Talk:E (mathematical constant)

Harlan Brothers' papers

Recently, Hjb annotated the following formula with a reference:

$e=\lim _{n\to \infty }\left[{\frac {(n+1)^{n+1}}{n^{n}}}-{\frac {n^{n}}{(n-1)^{n-1}}}\right]$

The reference is "H. J. Brothers and J. A. Knox, New closed-form approximations to the Logarithmic Constant e. The Mathematical Intelligencer, Vol. 20, No. 4, 1998; pages 25-29.". Presumably Hjb is Harlan J. Brothers himself.

I would have no objection to the reference, except that I have reviewed the paper, and it does not appear to contain the specified formula. It does contain many equivalent formulas, but (as has been noted on this page and elsewhere several times before) many formulas are equivalent to this one, including the definition itself:

$\lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n}$

In any case, it is not clear to me what value there is in citing the formula with a paper that does not actually mention that formula.

It is also possible that I am missing something; perhaps the formula appears in the paper and I didn't see it. I would welcome corrections.

Any discussion on this? -- Dominus 22:44, 11 March 2007 (UTC)

Hello. Thanks for your attentiveness. However, the formula, dubbed the "Power Ratio Method," appears on page 26, approximation number 4.

Hjb 07:35, 12 March 2007 (UTC)

It does indeed. I wonder now how I missed it. Thank you! -- Dominus 11:12, 12 March 2007 (UTC)

relation with pi

see: Talk:Pi#T-shirt_equation

— Xiutwel (talk) 10:51, 2 April 2007 (UTC)

Approximate mathematical "coincidences" like this one between powers of π and powers of e are a dime a dozen. Come back to me when you find one where they match to one thousand places. Then we can look for a proof that they are the same. JRSpriggs 09:32, 3 April 2007 (UTC)

Known digits

I disgree in cutting out the calculation of known digits. I think it is interesting and relevant (however it does need to be referenced). I would also like to see a similar section in pi. — Preceding unsigned comment added by Jim77742 (talk • contribs)

I agree it's interesting and relevant. But each item needs a good reference. I tried to find one on von Neumann, and can see why someone marked it dubious. Work on it some before putting it back, or add cn tags to ones you can't find refs for, so someone will know to work on them. I calculated e to 5000 digits with a basic program once in 1970, but had an error due to not leaving enough digits for carries at some point; time to get cracking on the 50 billion or so record... Dicklyon 05:35, 16 April 2007 (UTC)

The known digits table is incomplete, there is no entry for 1,000,000 million digits, which were calculated by Robert AH Prins in 1992 using a PL/I program running on an IBM 3048 at his then employer Willis Corroon in Ipswich - needlessly to say they were not pleased he did this. Robert AH Prins 26 June 2007

revert of edit

Why did you guys rever the edit of the external link back to the link that is broken, which I put in there like a year back in this here --BorisFromStockdale 02:07, 7 May 2007 (UTC)

It fails WP:RS, and seems unnecessary as WP:EL, even if we had some external assurance it was correct. It's your site, isn't it? — Arthur Rubin | (talk) 02:31, 7 May 2007 (UTC)

Shouldn't we also remove the IP site with 10 million digits for the same reason? I can never get it to load, so I can't even see if it's attributed or reliable, but I expect not if it doesn't even have a domain name. Dicklyon 03:13, 7 May 2007 (UTC)

Yes. — Arthur Rubin | (talk) 03:25, 7 May 2007 (UTC)

Yes, it is my site, but you guys let it in in 2006 and let it stay there for a year. --BorisFromStockdale 03:51, 7 May 2007 (UTC)

Thanks for pointing that out. We'll try not to make the same mistake twice. Dicklyon 04:21, 7 May 2007 (UTC)

By the way, what exactly does it violate in the [WP:EL]]? --BorisFromStockdale 03:55, 7 May 2007 (UTC)

Wp:el#Advertising_and_conflicts_of_interest where it says "You should avoid linking to a website that you own, maintain or represent, even if the guidelines otherwise imply that it should be linked. If the link is to a relevant and informative site that should otherwise be included, please consider mentioning it on the talk page and let neutral and independent Wikipedia editors decide whether to add it. This is in line with the conflict of interest guidelines." So, if you think it should be linked, please explain here, in terms on the criteria at Wp:el#What_to_link. Please also explain where the digits come from, and why we should see it as authoritative. -- Dicklyon 04:21, 7 May 2007 (UTC)

Basically the digits come form Mathematica 5.0
I used the commands:

abc = N[GoldenRatio, 20000000];
Export["C:\my_files\golden.txt", abc, "table"];

This produces a .txt file. Then zipped it using a regular zip compresstion built into Total Commander 6.56. This reduced the file sizes to about 50% of the original. this way I calculated the 3 constants on the website (e, golden ratio, pi). Well, the website is being served by google through their google page creator, so it should handle the bandwidth. Unfortunately I do not think that google allows the upload of more than 10 MB files, so 20 million digits is the limit for what I can put up in one peace... --BorisFromStockdale 08:31, 7 May 2007 (UTC)

I see. Wouldn't it be much simpler to just say in the article that anyone who needs lots of digits can get them trivially in a one-liner in Mathematica, and show the command? Dicklyon 14:24, 7 May 2007 (UTC)

And how many people can afford Mathematica? Fredrik Johansson 16:01, 7 May 2007 (UTC)

I was thinking of just aggregating the digits for several important constants on the same website. I think that having it all in one place might be useful for some people. Anyway, the website is still there, I will be upgrading(adding new constants) to it. If you guys want to incude it to this or to any other articles, Great. If not then ... --BorisFromStockdale 21:58, 8 May 2007 (UTC)

GA review

Overall the article looks to be in pretty good. A few notes:

• The Feynman quote needs a more complete citation, but is it really even necessary here? Seems a bit like fluff, but it could (should?) be moved into a footnote for "one of the most important formulas in mathematics:".  Done
• The maximum value "f(e)=..." seems a bit superfluous,  Done and perhaps the two properties related to f(x)=x^(1/(x^n)) and its special case could be combined.
• Why is one of the four equivalent definitions listed in the properties section as the most common? Just delete it, and perhaps add a note about the commonness in the definitions section above.
• All the continuous fractions and infinite series seem like overkill. Reduce it down to the most important ones.
• The "Non-mathematical uses of e" seems like a trivia section that should be incorporated or deleted.  Done

I'll put the nomination on hold until these are resolved one way or the other. --Flex (talk/contribs) 03:05, 9 June 2007 (UTC)

Here are some book sources for the Feynman quote if anyone wants to add one. Dicklyon 04:24, 9 June 2007 (UTC)

I found a citation for it on the article for Feynman. —Disavian (^talk/_contribs) 05:02, 9 June 2007 (UTC)

I don't see a place where a certain definition is listed as the most common. Are you talking about the beginning of the section, where it discusses the "e is its own derivative" property? I think that's just stating the property by using one of the definitions above. —Disavian (^talk/_contribs) 05:18, 9 June 2007 (UTC)

I don't know which infinite sums are important. Would someone else like to make that decision? —Disavian (^talk/_contribs) 08:46, 9 June 2007 (UTC)

Would you be willing to restore the "Non-mathematical uses" section? (Perhaps under a different section heading?) It didn't feel like trivia, at least not in the sense that these facts needed to be merged into the History section. I think the reviewer may have been objecting to the organization of the material in this section as a bulletted list, although with a little effort I'm sure it can be tied together in prose. Also, I don't feel that the History section is an appropriate place for Google to be incorporated, as the company had nothing to do with the history of e aside from paying homage to the number. An unrelated note: At WP:WPM, User:Geometry guy made a promising suggestion of moving the "Representations of e" out as a subarticle. I think this is an excellent idea. Silly rabbit 10:48, 9 June 2007 (UTC)

There is a reference in the "Notes" section S. M. Ruiz 1997 which doesn't attach to anything. Silly rabbit 11:01, 9 June 2007 (UTC)

Regarding the "Non-mathematical uses" section: it felt like trivia to me because it was a list of unconnected factoids that don't really have anything to do with definitions or uses of e proper. (It's like if someone named a character in their novel after Alex Trebek -- that would likely be a relevant fact to incorporate into the article on the novel, but it would not be appropriate on his page. Substitute e for Alex and Google for the novel.) As it stood, this section seemed little different than a "Miscellanea" or "Cultural references" section (cf. WP:TRIVIA and Wikipedia:Handling_trivia#Trivia_and_lists), but renamed it seems slightly better. I'd still say it should be deleted, but I'll leave it up to you all.

Also, are there naturally occurring instances of e in biology or other fields besides finance and math proper? If so, perhaps the compound interest section could be expanded into an "Applications" section to reflect that. --Flex (talk/contribs) 15:48, 9 June 2007 (UTC)

See exponential growth. Septentrionalis PMAnderson 23:41, 9 June 2007 (UTC)

Reply to first paragraph: I think a "Pop culture" section can have some value if it's done properly, and I have provided some rather limited connection between the three facts that were listed as bullets before. I would personally like to see expansion rather than deletion, but in some kind of more encyclopedic direction. Anyway, merging the google references into the "History" section was not the way to go about incorporating this into the article in a harmonious way: in fact, it had the opposite effect (from Euler to Google?!) Nevertheless, if the section isn't headed anywhere, interesting though it may be, perhaps you're right that it should be deleted.

On the second point, yes: it would be nice to find some applications of e. The trouble is that most applications seem to focus on the natural exponential and logarithm. The significance of the numerical constant e is difficult to disentangle from these ideas. I, too, am eager to see suggestions and edits in this direction, though. Silly rabbit 16:01, 9 June 2007 (UTC)

I think the article's looking better and better. On the motivation section, I'd suggest that it seems too specific to e's calculus properties and that there are other motivations (e.g., e's occurrence in certain natural and probability problems). This could perhaps be resolved by putting the history section first or expanding the motivation section (and/or the history section?) to include it's other motivators. --Flex (talk/contribs) 14:26, 11 June 2007 (UTC)

Thanks for the suggestion. In hindsight, this is the "obvious" thing to do, but I couldn't quite see how to suitably organize the article beforehand. Please let us know if you have any other organizational suggestions! Silly rabbit 14:50, 11 June 2007 (UTC)

The article seems to be in a bit of flux right now, thanks in part to my review, I suppose (but cf. also Septentrionalis's comment below). If it settles in the next day or three, please leave a note on my talk page, and I'll come back and take another look to finish the GA process. I think it's pretty close to GA, but stability is also a factor. --Flex (talk/contribs) 18:48, 11 June 2007 (UTC)

I failed this article for now. Feel free to renominate it when it gets to a good resting place. --Flex (talk/contribs) 17:00, 19 June 2007 (UTC)

Representations of e

Representations of e is now live, so the material here should be summarized in prose, with one or two supporting formulas. I don't know enough about the history, level of interest, and applications of these techniques to comment on them, aside from the sophomoric "There are many ways to represent e..." (etc.) Is there an expert among us? Silly rabbit 11:13, 9 June 2007 (UTC)

Good split. It really cleaned up this (main) article. —Disavian (^talk/_contribs) 20:28, 9 June 2007 (UTC)

This is a serious loss to the article; it would be better to withdraw the Bad Articles nomination, and proceed directly to Wikipedia:Scientific peer review than to disfigure it in this manner. Septentrionalis PMAnderson 23:45, 9 June 2007 (UTC)

Probability application

Consider a slot machine that pays off one time in a million. If you play the slot machine one million times, you can expect to win once. But you have a 1/e probability of winning nothing.

Perhaps this is worth mentioning as a natural appearance of e in a fairly simple problem not obviously related to compound interest. -- Dominus 05:36, 10 June 2007 (UTC)

Good one. I had thought about including an application of e (as opposed to its relationship with exponential growth). I came up with derangements, but this is much easier. Silly rabbit 10:43, 10 June 2007 (UTC)

Actually, they are much the same problem; the derangement problem is a lottery which one guest may be expected to win by getting his own hat. There is a real difference: no two guests can get the same hat, but that's a second-order term. Septentrionalis PMAnderson 17:09, 10 June 2007 (UTC)

first citation is bogus

" The number e is one of the most important numbers in mathematics" is backed up by the citation: It was described by Richard Feynman as "[...] the most remarkable formula in mathematics [...], our jewel." Source: Feynman, Richard [June 1970]. "Chapter 22: Algebra", The Feynman Lectures on Physics: Volume I, p.10.

e is a number not a formula. --Chan-Ho (Talk) 00:23, 12 June 2007 (UTC)

As I recall, that passage from Feynman is talking about the Euler identity e^{i\theta} = \cos \theta + i \sin \theta. Probably somewhere in there he mentions how all the important constants of mathematics are in that identity when you plug in \theta = \pi. So it may be possible to fix the cite. --Chan-Ho (Talk) 00:36, 12 June 2007 (UTC)

I copied the ref from Richard Feynman. If it's wrong, then it should be fixed there too. —Disavian (^talk/_contribs) 03:29, 12 June 2007 (UTC)

I can't find any corresponding comment or ref in the Feynman article. But it's pretty clear that the quote is doesn't fit the way it's used here, so I'll take it out until someone who has the source consults it and figures out a more appropriate use for his jewel comment. He also had another jewel comment about QED, which you can find in GBS. Dicklyon 04:57, 12 June 2007 (UTC)

My bad, the article was Leonhard Euler; it's also used on Contributions of Leonhard Euler to mathematics. —Disavian (^talk/_contribs) 16:58, 21 June 2007 (UTC)

Peer review

I'm transcluding the peer review to this page so it will gather more attention from the many editors who frequent this talk page. Per typical Peer Review ettiquite, respond to and/or implement the reviewer's suggestions resonably quickly so that reviewers can identify and comment on new issues. —Disavian (^talk/_contribs) 16:16, 21 June 2007 (UTC)

E (mathematical constant)

I'm attempting to get this core article to GA status. I previously nominated it for GA status, and the article went through several dramatic changes before being (temporarily) failed for lack of stability. I'd like to get additional feedback and suggestions for the article before I renominate. Thank you. —Disavian (^talk/_contribs) 14:12, 21 June 2007 (UTC)

Some comments:

• I have a slight issue with the first sentence, because it appears to be using circular reasoning for the definition. e is the base of the natural logarithm, but the natural logarithm is defined as a logarithm to the base e. That doesn't seem very informative to me. The image to the right of the lead does a better job, I think, so perhaps e could also be defined in terms of the exponential function within the lead?
  • Not circular, merely complicated. "Natural log" is the basic concept here; e, and bases, can be defined in terms of it. Septentrionalis PMAnderson 22:09, 21 June 2007 (UTC)
• It might be helpful if the "compound-interest problem" section showed how the result extended to such real-world examples as population growth, the spread of disease, and radioactive decay.
• A substantial portion of the text consists of mathematical formulae that may not be of general interest. But I'm not sure how that could be addressed.
• There is some redundancy between the "Alternative characterizations" and "Representations of e" sub-sections. Should they be consolidated? — RJH (talk) 15:36, 21 June 2007 (UTC)

I hope this was somewhat helpful. Thanks. :-) — RJH (talk) 15:36, 21 June 2007 (UTC)

I have made comments below on two of the points. These certainly merit further discussion, so I have sectioned them off accordingly. Silly rabbit 16:39, 21 June 2007 (UTC)

Exponential growth and decay

All points are worth addressing, in my opinion. But allow me to zero in one the second bullet point for a moment. While it is certainly true that exponential functions play the fundamental role in all exponential growth and decay models, it is difficult to justify in general terms why one should use the peculiar base e. This is one reason for focusing on the probability applications rather than those manifestly involving exponential growth and decay: the number e arises quite naturally. It may be reasonable to include a mention of the applications of exponential functions (these are dealt with in other articles), but I would resist placing any emphasis on them here unless someone can come up with a convincing example why one would use e as the base rather than some other number. It's important to bear in mind that this article focuses on the number e rather than the function e^x. Silly rabbit 16:39, 21 June 2007 (UTC)

But wouldn't e naturally arise as the necessary base of the solution to certain differential equations? (E.g. Radioactive_decay#Decay_timing.) Especially since the article spends an entire section on e in calculus. — RJH (talk) 22:02, 21 June 2007 (UTC)

I see. Yes, certainly. If we are allowed to pursue the differential equations route, this could easily be worked into the e in calculus section. Silly rabbit 22:19, 21 June 2007 (UTC)

I started to bring in the radioactive decay timing example you suggested, but it did not seem to be popular with the other editors. Silly rabbit 16:22, 23 June 2007 (UTC)

No problem. These are only suggestions, after all. — RJH (talk) 18:06, 23 June 2007 (UTC)

Mathematical formulas

It's going to be hard to get rid of the mathematical formulas in the text. Already many formulas were moved to the Representations of e article. The trouble with e is that it is so intimately tied up with ideas of calculus, and to give a proper discussion seems to involve using formulas. There are levels of general interest to consider too. I doubt there is any way to make a compelling case for the number to someone who is unfamiliar with the basic ideas of differentiation, integration, and/or limits. The derangements example may come close, but that is mathematically sophisticated in other ways. Silly rabbit 16:39, 21 June 2007 (UTC)

Just to clarify, I don't have an issue with the presence of the formulae in the text. But they may deter some readers. So additional clarification may be needed. — RJH (talk) 22:04, 21 June 2007 (UTC)

Clarification is always good. But the equations should not be trimmed; this is an encyclopedia, not Richard Feynman's publisher, who told him that every equation would halve his sales. Septentrionalis PMAnderson 22:07, 21 June 2007 (UTC)

I concur with Septentrionalis' points there. —Disavian (^talk/_contribs) 01:01, 22 June 2007 (UTC)

Redundancy

With regard to the redundancy, I'm not sure how to tackle this problem. I would like to get rid of the two redundant representations of e, since these are already discussed at length during the preceding sections. However, that would leave only the continued fraction representation, and this gives a rather misleading impression to the reader about its relative importance. It may be appropriate to reassess the inclusion of a few select candidates from the Representations of e article. It would be nice if we could say why the selected representations are important as well. Silly rabbit 10:42, 23 June 2007 (UTC)

Perhaps then the article could give a mathematical representation of the software algorithm used to compute the digits e? (Presumably because it is the most efficient known means to compute said digits.) I think I would find that of interest. Thanks. — RJH (talk) 17:52, 24 June 2007 (UTC)

Yes, I thought that was a rather odd ommission as well, given that there is a big table of the number of digits computed. ;-) Silly rabbit 18:05, 24 June 2007 (UTC)