Talk:e (mathematical constant)/Archive 3

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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)

Thanks for seeking advice on this question! I would be against including this formula. e is such a fundamental constant that it must occur in hundreds of similar formulas. including them all would make the article impenetrable and detract from understanding rather than contribute to it.

i do however support inclusion of the classic formula (1 +1/n)^n and it's fair to ask why. here are few reasons 1) a single formula, rather then burdening the intellect, is likely to have an illuminating impact. this is the right level of detail to strive for in an encyclapedia article. 2) the classic formula is taught in the schools. its presence then will act more as a reminder to readers, rather than representing new information requiring assimilation. 3) the classic formula is a very general result and is largely responsible for e's importance. the suggested formula - while it may be the right tool to use for a special class of problems - hasn't been shown to have the same impact in a broad range of cases.

--Philopedia (talk) 08:46, 5 May 2008 (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 —Preceding unsigned comment added by 193.244.32.140 (talk • contribs) 08:25, 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:". Y Done
• The maximum value "f(e)=..." seems a bit superfluous, Y 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. Y 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. --C S (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. --C S (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)

First sentence seems a little weak?

The article begins with:

"The mathematical constant e is the unique real number such that the tangent line to the graph of the exponential function y = e^x at x = 0 is the line y = 1 + x."

While no doubt true, this seems a fairly arbitrarily-chosen property to lead off with, and not one which gives readers (especially the less technical readers) any real feel for what e is all about. Also, the definition is circular because "the" exponential function is defined in terms of e, so a definition of e shouldn't really refer to it. I feel this intro could be improved. Matt 22:34, 6 July 2007 (UTC).

The original first sentence defined e to be the base of the natural log. This was rejected because it appeared to be circular. (I didn't feel it was, but trying to explain why this so to non-mathematicians seems impossible.) This sentence was suggested because it makes a link with the displayed image. Also, you don't need e to define an exponential function. For instance, y=2^x doesn't involve e. What makes e special is that the associated exponential function has unit slope at x=0. Silly rabbit 22:45, 6 July 2007 (UTC)

You are talking about an exponential function; I emphasised the exponential function, which is the term used in the definition and AFAIK always means exp(x). In fact, my original statement that the exponential function depends on already having defined e is nonsense, because obviously you can define it via the series. However, I still think the current wording is wrong, or at least misleading, because it implies that there are lots of different forms of the exponential function (effectively a^x for any constant a), and that e is the constant we need to use to satisfy the slope condition. I am not sure that this is correct.

Ah, this is then a miscommunication over the use of the word "the" in the first sentence. Point taken. Silly rabbit 00:13, 7 July 2007 (UTC)

No, it's not about the word "the", it's about a circular definition. Why not just say "The mathematical constant e is the unique real number y such that y = e¹"?John Lawrence 18:25, 12 September 2007 (UTC)

After thinking some more, I don't think it's circular. It would be more clear that it was not circular if it was written as

"The mathematical constant e is the unique real number a such that the tangent line to the graph of the exponential function y = a^x at x = 0 is the line y = 1 + x."

The way it is currently written, it looks like you need to know how to calculate e^x. Written this way, it is now clear that you need to know how to calculate a^x for an arbitrary a, how to find the tangent line at x=0, then locate the unique value of a that makes the tangent line what you want. However, if you don't know how to find the tangent line, good luck finding e. I think a more constructive definition such as either

a) the unique real number $a$ such that $\lim _{h\to 0}{\frac {a^{h}-1}{h}}=1$

or

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

are better. Although these definitions require an understanding of limits, either seem more accessible than the definition now. In fact, the key concept behind the definition written now lies within suggestion a). Since both definitions are equivalent and I think b) is the easier to understand, I suggest to replace the sentence with

"The mathematical constant e is the real number defined by $\lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n}$. An approximate value of e can be found by calculating this expression for a large value of $n$, for example $\left(1+{\frac {1}{1000}}\right)^{1000}=1.001$ multipled by itself $999$ times $=2.71...$"John Lawrence 16:38, 15 October 2007 (UTC)

On the second point, your phrasing referring to the slope seems much better to me than stating the equation of the tangent line. The slope is the key thing (linking in to the importance in calculus), and the equation of the tangent line is just a relatively unimportant consequence of that. Matt 00:04, 7 July 2007 (UTC).

To condense that into a concrete suggestion, then, you would like to see the sentence rephrased in terms of the slope? Ok. But this alone doesn't seem to justify such a lengthy debate. How would you define e in the first line? The original version, which I favor, is here. Silly rabbit 00:13, 7 July 2007 (UTC)

Well, on the first point I wrote one sentence querying the use of the tangent line equation in the definition, you suggested something else, and I wrote two sentences saying I preferred your version. I don't see how that constitutes a "lengthy debate". Wd u rthr i rote lk ths? I would define e as equal to 1/0! + 1/1! + 1/2! + ..., but I didn't want to suggest that as it is seems less accessible than what we have at the moment. There are potential pitfalls with any definition that relies solely on expressions such as a^x, because to know what that means we first have to define what the "^" operator means for real numbers, and usually that itself involves defining exp and log first (though you could get around it by using the limit of rational approximations to x I guess). I am no mathematician though. Matt 00:53, 7 July 2007 (UTC)

I'd suggest changing it to be more like that caption, omitting "the exponential function" and the equation for the tangent line. Dicklyon 00:32, 7 July 2007 (UTC)

Again, without referencing the exponential function, how are we supposed to know what something raised to the power of x means when x is not a rational number? Is it OK to just gloss over this? Matt 00:58, 7 July 2007 (UTC).

In e^x it doesn't matter that e is irrational, and the x can be dealt with by taking the slope using a limit approached by a sequence of rational values of x that approaches 1, like e^((N+1)/N). So, sure sweep it under the rug is OK by me. Dicklyon 01:23, 7 July 2007 (UTC)

With my limited experience, I would expect a differentiable function to be defined for all real x (in the relevant range), but technically I don't see why your method shouldn't work. I've changed the intro along the lines suggested to refer to the slope rather than the equation of the line, and remove the reference to "the exponential function". Please feel entirely free to change further as you see fit! Matt 02:05, 7 July 2007 (UTC).

Yes, of course. But if you only know how to do rational powers, you can define the value at other points as limits of those, or you can define the deriviative using limits of differences using only rational args. Many ways to get there... Dicklyon 03:06, 7 July 2007 (UTC)

Right. After looking again at the new intro, I've reinstated the link to exponential function, as I think having mentioned f(x) = e^x we really ought to say what that's called. I've tried to do it in such a way as not to give the impression that "the exponential function" is a whole family of functions, which was my quibble with the original text in this respect. Matt 19:47, 7 July 2007 (UTC).

GA review Mk.2

The reviewer(whoever he/she will be) may add comments here.--Cronholm¹⁴⁴ 06:31, 18 July 2007 (UTC)

Well, it looks like we were passed without any comments. —Disavian (^talk/_contribs) 20:36, 18 July 2007 (UTC)

That's kind of anticlimactic. :P Silly rabbit 20:40, 18 July 2007 (UTC)

Now hold on just one minute. Did you actually think i wouldn't make a statement? ;-) I would like to start off with excellent job everyone. Give yourselves a pat on the back. I noticed that you went into depth and included plenty of clear equations showing how E works. This and the writing prose made it over all a good article. Consequently the article gets a shiny. Dagomar 21:39, 18 July 2007 (UTC)

Another graphical example

I have always liked the function $x^{y}=y^{x}$ . It is multi-valued, with the trivial solution being x=y. But there is a second curve that looks somewhat like the first-quadrant rectangular hyperbola of y=1/x (See Image:HyperbolaRect01.png), except that it asymtotically approaches x=1 and y=1 instead of x=0 and y=0. The straight line and the curved line intersect at (e,e). It is probably not very satisfying to the purist, but graphically it is very compelling. Perhaps we should include this in the article. Objections?--SallyForth123 07:48, 4 August 2007 (UTC)

I like that function as well—I'm trying to remember the name of the paper I read that discussed that function in some detail, in case you need it as a ref. Perhaps it will come to me. In any case, make a picture and add a paragraph to the article!

CRGreathouse (t | c) 01:09, 14 August 2007 (UTC)

Exponential function definition

I was looking through this page to verify that $e^{x}$ can also be defined as $lim_{x\to \infty }(1+{\frac {x}{n}})^{n}$. I couldn't find this information for quite a while until I came upon the exponential function page ( http://en.wikipedia.org/wiki/Exponential_function ). Since the series definition for the exponential function, e^x, is included on this page, it might be relevant to include this second definition as well. At the very least, a "see also" section containing a link to the exponential function on wikipedia would be useful. Jamned 04:04, 16 August 2007 (UTC)

missing limit formula

From the limit formula

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

follows the useful fact that

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

I miss this formula in the article.

The two formulas may be written together as.

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

Bo Jacoby 07:03, 17 August 2007 (UTC).

Definition is circular?

The definition given of "e" in the opener is circular as it involves e^x. With that we don't need to bother with a derivative, we can just say what's the value of e^x at 1 since anything to 1 is itself? But that's of course still just as circular. Although "e^x" can be defined without first defining "e", it just looks bad since we see the symbol "e" in both places. mike4ty4 18:27, 17 August 2007 (UTC)

Various people have already attempted to make a good non-circular one-line definition. To address your concern, the definition could be rephrased to something like: The number e is the unique value of the constant a for which the function f(x) = a^x has slope 1 at x=0. I believe this is already implicit in the definition, but if you prefer, it can also be spelled out more explicitly. (An earlier version defined e in terms of the natural logarithm. This was also not circular, but other editors seemed to feel that it was.) Silly rabbit 20:29, 17 August 2007 (UTC)

A similar thing is done with the definition of natural logarithm when the author states:

In simple terms, the natural logarithm of a number x is the power to which e would have to be raised to equal x

Here, one function is defined by its inverse.

Zgozvrm 22:04, 24 August 2007 (UTC)

The definition is not circular unless the circle is closed, e.g. if ln were defined in terms of exp and exp in terms of ln. It is OK to define ln as the inverse function of exp provided that exp is defined in a way that does not depend on ln, e.g. define exp as the limit of a power series. JRSpriggs 01:59, 25 August 2007 (UTC)

GA status reviewed

As part of GA Sweeps, I have reviewed the status of this article. I believe this article is accurate due to my background in calculus, but requires more references. Therefore, I decided to keep this article as GA. OhanaUnited^{Talk page} 02:26, 1 September 2007 (UTC)

infinite series

number e is the sum of the infinite series e=1/0+1/1+1/2-------- in the summation 1/0 is it not equal to infinity. am i missing something70.131.68.114 21:44, 2 October 2007 (UTC)g.g.subramanian

That's $e=\sum _{n=0}^{\infty }{\frac {1}{n!}}$, where the "!" represents the factorial function. — Arthur Rubin | (talk) 22:04, 2 October 2007 (UTC)

First Sentence

"In mathematics, e denotes one of the most important irrational numbers in mathematics,[1] along with the additive identity 0, the multiplicative identity 1, the imaginary unit i, and π, the ratio of the circumference of any circle to its diameter. e can be defined in a number of ways, several of which are shown below."

To me, this sentence seems to imply that 0, 1 and i are irrational numbers, which they are not. Or am I missing something? 222.153.7.37 (talk) 15:41, 7 December 2007 (UTC)

Suggestion from history

More clarity may be obtained by using #5 in the alternative characterizations. After all, when Gregoire de Saint-Vincent set about making a quadrature of the hyperbola xy = 1, he discovered the facts we now associate with the Natural logarithm. Though we teach calculus with derivatives first, then integrals, in this case the integral and natural logarithm are the definitive concept. The number e just happens to be the projection point on the asymptote where the quadrature reaches one.Rgdboer (talk) 22:21, 13 December 2007 (UTC)

The Intro is inadequate

The current version of the Intro is weak and inadequate for all the reasons mentioned above, and more. I emphatically dissent from any claimed "consensus" and submit that the Intro should read as follows:

"e denotes one of the most important numbers in all of mathematics, The only numbers of comparable significance are the additive identity 0, the multiplicative identity 1, the imaginary unit i, and π, the ratio of the circumference of any circle to its diameter. e can be defined in a number of ways, several of which are shown below.

"e is one of the two most important irrational numbers in mathematics (π being the other). Hence e cannot be stated as a finite or repeating decimal. The first 20 digits in the decimal representation of e are:

2.71828 18284 59045 23536...

"e is also transcendental.

"e occurs frequently in calculus, differential equations, analysis, the theory of complex numbers, probability, statistics, physics, chemistry, engineering, and economics. e appears in the exponential function e^x, one of the most common functions in all of mathematics. This function is potentially useful whenever change over time, or growth and decay, are treated mathematically.

"e is occasionally called Euler's number to honor the Swiss mathematician Leonhard Euler who first proposed the notation e and who discovered many results that involve it. e has also been called or Napier's constant in honor of the Scottish mathematician John Napier who introduced logarithms. (e is not to be confused with γ – the Euler–Mascheroni constant, sometimes called Euler's constant.)"

This entry also does not convey much of the richness and beauty of Eli Maior's remarkable book. Getting this entry right is very important, because while most of us finish secondary school with some idea of the centrality of π, that is not the case for e. Only in grad school did I see the light re e.132.181.160.42 (talk) 00:46, 14 December 2007 (UTC)

right align column

on the column of known digits at certain times, I believe we should right align it to allow easier comparisons of order of magnitude —Preceding unsigned comment added by Michael miceli (talk • contribs) 04:42, 2 January 2008 (UTC)

Rational approximation of e

Maybe I'm not looking hard enough, but is there a rational approximation of e, such as 22/7 for pi? --Steerpike (talk) 16:36, 20 January 2008 (UTC)

The continued fraction doesn't have early big terms like the one for pi has, so you don't get any amazing accurate convergents. But they're all tabulated: http://www.research.att.com/~njas/sequences/A007676 and http://www.research.att.com/~njas/sequences/A007677, which has:

Numerators: 2, 3, 8, 11, 19, 87, 106, 193; Denominators: 1, 1, 3, 4, 7, 32, 39, 71; which gives 2/1, 3/1, 8/3, 11/4, 19/7, 87/32, 106/39, 193/71, the last two of which are 2.71794872 and 2.71830986. The next one needs much better integers to get another half digit of accuracy: 1264/465 = 2.71827957. Dicklyon (talk) 18:38, 20 January 2008 (UTC)

Thank you! --Steerpike (talk) 20:03, 20 January 2008 (UTC)

Also see Talk:E (mathematical constant)/Archive_2#Inaccurate_value.3F -- Dominus (talk) 14:37, 5 March 2008 (UTC)

From the cited continued fraction in the main article

$e=[[1,{\textbf {0}},1,1,{\textbf {2}},1,1,{\textbf {4}},1,1,{\textbf {6}},1,\ldots ]]\,$

e clearly has a pattern that pi lacks. After the 1st term we have 1/1; the 4th, 3/1; the 7th, 19/7; and the 10th, 193/71. To get the next important rational approximation to e, note the following patterns:

1 + 3*6 = 19 and 1 + 1*6 = 7. Also, 3 + 19*10 = 193 and 1 + 7*10 = 71.

As Dicklyon points out, the next fraction is 1264/465; this is followed by (193+1264)/(71+465) = 1457/536, followed by the great approximation (1264+1457)/(465+536) = 2721/1001. Guess what?

19 + 193*14 = 2721 and 7 + 71*14 = 1001. Clearly the next important one is

193 + 2721*18 = 49171 and 71 + 1001*18 = 18089. This continued fraction gets them all:

$e=1+{\cfrac {2}{1+{\cfrac {1}{6+{\cfrac {1}{10+{\cfrac {1}{14+{\cfrac {1}{\ddots \,}}}}}}}}}}$

Glenn L (talk) 09:42, 5 March 2008 (UTC)

First sentence (again)

I hesitate to rehash old discussions, but the first sentence does strike me as unideal. Admittedly, there is a chicken-or-egg thing between which is the more primitive definition, that of e or that of raising to a power. But I point out that for general (complex) a, a^x is defined to be e^{ln(a) x}, so there may be some ambiguity here. I always thought e was defined as the root of

$\int _{1}^{x}{\frac {\mathrm {d} z}{z}}=1\;\!$

and that the properties of the exponential function (including its derivative) followed from that.

It's been a while though, so I admit perhaps my memory isn't quite up to the task here. Thoughts? Baccyak4H (Yak!) 19:19, 21 March 2008 (UTC)

This is old ground. Many moons ago, e was defined (in the first sentence) as the base of the natural logarithm. To my mind, this is the cleanest approach to the constant, and the most compelling motivation. However, the inevitable chorus arose, claiming that the definition is "circular" because, it was claimed, the natural logarithm requires the exponential function to define it. Well, of course it isn't true, but one can never satisfy the naysayers. Ultimately, the present suboptimal version of the first sentence was settled on. You're welcome to try your hand at improving it, though. silly rabbit (talk) 19:38, 21 March 2008 (UTC)

Hmm, I guess I would agree, as my so called definition coincides with it being the base of the natural logarithm (and avoiding circularity to boot). But I am not in a boat rocking mood. If I become so, I'll chime in here first. Thanks for the feedback. Baccyak4H (Yak!) 02:02, 22 March 2008 (UTC)

There are many equivalent definitions. But for the base of the natural log be to one of them, you have to have an independent definition of the natural log. You can say it's the log base such that the derivative of ln(x) at 1 is 1, that would be equivalent to the current definition we have, but not as direct. Or you can say that ln(x) is the antiderivative of 1/x, which leads to that integral definition above. Pick one. I like the one we have now because I could see how to illustrate it. Dicklyon (talk) 02:04, 22 March 2008 (UTC)

Yes, I always have taken the natural log to be defined to be the antiderivative as above, which prompted my query here. However, your point about illustration is very well taken, so I will not make the perfect the enemy of the good. Baccyak4H (Yak!) 02:42, 22 March 2008 (UTC)

Defining e

Holly smokes thank you!!! I have been trying to figure out the significance of e for a decade, trying to figure out why it always seems to be the default base in the equations I was assigned in college. I knew the limit definition, but no one ever told me it was the value A for which the slope of A^x at x=0 equals 1. That now explains to me why it was always the default base. Freaking 30 professors at my school couldn't answer this... Thank you sincerely. —Preceding unsigned comment added by 199.197.2.156 (talk) 18:22, 15 March 2011 (UTC)

We need a simpler way to calculate e

I can't figure out most of the math on the page. Can anybody help me? —Preceding unsigned comment added by 66.167.177.92 (talk) 01:59, 6 February 2010 (UTC)

e = (1+1/n)^n where n is infinite, that should probably be included. 64.233.227.21 (talk) 19:33, 23 March 2010 (UTC)

A precise statement of this is already on the page. Note that it does not make sense to say that "n is infinite", since you can't do arithmetic with this kind of infinity. Ozob (talk) 23:52, 23 March 2010 (UTC)

There isn't really a set description which has a value of infinity, I was offering that equation as it is the compound interest equation's result, where n is as high as possible.64.233.227.21 (talk) 18:59, 24 March 2010 (UTC)

You may enjoy reading limit of a sequence. Also, you might consider taking a course in real analysis, where this idea would be discussed in detail. Ozob (talk) 23:52, 24 March 2010 (UTC)

e = 1 + (1/1!) + (1/2!) + 1/3!) +.......

 — Preceding unsigned comment added by 198.86.240.147 (talk) 20:08, 14 September 2011 (UTC) 

Italics

Is there any reason why e is set in italics? See for example the IUPAC guidance. —DIV (128.250.80.15 (talk) 04:15, 7 April 2008 (UTC))

Yes, I'd say that's why. Are you reading it differently than I am? Dicklyon (talk) 04:27, 7 April 2008 (UTC)

Paragraph 7 says "The symbols π, e (base of natural logarithms), i (square root of minus one), etc. are always roman". It really bothers me to see stuff such as $\displaystyle e^{\frac {eV}{k_{\scriptscriptstyle {\text{B}}}T}}$ where the two es refer to different things. -- Army1987 (t — c) 01:17, 11 October 2008 (UTC)

Yep, looks like he was reading it differently that I was; I missed that detail. Should fix... Dicklyon (talk) 03:53, 11 October 2008 (UTC)

OK, I changed e to roman. We'll see if this get supported, or reverted. Support would mean help converting the ones in math mode. Dicklyon (talk) 04:25, 11 October 2008 (UTC)

Judging by a random selection of mathematics texts I've just looked at, it's fairly widespread practice to put e in italics. Additionally, Wikipedia:Manual of Style (mathematics) has this to say:

"On the other hand, for the differential, imaginary unit, and Euler's number, Wikipedia articles usually use an italic font".

Adding in the fact that we now have two competing styles in the same article, roman and italic (which is the sort of inconsistency style manuals advise against), I'm afraid I'm inclined to revert this. I'm sure there are other guides out there that say this is wrong, but the writers of said guides may not take into account that typing "\mathrm{e}" is not going to be popular with people writing formulae in TeX on a near daily basis.

There's an informative discussion on this issue here in which a number of people argue the pros and cons of italics vs roman type, if anyone's interested. Ash (talk) 16:59, 12 October 2008 (UTC)

Very interesting; looks like Tufte decided on the roman in the end; perhaps because of the standard this guy pointed out: "I have just discovered, through downloading a PDF file which Google pointed out under the URL http://www.tug.org/TUGboat/Articles/tb18-1/tb54becc.pdf that not only is making e=2.71828... roman instead of italic no longer forbidden, it is now compulsory... if one wishes to be compliant with ISO standards and recommendations made by the International Union of Pure and Applied Physics." – Does anyone have access to this ISO standard that's mentioned therein? Is the IUPAP same as IUPAC? Dicklyon (talk) 17:58, 12 October 2008 (UTC)

I can answer the last one: International Union of Pure and Applied Physics (IUPAP) and International Union of Pure and Applied Chemistry (IUPAC) are separate organisations. And I believe this is the ISO standard that's being referred to in the discussion. (Funny, I spent some fruitless minutes trying to search iso.org when I should've just checked Wikipedia.) Ash (talk) 19:12, 12 October 2008 (UTC)

Thanks for finding ISO 31-11#Exponential and logarithmic functions. It links a NIST document that explicitly calls for e and pi to be Roman. Of course, there's still the issue of usual practice that we need to reconcile. Dicklyon (talk) 23:40, 12 October 2008 (UTC)

I'd say that the IUPAC is totally wrong here. First of all, this is a mathematics article, not a chemistry article. So the applicability of this guideline is rather tenuous. Secondly, there is no such standard in mathematical literature. On the contrary, as has been discussed in other contexts here on Wikipedia, the recommendation to use roman fonts for mathematical operators and symbols runs completely against well-established almost universally adopted norms in mathematics. siℓℓy rabbit (talk) 17:20, 12 October 2008 (UTC)

That's not my impression. It's true that italic e is pretty widespread, due to the ease of doing it that way in TeX. But roman e is also widespread, where other variables are italic. So there's at least evidence that many people are willing to go to the trouble of following a standard that explains a good reason to distinguish mathematical constants from variables. Dicklyon (talk) 18:00, 12 October 2008 (UTC)

What about using ℯ ("U+212F SCRIPT SMALL E = error; natural exponent") so that people will see whatever their favourite font uses? (Unicode also has ⅇ ("U+2147 DOUBLE-STRUCK ITALIC SMALL E • sometimes used for the natural exponent"), but I've never seen it used anywhere.) -- Army1987 (t — c) 13:05, 18 October 2008 (UTC)

I think it's just likely to get changed back by someone who doesn't have fonts properly installed, like this recent incident over at Sobolev space illustrates. siℓℓy rabbit (talk) 14:27, 18 October 2008 (UTC)