a small dictionary for high-school students. Also I felt useful to know this and wondered if someone could be precise on it. The exact text of the source is

Napier ... da' ... i logaritmi dei seni degli angoli, i quali sono riferiti non alla base e ma alla base 2.71828_285

Napier computed the logaritms in base 2.71828_285 of the sines of the angles ...

pietro151.29.188.231 (talk) 17:40, 29 June 2019 (UTC)
 * For knowing what Napier computed, see History of logarithms and Napierian logarithm, where the exact relation between Napieran logarithm and natural logarithm are given. At the time of Napier, the constant $e$ was not yet defined nor the base of logarithms. So talking of base for Napier logarithms is an anachronism. D.Lazard (talk) 18:30, 29 June 2019 (UTC)

(Not so) hidden large prime
Did you know that if you cut off the first 2 digits of the decimal representation of e (2.7...), the next 69 digits, beginning with 1828... and ending with ...0429 are a prime number? https://twitter.com/fermatslibrary/status/1156540146159951874/photo/1 DaKine (talk) 16:04, 31 July 2019 (UTC)

Begging the question.
The first line of the lede is: "The number e is a mathematical constant that is the base of the natural logarithm: the unique number whose natural logarithm is equal to one."

I really hate this definition because it begs the question. The common definition of the natural logarithm is simply "the logarithm with base e", so this is kinda like saying "e is the unique number which, when taken to the first power, yields e". It sounds informative, but lacks context to be, and provides no illumination on why e is an important constant. Could we have something like "e is the unique number such that the derivative of e^x is e^x", or "the function e^x is the unique function of x equal at every point to its slope", maybe? TricksterWolf (talk) 16:50, 23 November 2019 (UTC)


 * I doubt we're really going to be able to explain why $e$ is important in the very first sentence. There are a number of equivalent definitions to start with, and I'm not sure why what you propose would be any better.  You say that a common definition of the natural logarithm is the base-$e$ logarithm.  This is true that it's a common definition, but it's not the only common definition.  Also common is $$\textstyle \log x = \int_1^x dt/t,$$ which makes no reference to $e$.  If you're worried about circularity, there are still problems anyway.  If you want to say something like "$e$ is the unique value of $a$ so that $$a^x$$ is equal to its own derivative, then you have to worry about how you even define the function $$a^x.$$ It's usually  to be $$e^{x\log a}.$$  There are ways around this, but they're awkward.  –Deacon Vorbis (carbon &bull; videos) 17:28, 23 November 2019 (UTC)


 * I thought a^x came as an extension of a^n. Saying "there are still problems anyway" implies that no one knows where e came from. --ilgiz (talk) 04:43, 19 January 2020 (UTC)


 * I agree that the text has no clear definition or explanation of e, but so far no-one has done anything to improve it. As you imply, the statement "..the unique number whose natural logarithm equals one" is a stupid circular definition since the logarithm of any number to its own base is 1. So I'll have a go. Rjdeadly (talk) 09:30, 25 July 2020 (UTC)
 * (1) This is definitely not circular. (2) It might be advisable to review the multiple discussions of the lead that took place two years ago, starting with Talk:E_(mathematical_constant)/Archive_7. —JBL (talk) 12:26, 25 July 2020 (UTC)

July 2020 changes to lead
I have destroyed 's version of the lead as the consequence of an edit conflict. Their version is not convenient, at least because of the too vague first sentence, and their lead is written for a specific public, which should not be the case of a Wikipedia article.

The are two problems is the previous lead. Firstly, the phrase "the unique number whose natural logarithm is equal to one" can be understood as a part of the definition of $e$, while is simply a parenthetical explanation of "base" therefore I have replaced it by a link to base of a logarithm.

The second issue is more subtle, and is related to the use of the word "definition": a constant has not to be defined, it is characterized by some property. It is the name of the number with this property that is subject of a definition. With my formulation, it becomes clear that readers can choose the characterization/definition which fits best with their background. Even if the first characterization may be viewed as circular (this depend on the chosen definition of the natural logarithm), those that follow are undoubtly not circular. I have kept as the first one the relationship with the natural logarithm, because it seems that for most students this is their first meet with $e$. D.Lazard (talk) 11:25, 25 July 2020 (UTC)


 * To start with:
 * "...at least because of the too vague first sentence"??? That has not changed.
 * "....their lead is written for a specific public" ??? not at all. see MOS:INTRO. The noteworthiness of e needs to be pointed out first of all in an accessible way.
 * "It is the base of the natural logarithm" does not belong as a key sentence in the introduction to explain e as that explains nothing. That is simply the definition of the natural logarithm.
 * a whole series of "characterisations" does not belong here when it is covered later; the summary is supposed to be general and brief.
 * "I have kept as the first one the relationship with the natural logarithm, because it seems that for most students this is their first meet with $e$". Doubtful that's true and not a good reason to keep it there.
 * "...related to the use of the word "definition": a constant has not to be defined, it is characterized by some property" So you say that Mathematical constant is completely wrong where it states "A mathematical constant is a number whose value is fixed by an unambiguous definition..." Are you going to change this wiki page?
 * Rjdeadly (talk) 13:55, 25 July 2020 (UTC)
 * That is simply the definition of the natural logarithm This is at best naive and at worst definitely wrong, as is explained in Deacon Vorbis's comment and in some of the discussions I linked above. --JBL (talk) 14:15, 25 July 2020 (UTC)
 * A few comments: the "unique number whose natural logarithm is equal to 1" bit is probably good to keep because it simply reinforces what the base of a logarithm is; a little redundancy is helpful here. Stylistically, the changes are rough – they leave the intro with several short, choppy (often one-sentence) paragraphs.  Neither this nor the "as described on this page" self-reference are appropriate.  Mathematically, simply stating that $e$ is defined to be the base of the natural logarithm  is misleading at best; it can be defined in multiple ways, as the current lead attempts to convey, and I think it's important for it to do so.  I'm not overly enthusiastic about some of the structure of this article, and there's probably some room for tweaking the lead, but what was there couldn't stand. –Deacon Vorbis (carbon &bull; videos) 15:47, 25 July 2020 (UTC)

The previous lead completely misses the point. e is not just another number like any other, but has a significance not just in maths but physically in the real world, just as Pi is a fundamental ratio inherent in all circles and all matters related. e is not an obscure, seemingly random number that can be "applied" to or derived from mathematical problems but represents something from the real world e.g. population, radioactive decay, interest calculations and the idea that all continually growing systems are scaled versions of a common rate. This is badly lacking in the previous introduction.Rjdeadly (talk) 15:55, 25 July 2020 (UTC)

Digits redux
(Disclaimer: I hold five world records of mathematical constants now.) I would like to respark this debate on whether to add digit records later than Steve Wosniak. I admit that since the Taylor Series has been the most efficient algorithm for hundreds of years, the pure mathematical relevance has diminished. However, world records of e currently being broken are still relevant in experimental mathematics scopes and computational improvements (such as binary splitting merged with Advanced Vector Extension multiprocessing) that facilitate astronomical digit computations are still of value to Wikipedia. Since the plight to the known digits of Pi is not necessarily purely because to expand mathematical significance, other mathematical constants that have the known digits after 1978 have relevance in public perceptions and understanding of mathematics. Not to forget that other major mathematics constants such as Apery, Catalan, Ln(2), and Euler-Mascheroni keep track of all recent computations. I hope that the mathematician moderators here understand that this isn't just a pure mathematics thing, and it is a debate on whether adding a few lines on this table helps public awareness on mathematics vs making the document longer unnecessarily. I don't think this is a thing that makes this document longer for no reason. 39.119.42.102 (talk) 08:30, 10 August 2020 (UTC)
 * Can you clarify two things? (1) Are you the same person as Owlice1 or Rick314?  (2) Are you proposing to do something different from what they were proposing, and if so, then what? --JBL (talk) 11:09, 10 August 2020 (UTC)
 * Answer: No, I am my own person and this is my first time writing. I would like to add in the Known digits section more recent known digit values and also add in some information about the development of multiprocessing capabilities assisted by the binary splitting of Taylor series, since the Euler-Mascheroni Constant, Catalan's Constant, Apery's Constant, etc all have recent known digit records on its page.39.119.42.102 (talk) 16:05, 10 August 2020 (UTC)
 * I've moved this to a new section even though it's related to the old discussion to help keep things organized. I've also indented your replies; see WP:THREAD for more info. I'm still pretty tepid on the digit records.  On the other hand, if there are sources that discuss new algorithmic improvements to digit computation, that could be worth a short mention at least.  Can you be any more specific? –Deacon Vorbis (carbon &bull; videos) 16:25, 10 August 2020 (UTC)
 * 1. [] 2. [] 3. []. Important points are that the first-ever multithreaded program to compute both the world record of Pi and e was developed in the late 2000s by [Alexander J. Yee] and binary splitting coupled with FFT has been a very crucial component. e is the simplest form of the Taylor Expansion, so binary splitting for e was one of the first computationally implemented as a proof of concept for the usage of binary splitting for multiprocessing. The rapid progression of digits would have been impossible with multiprocessing and this has influenced binary splitting algorithms of more complex mathematical constants. This is why I think recent digits are still meaningful. I also repeat that Apery, Catalan, Ln(2), and Euler-Mascheroni documents keep the recent history starting from the 1700s to just a month ago and multiprocessing of such constants would have never been possible without the implementation of binary splitting for e. Last but not least, it's still worthy to keep track of digit history for the general audience not only as a mathematics encyclopedia, since this is Wikipedia. 39.119.42.102 (talk) 15:11, 11 August 2020 (UTC)

First image in the article
Shaded area cannot be 'equal to 1'. --5.43.102.127 (talk) 18:16, 1 September 2020 (UTC)
 * Why not? It is true that it equals 1. D.Lazard (talk) 19:00, 1 September 2020 (UTC)

Complex Number - Possible Edit War
Both Sine(x) and Cosine(x) can be expressed in terms of $$e^{ix}$$ and $$e^{-ix}$$ The expression for Sine is divided by 2i whereas the expression for cosine is divided by 2. (Not 2i)

There is a disagreement between editors. Twice an attempt has been made to write $$\cos x = \frac{e^{ix} + e^{-ix}}{2i}$$ (sic) and twice this has been reverted to the correct expression $$\cos x = \frac{e^{ix} + e^{-ix}}{2}$$.

Please if you disagree then discuss it here. Failing that I shall revert any attempt to change these expression (unless someone else gets there first). OrewaTel (talk) 02:21, 12 October 2020 (UTC)

(and hence one may define e as f(1)
maybe this has been flogged to death already, but that statement confuses me beyond belief. Does it really have any meaning? As far as I can see f(1) means e to the power 1. and defining any number as itself to the power one seems bizarre. I would just delete it, but I don't want to interfere in contentious waters. Maybe you could explain it to me here. JohnjPerth (talk) 15:44, 9 October 2020 (UTC)JohnjPerth


 * It seems fine: define $e$ to be the function which is equal to its own derivative such that $f(x)$, and then define $e$ to be $f(0) = 1$. Proving that such a function exists and is unique can be done, as can showing that it's equivalent to other definitions of the exponential function. –Deacon Vorbis (carbon &bull; videos) 16:02, 9 October 2020 (UTC)
 * In fact, pretty much any definition of $e$ can be stated as something like: Define the exponential function to be ____, and then define $e$ to be the value of that function evaluated at $f(1)$. –Deacon Vorbis (carbon &bull; videos) 16:18, 9 October 2020 (UTC)

Thank you Deacon. To say that 2 is defined as f(1) where f(x)= 2 exp x would be silly, and those were the terms that I was thinking in. I was not realizing that 'we know what 2 is', but we don't know what 'e' is in the same way. 'e' is only defined by things like the properties of f(x)=e exp x and therefore the function defines 'e' rather than 'e' defines the function. Your answer did not seem to explicitly say that, but it brought me to that understanding. Thank you. JohnjPerth (talk) 08:35, 31 October 2020 (UTC)JohnjPerth

Taylor Series
The introduction states that :$$e = \sum\limits_{n = 0}^{\infty} \frac{1}{n!}$$ whilst this is universally accepted, two editors differ as to how the expansion should be written. The variants are:
 * :$$e = \sum\limits_{n = 0}^{\infty} \frac{1}{n!} = 1 + \frac{1}{1} + \frac{1}{1\cdot 2} + \frac{1}{1\cdot 2\cdot 3} + \cdots$$
 * :$$e = \sum\limits_{n = 0}^{\infty} \frac{1}{n!} = \frac{1}{1} + \frac{1}{1} + \frac{1}{1\cdot 2} + \frac{1}{1\cdot 2\cdot 3} + \cdots$$

It does no good for the editors to continually revert each others work. Let's sort it out here. My preference is to be explicit.
 * $$e = \sum\limits_{n = 0}^{\infty} \frac{1}{n!} = \frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \cdots = \frac{1}{1} + \frac{1}{1} + \frac{1}{1\cdot 2} + \frac{1}{1\cdot 2\cdot 3} + \cdots$$

Your thoughts please. OrewaTel (talk) 21:10, 30 November 2020 (UTC)
 * IMO, the version that only uses base operations and expands out the ! function should be shown. Not simplified, but would be more accessible to the reader.  Eve rgr een Fir  (talk) 21:12, 30 November 2020 (UTC)


 * There are at least four editors involved, not two. The objection of Pedro Fonini to the formula $$\frac{1}{1} + \frac{1}{1} + \frac{1}{1\cdot 2} + \frac{1}{1\cdot 2\cdot 3} + \cdots$$ is compelling, and has been endorsed by me and by D.Lazard.  No one has articulated a similar-quality rebuttal.  (To re-hash: in the sequence 3! = 1 * 2 * 3, 2! = 1 * 2, 1! = 1, the term 0! is an empty product; writing it as 0! = 1 is true but obscures the pattern.  Removing the fraction is more in keeping with the fact that it is an empty product and also marks the term as exceptional, making the pattern of the other terms clearer.)  --JBL (talk) 21:41, 30 November 2020 (UTC)
 * My take on it is that when (for educational reasons, to show the pattern) we write $$\frac{1}{1!}$$ as $$\frac{1}{1}$$, then (for even more compelling educational reasons, to show the definition of 0!) we should also write $$\frac{1}{0!}$$ as $$\frac{1}{1}$$. - DVdm (talk) 22:41, 30 November 2020 (UTC)
 * Yes, $$\frac{1}{0!}$$ should be written as $$\frac{1}{1}$$. The possible simplification of $$\frac{1}{1}$$ to $$1$$ is unrelated to the fact that 0! is an empty product, and I doubt very much that the reader would understand that the simplification is done because 0! is an empty product while the simplification is not done for $$\frac{1}{1!}$$ because 1! is not the empty product; I have never seen such a rule. And the term $$1/0!$$ is not exceptional: it is the generic term for $$n = 0$$. If just $$\frac{1}{0!}$$ is simplified to 1, this gives the impression that some new term (not the generic term) has been added, thus this is confusing. — Vincent Lefèvre (talk) 23:22, 30 November 2020 (UTC)
 * In the sequence ..., 1, 1*2, 1*2*3, ..., the previous term is, not 1.  There is no pattern to 1, 1, 1*2, 1*2*3.   Likewise there is no pattern in 1/1, 1/1, 1/(1*2), 1/(1*2*3).  Replacing the first 1/1 with 1 correctly indicates to the reader (at least, any reader who notices -- I can't imagine this matters much one way or the other) the violation of the pattern being illustrated. --JBL (talk) 23:42, 30 November 2020 (UTC)
 * The empty product is represented by the neutral element 1, never by a blank. — Vincent Lefèvre (talk) 23:50, 30 November 2020 (UTC)
 * What do WP:SECONDARY say? We can have our preferences about patterns and 0!, but we must follow what sources say.  Eve rgr een Fir  (talk) 23:54, 30 November 2020 (UTC)
 * EvergreenFir, this is an interesting question; I would guess that few secondary sources choose the specific approach here to define the factorial merely by illustration -- but that's because secondary sources will be different kinds of documents (notably, textbooks). I don't think they bind us for stylistic questions like this. (Everyone here agrees about what the content of the equation is.) --JBL (talk) 00:07, 1 December 2020 (UTC)
 * Perhaps I should have said, "(for educational reasons, to show the pattern that starts with the second term)". - DVdm (talk) 23:57, 30 November 2020 (UTC)
 * Vincent Lefèvre, I was extending my comment when yours came in, I hope you do not object -- the new part follows the edit conflict tag.  Your comment is completely non-responsive to the point -- empty products are denoted by 1 in isolation because that's what they evaluate to, but here we are explicitly illustrating the unevaluated products; that creates a conflict.  Not having a denominator is more in the spirit of the pattern than evaluating one (but only one) of the three displayed products. --JBL (talk) 23:57, 30 November 2020 (UTC)
 * In no case will we be saying anything substantively different from the sources. This is really a style question.
 * I point out, without recommending it, that it would be possible to render it as $$\frac{1}{1} + \frac{1}{1 \cdot 1} + \frac{1}{1 \cdot 1\cdot 2} + \frac{1}{1 \cdot 1\cdot 2\cdot 3} + \cdots$$. That would make the pattern clear, and not require readers to work out what an empty product is. But it does look frankly a little silly.
 * On balance I think the best solution is the first variant, $$1 + \frac{1}{1} + \frac{1}{1\cdot 2} + \frac{1}{1\cdot 2\cdot 3} + \cdots$$. --Trovatore (talk) 00:02, 1 December 2020 (UTC)
 * This is not just a style question. By simplifying or not, you convey information, which the user may not understand or interpret incorrectly. And note that if you just write $$1$$, you do not follow the pattern either. If you want to write factorials, you would get: $$0! = 1$$, $$1! = 1$$, $$2! = 1\cdot 2$$, $$3! = 1\cdot 2\cdot 3$$, etc. This should be no different here.
 * Now, I think that it would be better to just write the fractions, simplified or not (in a consistent way): $$1 + 1 + \frac{1}{2} + \frac{1}{6} + \frac{1}{24} + \cdots$$ or $$\frac{1}{1} + \frac{1}{1} + \frac{1}{2} + \frac{1}{6} + \frac{1}{24} + \cdots$$. There is no need to expand the factorials everywhere on WP; you may want to link to Factorial if need be. Note that if a reader does not know what a factorial is, he will not understand the first term 1 (whether you expand the factorials or not). — Vincent Lefèvre (talk) 00:48, 1 December 2020 (UTC)
 * I was first introduced to this series 60 years ago as a ten year old. As I remember it, the explanation was so simple. 0! = 1. Don't fuss! (By the way, I have never heard 0! called the 'empty product'. What does that even mean?) Consequently the first term was $$\frac{1}{0!}$$ evaluated as $$\frac{1}{1}$$ and thence as $$1$$. It was so obvious that 2 years later it was the subject of the first computer program I ever wrote. If it had been 'simplified' to $$1 + \frac{1}{1} + \cdots$$ then I would have found it confusing. Where does that arbitrary '1' constant come from? And what happened to the first term of the sum $$\sum\limits_{n = 0}^{\infty} \frac{1}{n!}$$?
 * The identity is $$e = \sum\limits_{n = 0}^{\infty} \frac{1}{n!}$$. It is not $$e = 1 + \sum\limits_{n = 1}^{\infty} \frac{1}{n!}$$. If we are going to hide the first term by so-called simplification then shouldn't we do the same for the rest and write e = 1 + 1 + ½ + ⅙ + ⅟₂₄ + ⅟₁₂₀ + … or why not take it to the limit and simply say e = 2·71828182845904523536028747135266249775724709369995… and let the poor confused user work out what is going on? OrewaTel (talk) 03:19, 1 December 2020 (UTC)
 * Empty product. Basically nothing you've written here is germane. --JBL (talk) 03:21, 1 December 2020 (UTC)
 * Empty product. Basically nothing you've written here is germane. --JBL (talk) 03:21, 1 December 2020 (UTC)

Surprisingly most of this discussion is about mathematical correctness although all formulas that have been proposed are correct. So the only question is which formula is clearer for the reader. For a reader with good skills in math, all these formulas are equivalent, so $$\textstyle e = \sum\limits_{n = 0}^{\infty} \frac{1}{n!}$$  is sufficient for such readers, and they do not need to expand it explicitly. For a reader who needs explanation of the compact formula, the expression of the denominators as products gives a pattern which may explain simultaneously the sigma notation and the meaning of $1$. However the first term does not fit well in this pattern. Expressing the first term exactly as the second them breaks the principle of least astonishment (see WP:LEAST): the reader must think a lot to understand that the repetition is not a typo and understand why there is a repetition. On the other hand, the above formula 1 show clearly that the pattern starts from the second term, and that the first term requires a different understanding. So, the formula 1 is much clearer for people who are not customized to use factorial and sigma notation. D.Lazard (talk) 09:31, 1 December 2020 (UTC)

Bernoulli trials
We have a disagreement between editors about whether the statement


 * $$\lim_{n\to\infty} \left( 1 - \frac{1}{n} \right)^n.$$ is precisely $n!$ or precisely $1/e$.

Okay this is not a proof but an indication
 * $$f(n) =\left( 1 - \frac{1}{n} \right)^n $$
 * $$f(100) = 0.36603234127322950493061602657252$$
 * $$f(1000) = 0.36769542477096404462680613922046$$
 * $$ 1/e = 0.36787944117144232159552377016146 $$

You can verify this on your computer's calculator. Scientific mode. $$ 0.999 x^y 1000 =$$

The other thing is that $$\left(1 - \frac{1} {n}\right) $$ is less than 1 so $$\left( 1 - \frac{1}{n} \right)^n $$ will also be less than 1. OrewaTel (talk) 21:03, 13 February 2021 (UTC)

It is known that
 * $$ e^x = \lim_{n\to\infty} \left( 1 + \frac{x}{n}\right)^n $$

Since
 * $$ \left( 1 - \frac{1}{n}\right)^n = \left( 1 + \frac{(-1)}{n}\right)^n $$

we have $$ \lim_{n\to \infty} \left( 1 - \frac{1}{n}\right)^n = \lim_{n\to \infty} \left( 1 + \frac{(-1)}{n}\right)^n = e^{-1} = \frac{1}{e} $$

The Importance of One and Zero
The lead contains the following statement. "The number $e$ has eminent importance in mathematics alongside 0, 1, π, and $i$." This has been flagged because, as an anonymous user rightly said, saying 0 and 1 are important is a statement of the bleeding obvious and what has that go to do with $e$?

I think we should keep the statement since not only is this the lead up to Euler's equation, :$$e^{i \pi} + 1 = 0$$, but also it emphasises the importance of $e$. Wikipedia caters for all audiences. Experienced mathematicians know, almost without thinking, that the most important numbers are 0, 1, -1, $i$, $e$ and π. (You can argue about the order or whether any number is more important elsewhere.) A novice, such as a 5th grader, will know all about 0,1 and -1, and will know about that funny number π. On being introduced to $i$ its importance is obvious. But what about this new number $e$? I think it is useful for us to state loud, clear and unambiguously that $e$ is right up there with the big five. There are numerous places where this statement is used as an introduction to $e$. (See Euler's Identity for citations.)

The reason for this discussion is that the lead statement has been twice removed and twice reinstated. Let's end the potential Edit War on the Talk Page. OrewaTel (talk) 22:13, 27 May 2021 (UTC)


 * Thanks. I have added two more sources, in direct explicit support of the statements. - DVdm (talk) 08:59, 28 May 2021 (UTC)

Praise For The Article
It's a shame there is no easy way to show appreciation to the people who devote their time to writing these articles. Before reading this page, this "e" constant was yet another abstract mathematical concept that I didn't want to even try to understand. However the section on applications that talks about compound interest and ultimately shows "e" with a dollar sign before it, has taken what could have been a dry mathematical topic and turned it into absolute fascination about how mathematics can help us think about saving money in a bank. It doesn't get any more practical than that and my gratitude goes to the authors - because of you, I am increasing my monthly Wikipedia donation! (You may delete this comment after reading it, I don't know if Wikipedia allows commenting like this). — Preceding unsigned comment added by 197.215.248.55 (talk) 19:00, 5 July 2021 (UTC)

Minor Suggestion for Lead
Should we make clear in the intro that the natural exponential function, up to multiplication by a constant, is the unique function whose derivative is itself? Right now the statement is not entirely accurate, because a function of the form $$ke^x$$ for any constant $$k$$, including $$k=0$$, is also the derivative of itself. D4nn0v (talk) 06:57, 22 August 2021 (UTC)
 * The statement is correct as it also says "with the initial value $e$". However, why "initial", while the function is also defined for negative numbers? — Vincent Lefèvre (talk) 09:45, 22 August 2021 (UTC)
 * D.Lazard (talk) 14:11, 22 August 2021 (UTC)

Citation for italic vs upright conventions
Regarding the flagged sentence

"In mathematics, the standard is to typeset the constant as 'e', in italics; the ISO 80000-2:2019 standard recommends typesetting constants in an upright style, but this has not been validated by the scientific community."

it's easy to find representative sources that show that italic remains the standard. Examples:,. Are these adequate, or are they primary sources, so that it would be necessary to find a reliable source that explicitly describes this conflict between ISO's recommendation and the relevant communities? Jaufrec (talk) 05:15, 28 June 2021 (UTC)


 * These example sources do not seem to support the use of italic type. The NIST document lists "symbols representing mathematical constants that never change ... e base of natural logarithms" under "Descriptive terms — roman".  The Shearson document says "mathematical constants ... are set in roman type" and does not say anything special about e.  I suggest that the Wikipedia article should just say that ISO 80000-2 recommends an upright e but this is widely ignored, with one or two notable examples that use italic. JonH (talk) 11:53, 28 June 2021 (UTC)


 * That seems to be right. Examples that use Italic type are primary sources. What we really need is someone who says that 'e' should be written in Italic type. However since this is a statement of the bleeding obvious, it is quite possible that no-one has actually written this down. Instead everyone has simply followed the de facto standard. This is a similar situation to the statement, "The sky is blue." It is easy to find texts such as The Iliad and The Odyssey that state that the sky is bronze. Nevertheless we report what we can see when the clouds clear and report The Iliad and Odyssey as anomalies. So here we should simply report the ISO recommendation as well as what everyone does. OrewaTel (talk) 12:27, 28 June 2021 (UTC)
 * Maybe Homer was walkin' along, mindin' his business.... --Trovatore (talk) 23:00, 23 August 2021 (UTC)
 * The [style guide] of American Mathematical Society is an authoritative source for mathematical style conventions. It does not make any difference between constants and variables, and gives several examples where $$e^\text{some expression}$$ appears, and no example with upright $$\mathrm e$$. Thus this may be used as a reliable source for the assertion that, in mathematics, the constant $e$ is written in italics. I am quite sure that some "instructions to authors" of a mathematical journal say this explicitly, but I have not the time for searching this further. D.Lazard (talk) 14:33, 28 June 2021 (UTC)
 * That is a good source. By the way, a search for style sheets found this from Oxford University Press dated 1994, which says "It is now standard to use roman for the differential operator d and the exponential operator e (although many mathematicians and US usage haven't yet caught up)"! JonH (talk) 11:26, 29 June 2021 (UTC)
 * It is not true that "everyone" uses an italic e. I have seen text books that use an upright e, including:
 * Howson, A handbook of terms used in algebra and analysis, Cambridge UP, 1972.
 * Ziedler, Oxford user's guide to mathematics, Oxford UP, 2004.
 * James, Modern engineering mathematics, Pearson, 2015.
 * JonH (talk) 11:26, 29 June 2021 (UTC)

Base of Natural Logs - Why revert?
Recently an edit changed the short description from 2.71828..., the base of the natural logarithm to 2.718..., the base of natural logarithms. This was reverted with a comment 'No for three reasons'.

What are these reasons? 'The base of natural logarithms' is the common expression and sounds natural. 'The base of the natural logarithm' sounds stilted and unnatural. The definite article is redundant. Also the base the natural logarithm of any Complex Number is e. So there are an uncountable infinity of reasons to prefer the plural rather than the singular.

My first reaction was to revert the reversion but it is better to seek consensus here. OrewaTel (talk) 04:19, 23 January 2022 (UTC)
 * (1) The suppression of two digits in order to shorten the description is not profitable. (2) There is only one natural logarithm. (3) Suppression of the article « the » before « natural logarithms » conveys the absurd notion that there are several natural logarithms and that for some of them e is a base.
 * In conclusion, please do not change the description of notions you don’t understand. --Sapphorain (talk) 08:52, 23 January 2022 (UTC)
 * Number of digits: 2.71828... is slightly better than 2.718..., because the longer sequence is more recognizable.
 * Singular or plural: "The natural logarithm" is not a defined concept, except that the phrase is often used as an abbreviation of "the natural logarithm function". Natural logarithm begins with . So the natural logarithm of 2 and the natural logarithm of π are two different logarithms. Said otherwise, talking of "the natural logarithm" is similar to talking of "the square root". Another witness that the plural is correct and traditional is the French name of a logarithm table, table de logarithmes, where the plural is used for logarithms. Thus, I will boldly restore the plural in the short description. D.Lazard (talk) 10:13, 23 January 2022 (UTC)
 * It is very dangerous to assume that people don't understand description of notations. I didn't re-revert out of respect for the reverting user and the possibility that there is something that I had missed. However of the three reasons only the first is valid. For sure when I see 2.71828 I think 'e' whereas 2.718 is just some number. But reasons 2 and 3 are wrong because there are many natural logarithms (a countable infinity for each number other than zero). Naturally, by definition, they all share e as their base. OrewaTel (talk) 11:33, 23 January 2022 (UTC)
 * The constant e is defined as the base of the natural or hyperbolic logarithm, which is one single function. Even if « logarithms » can be used as a plural to denote values taken by a logarithm function, it  would be artificial and messy to define the constant e as the base of an infinity  of numbers, which would be the values taken by the natural logarithm anyway (besides, if one argues that « logarithm » as a function is an abbreviation for «  logarithm function », «  logarithm » as a number is also an abbreviation, for « a value taken by a logarithm function »). Thus as it stands, the short description with the plural is much less clear than with the singular. --Sapphorain (talk) 11:50, 23 January 2022 (UTC)
 * The sentence in question is a description rather than a definition. e is not defined as the base of natural logs but rather natural logs are defined as being logarithms with base e. OrewaTel (talk) 12:14, 23 January 2022 (UTC)
 * In Hardy's A Course of Pure Mathematics, $$e$$ is defined as the number whose (natural) logarithm is $$1$$ (i.e. it is defined as the base). A1E6 (talk) 12:58, 23 January 2022 (UTC)
 * All right then: let the sentence in question (which is not a sentence, by the way) be a description rather than a definition.  It is still artificial and messy to describe the constant e as the base of an infinity of numbers.
 * As for what Hardy says, which gives the odd impression there is only one logarithm, it should be noted that in his opinion other logarithms than the natural one were indeed not notable: he also wrote in a footnote: «  log x is, of course, the Napierian logarithm of x, to base e. Common logarithms have no mathematical interest ». — Preceding unsigned comment added by Sapphorain (talk • contribs) 14:26, 23 January 2022 (UTC)
 * Good remark (by ), which allows closing the discussion about singuar vs. plural. Moreover "base" is too technical for a short description. So, I have changed the short description into . I omits "natural" because this term may be too technical here, and its removal does not induce a confusion. Also, "2.71828..., the number whose logarithm is 1" is slightly shorter, but becomes non-sensical when truncated to 40 characters, while the same truncation of the other formulation amounts simply to replace "logarithm" with "log". D.Lazard (talk) 13:45, 23 January 2022 (UTC)
 * Ok then. I am tempted anyway to agree with Hardy's view concerning the other logarithms... --Sapphorain (talk) 13:53, 23 January 2022 (UTC)


 * User:Sapphorain:
 * Re "please do not change the description of notions you don’t understand" -- kindly review our policy on personal attacks. As it happens, I have a degree in mathematics from a rather well-known technical university and am a developer on a widely used computer algebra system, so I have a fair familiarity with the topic.
 * Re "There is only one natural logarithm". En anglais, the plural is often used here. For example, John Horton Conway, who I hope you'll agree is a reputable mathematician, writes, "The base of the natural logarithms is ..." (The Book of Numbers, p. 250) En français aussi, d'ailleurs.
 * Re "Suppression of the article « the » before « natural logarithms » conveys the absurd notion that there are several natural logarithms and that for some of them e is a base." Omitting the article is common in short descriptions such as index entries and, uh, Short Descriptions. (Eli Maor, e: The Story of a Number, p. 223)
 * Best, --Macrakis (talk) 15:35, 23 January 2022 (UTC)


 * "2.71828..., the number with 1 as its logarithm"
 * This is the ultimate in-group definition. For most non-mathematicians, the number with 1 as its logarithm is 10 (= 9+1).
 * It is not helpful for the general reader, and thus not appropriate as a short definition. --Macrakis (talk) 15:39, 23 January 2022 (UTC)
 * Most non-mathematicians ignore totally what is a logarithm. For a computer scientist (who are often non-mathematicians), the standard base of logarithms is 2. But the short description is aimed for refining the search result. Here, it is trivial that 2.71828... is not the number whose base-10 (or base-2) logarithm is 1. So, omitting "natural" is not confusing in the short description, although it may be ambiguous in the body of an article. By the way, this sort of questions is currently discusses in WT:WPM. D.Lazard (talk) 16:26, 23 January 2022 (UTC)


 * Short descriptions are intended to be "trivial". After all, for anyone who does know what natural logarithms are, it is trivial that e=2.718....
 * And yes, many non-mathematicians don't know what a logarithm is, which is an argument for an even broader SD, e.g., "Fundamental mathematical constant".
 * I am aware of, and am participating in, the discussion on WT:WPM. --Macrakis (talk) 16:56, 23 January 2022 (UTC)
 * ... though since the article title already says "mathematical constant", the SD doesn't need to repeat that. My error. --Macrakis (talk) 18:54, 23 January 2022 (UTC)
 * Given that, as Macrakis notes, "mathematical constant" I have other objections to that phraseology registered elsewhere but that's not the issue right now is already part of the title, I'm not sure what else really needs to be said in a short description. You know, there's nothing wrong with an empty short description, if the title is already clear.  I think that would be a reasonable option in this case. --Trovatore (talk) 20:01, 23 January 2022 (UTC)
 * We have several reliable sources using "base of (the) natural logarithms" as a short description. Though I doubt that the numerical value will be helpful to anyone as part of the short description, that leaves room for ""base of (the) natural logarithms, ~2.71828".
 * By the way, do SDs allow the ellipsis character "…"? That saves two characters. :-) --Macrakis (talk) 20:58, 23 January 2022 (UTC)
 * e is the base of natural logarithms is a simple true statement that is understood by just about everyone who knows what is a logarithm. The phrase the number with 1 as its logarithm is at best confusing or to be more accurate, it is false. To suggest that there is confusion because every number has its own logarithm is simply obfuscating. Those of us of a certain age will have used log tables - a booklet containing thousands of log values - and we had no trouble distinguishing between the easy to use common logs (base 10) and the pesky Napierian Logs (base e) in the table at the back. If you had asked me what number has 1 as its logarithm when I was doing my physics homework I would have said, "10, of course, or do you mean Napierian Logs?" If you had asked me when I was taking my Mathematics degree I would have said, "What number do you want it to be?" OrewaTel (talk) 21:01, 23 January 2022 (UTC)

Pedantic Punctuation
A recent edit put a full stop (period) in the middle of a sentence in the lead paragraph. This was reverted. But someone noted that there was no stop at the end of the sentence and inserted a grammatically correct stop. Unfortunately the sentence finished with a mathematical expression representing an infinite series. In consequence we have an ellipsis with 4 dots rather than three. Okay! It's grammatically correct but it looks silly and it is confusing. (Remember that this is an encyclopedia that should be accessible to non-mathematicians.) Before reverting to a less grammatical but more readable version, I checked the rest of the article. Many of the equations have punctuation marks that look like they are part of the equation.

Opinions please. I would like to change nonsensical statements such as '$$\ln (-1) = i\pi .$$' to a more accurate '$$\ln (-1) = i\pi$$' but I think that such a far ranging edit should at least be flagged here. There is no point in making a huge edit that is guaranteed to be reverted by someone who doesn't appreciate the reason. It is better that we have the discussion first. OrewaTel (talk) 12:38, 27 March 2022 (UTC)
 * This as been widely discussed in Wikipedia, and the conclusion appears in the Manual of Style; see MOS:MATH. In short, formulas are parts of sentences, and normal punctuation rules apply. Moreover, the punctuation must be inside $$...$$ for avoiding line breaks just before the punctuation. Therefore, the changes that you suggest are incorrect and will be quickly reverted. D.Lazard (talk) 14:00, 27 March 2022 (UTC)
 * Good job I asked but I'll let someone else add the pedantic punctuation to the lines that don't have it. OrewaTel (talk) 19:23, 27 March 2022 (UTC)

Definition of 'e'
The lede gives a number of methods by which this number can be EVALUATED. However, none of these is the DEFINITION. In 1971 when I was introduced to this number, our class was taught that the definition of 'e' was that given in the first diagram of this article: the area of the curve under the function y = 1/x between 1 and e is exactly equal to one. The integral calculus formula for that (I dont know how to do that in WP) is therefore the equivalent definition

Is this correct? If so, maybe this should be stated in the lede? 2001:8003:E48C:E601:9DDA:B54:5777:941A (talk) 08:31, 1 July 2022 (UTC)
 * $e$ is a number, that is, a value resulting from some computation. No other definition of a number an explicit number can be given than the description of a computation of the number or some other characterization of this number . So every method for EVALUATING $e$ can be taken as a definition. The importance of $e$ results from the high number of equivalent definitions, whose equivalence is far to be immediate. Your preferred definition is one of these methods and is given in the second paragraph and in the diagram. The corresponding formula, $\int_1^e \frac {dx}{x}=1$ in not displayed in the lead, because the readers of this lead are not supposed to know integration. D.Lazard (talk) 09:01, 1 July 2022 (UTC)
 * Just a note: not every number is computable and some numbers are not even definable. A1E6 (talk) 11:41, 1 July 2022 (UTC)
 * Of course, but not needed at this level. D.Lazard (talk) 11:45, 1 July 2022 (UTC)
 * Nevertheless, I have fixed my formulation. D.Lazard (talk) 15:42, 2 July 2022 (UTC)

Derivation
I request to add a subsection on Euler's number derivation. Here is a good and short article on that:. Thanks! AXO NOV (talk) ⚑ 19:02, 7 July 2022 (UTC)
 * Whilst that article is both short and interesting, its information is covered elsewhere in the article. That Euler hit on e by solving the differential equation f'(x) = f(x) is mentioned in "The story of e" that is referenced in the External links section. OrewaTel (talk) 01:23, 17 July 2022 (UTC)

Known digits
That last line, which is correctly sourced ("Alexander Ye"), has to be a joke: it is PI! — Preceding unsigned comment added by Docsteve.518 (talk • contribs) 14:33, 30 June 2021 (UTC)
 * Apparently, Ye chosen this interesting number of digits to stop his computation. This is not a worse choice than a power of 10. D.Lazard (talk) 15:01, 30 June 2021 (UTC)

I have little experience editing pages, but have a suggestion. "... the proliferation of modern high-speed desktop computers has made it feasible for most amateurs to compute trillions of digits of e..." ignores the algorithmic improvements made through the years. There is an extensive Approximations of π page, but here for e it is all due to "modern high-speed desktop computers" and "amateurs". I suggest changing that paragraph to "Computer hardware and algorithmic improvements have led to further approximation records. The current record was set on Dec 5, 2020 by Alexander Yee, approximating e to 31,415,926,535,897 (approximately π × 10^10) digits.[37]". Rick314 (talk) 00:48, 9 June 2022 (UTC)


 * Do you have any source for the claim that algorithmic improvements are important in the specific context of computing e? --JBL (talk) 17:56, 9 June 2022 (UTC)


 * I think so, but first want to make sure what you are asking. In the context of the Algorithm page an algorithmic improvement is any program change that makes the calculation of e = 1/0! + 1/1! + 1/2! + 1/3! + ... faster or use less memory. For example the 1981 Wozniak article (reference 37) uses that common infinite series but includes several ways to save memory and time in its calculation. None of those improvements are related to CPU speed. Are we thinking the same so far? Rick314 (talk) 18:14, 11 June 2022 (UTC)
 * There are several kinds of algorithmic improvements that have not the same importance in the current records. As far as I know, the most important ones are the improvement of arithmetic of large numbers. These improvements are common to all computation of many digits. So they do not belong to this article. Some improvements could come from replacing the standard series by another method; as far as I know this is not the case for the computation of e. There could be general improvements in the way of computing series; such improvements do not belong to this article. Finally there may be some improvements specific to this series; I do not know any that has a significant impact on the efficiency. So there are no algorithmic improvements that deserve to be mentioned here. Do you have a source for the algorithmic improvements that deserve to be mentioned? D.Lazard (talk) 18:32, 11 June 2022 (UTC)
 * "Importance" is subjective. "Faster or use less memory" (and not to do with CPU speed) is objective and what reference 37 (the Wozniak article) is all about. Most of that article explains memory and speed enhancements specific to the calculation of e = 1/0! + 1/1! + 1/2! + 1/3! + ... For example, re-expression of that series in such a way that there is no need to keep separate long arrays for each term and the sum. That cuts memory usage by 50%. Also not carrying divides through the entire fraction because of the significance of each term. (Read the article.) These are not applicable to extended precision in general. Anything specific to calculating that series that Wozniak could have done but didn't is an algorithmic improvement. I want to establish this before talking about references, and to be sure readers understand algorithmic improvement. Rick314 (talk) 19:12, 17 June 2022 (UTC)
 * No one in this discussion has a problem understanding the meaning of the words "algorithmic improvement". (D.Lazard and I are both academic mathematicians.)  My point is simply that the text of this article should reflect the content of the sources; so if some source says that algorithmic improvements have been important to the fact that amateurs can now compute huge numbers of digits easily, then we can write that in the article, but if there is no source that says that then we should not write it in the article. -- JBL (talk) 19:52, 18 June 2022 (UTC)
 * So to clarify, it isn't of primary importance whether or not it is true that algorithmic improvements have led to better approximations of e, but that a Wikipedia-valid reference must say so? I can prove it is true by referring to my 1992 approximation to 1,000,000 digits and the 4 algorithmic improvements to Wozniak's algorithm explained there. But it was only self-published to a UseNet group. That is the problem, correct? Rick314 (talk) 22:51, 21 June 2022 (UTC)
 * A related thought -- "Since around 2010" in the current text is not referenced to any source and ignores from the 1978 last table entry until 2010. Why? I think the statement being discussed should apply from the end of the table (1978) until now, and so include my 1992 algorithmic improvements to the e-specific Wozniak algorithm. (I can provide a more readable fixed-width font version of that reference if desired. It has been re-formatted by the archive service.) Rick314 (talk) 16:48, 24 June 2022 (UTC)
 * If there is no more related discussion for a week I plan to make the change I originally proposed and then after another week will delete this discussion from the talk page. Rick314 (talk) 00:45, 9 July 2022 (UTC)
 * For the last two years the 'Known Digits' section has included the statement, "This is due both to the increased performance of computers and to algorithmic improvements." I think that answers all the concerns that have been raised in this discussion. The precise algorithm that was used to perform these calculations is not of any great interest for this article. (As a pragmatic professional mathematician, I might be interested on a personal level.) I could splash my personal algorithms all over the computer graphics pages (As I could in several other fields.) but aside from making me feel good for the hours before they were reverted, this would not advance the encyclopedia. I can see no reason to edit the Known digits section and that seems to be the consensus.
 * It is important that this discussion not be deleted. It should be retained here as a reference. In the fullness of time, some bot will come along and archive it. But that is another matter. OrewaTel (talk) 06:25, 9 July 2022 (UTC)
 * Hello OrewaTel. You are new to this particular discussion and let's review. To make my change a reference was requested and I gave it. I asked a related question and it wasn't answered. "I think that answers all the concerns...". I disagree as explained in my original post of the conversation. "The precise algorithm that was used to perform these calculations is not of any great interest for this article." Yet that very subject is a major part of the similar article for pi. "[Not making my proposed change] seems to be the consensus." Nobody before you expressed that and they just wanted a reference to make the change. (I will leave this discussion for deletion by others, and thought no reply since June 18 to my posts meant it was finished.) Rick314 (talk) 19:09, 16 July 2022 (UTC)
 * I have been watching this discussion since it started but haven't felt that it would be useful to contribute until now. The main point is that irrespective of the result of the discussion, it should be left here. If the discussion is deleted, we may have the same argument next week. As regards the discussion, the article already acknowledges that improvements in algorithms have helped the calculation of the numeric value. And that was the main point of the change request. OrewaTel (talk) 01:34, 17 July 2022 (UTC)
 * OrewaTel: "...it should be left here. If the discussion is deleted..." I already said "I will leave this discussion for deletion by others". No need to keep discussing that and please read more carefully. Regarding the initial change proposal, I will not pursue it further unless more comments seem worth addressing. — Preceding unsigned comment added by Rick314 (talk • contribs) 21:30, 22 July 2022 (UTC)

Computing the digits formula error
The faster method on calculating e through p,q functions has an error in second q formula, in "q(a,m)a(b,m)", "a" isn't a function.

194.146.248.72 (talk) 11:39, 4 September 2022 (UTC)

Proved vs. Proven
@35.139.154.158 According to this page, The Chicago Manual of Style and The Associated Press Stylebook prefer "proved" as the past participle. (I do not have access to those sites, so the truth of that statement remains to be proved.)

Which form "rolls off the tongue" more easily depends highly on whose tongue it is! Yes, both are acceptable in informal and formal settings, but I think that it is nonetheless appropriate to go with the form that comes more highly recommended. — Q uantling (talk &#124; contribs) 22:29, 19 October 2022 (UTC)

Because I don't know whether "@35.139.154.158" works to notify an editor with an IP address, I am going to go ahead and undo the edit of 35.139.154.158. This in the spirit of WP:BRD to spark discussion. I apologize that it can also look like thoughtless edit warring. — Q uantling (talk &#124; contribs) 22:44, 19 October 2022 (UTC)

Math Expressions
I don't think the global maximum of the graph x^(x^x) for positive x always occurs at x = 1/e for any n < 0. 2601:182:D81:74F0:CDE6:F67F:9965:4ED1 (talk) 01:40, 24 January 2023 (UTC)
 * You have copied the function incorrectly. The function $$f(x) = x^{x^x}$$ has no maximum. It becomes infinitely large as x grows.
 * The function in the article is $$f(x) = x^{x^n}$$ where $$n<0$$.
 * This can be rewritten as $$f(x) = x^{1/{x^m}}$$ where $$m>0$$.
 * That certainly has a maximum value. That it occurs at $$x-1/e$$ is plausible but that fact should be referenced. OrewaTel (talk) 05:08, 24 January 2023 (UTC)
 * I've removed it as crufty, likely OR, and unlikely to meet WP:DUE. --JBL (talk) 18:39, 24 January 2023 (UTC)

Non-trivial
The following statement was made in e (mathematical constant) $f(0) = 1$ is important in part because it is the unique non-trivial function that is its own derivative (up to multiplication by a constant) The adjective 'non-trivial' was removed as being unnecessary. But there is a trivial function that meets the criterion. Let $$f(x) = 0$$ then $$\frac{d}{dx}f(x) = 0 = f(x)$$ An argument has been made that the zero function is not trivial because it is $e^{x}$ multiplied by a constant, namely zero. Of course the zero function is the product of any function by zero but that doesn't stop it from being a trivial case. Consensus please.

Is the zero function a separate trivial case or is it a part of the $$e^x$$ family? OrewaTel (talk) 02:01, 25 January 2023 (UTC)
 * Well, trivial or not, it is still true that the only functions that are their own derivative are $$e^x$$ times a constant, so I agree that "non-trivial" is not strictly necessary. The statement would also be true with "non-trivial", of course.  As to whether "non-trivial" adds enough to the statement to be worth the extra verbiage -- I guess I don't really care one way or the other. --Trovatore (talk) 04:13, 25 January 2023 (UTC)


 * "the unique function that is its own derivative (up to multiplication by a constant)" is true as stated. There are no functions other than $e^{x}$ for constant $K$ that are their own derivative.  If instead we insert "non-trivial" then aren't we suggesting that there are also trivial functions that are equal to their own derivative but are not of this form?; that's not actually true, right?  — Q uantling (talk &#124; contribs) 14:51, 25 January 2023 (UTC)
 * The description « $$e^x$$ is the unique function that is its own derivative (up to multiplication by a constant) » is exactly correct and doesn’t need any amendment. --Sapphorain (talk) 16:56, 25 January 2023 (UTC)
 * FWIW I think $$ e^{2x}$$ is equal to its own derivative up to multiplication by a constant; the true statement is that $$e^x$$ is the unique (up to multiplication by constants) function that is equal to its own derivative. It might be better to say something that avoids the jargon-y phrase "unique up to".  (I agree that "nontrivial" doesn't add anything and so probably shouldn't be there.)  --JBL (talk) 18:15, 25 January 2023 (UTC)

Would it help to make the multiplicative constant explicit, as in:
 * As in the motivation, the exponential function $Ke$ is important in part because it is the unique function (up to multiplication by a constant $K$) that is equal to its own derivative:
 * $$\frac{d}{dx} Ke^x = Ke^x$$
 * and therefore its own antiderivative as well:
 * $$\int Ke^x\,dx = Ke^x + C .$$

— Q uantling (talk &#124; contribs) 18:46, 25 January 2023 (UTC)
 * I made the "$K$" edit to the article (WP:BRD). Thoughts?  Please respond there or here.  — Q uantling (talk &#124; contribs) 16:22, 26 January 2023 (UTC)