Talk:E (mathematical constant)

Italic vs. Roman

I removed the following chuck of recently added text:

Although it is not uncommon to see e printed in italic type ("e"), according to the recommendations of standards bodies such as ISO, NIST and IUPAC, it should not be (because it represents a fundamental constant, not a variable), and rather should always be printed roman ("e").

References

I find this rather opinionated and these references may be outdated, but they do state e should be roman and so may warrant a discussion here. -- [[User:Edokter]] {{talk}} 09:42, 20 February 2015 (UTC)

I agree that the text is problematic. For someone publishing a NIST document, one must obviously adhere to NIST standards. For someone publishing an AMS document, someone must adhere to those standards, etc. These standards are not the same. Contrary to what many believe, NIST does not actually dictate standards for all scientific best-practices. This is especially true of mathematics, which by necessity is rather flexible in the symbols that it uses. Overall such recommendations are irreflective of actual established practice in mathematics publishing. (More than that, in this case the recommendations do not even seem to be self-consistent: for example, in the IUPAC recommendation curl is bold-face but grad is standard face. Clearly mathematicians were not consulted in the preparation of these alleged "standards".) Sławomir Biały (talk) 11:24, 20 February 2015 (UTC)

WP obviously chooses its own conventions (MOS), as it should. Agreed, such a recommendation does not belong. However, a section about notations that occur in general, and which bodies recommend/mandate each notation would not be out of place. The arguments advanced by the references are not without merit, but these should be reported and not adopted as a recommendation in a WP article. —Quondum 18:19, 20 February 2015 (UTC)

unfortunate page title

Isn't it true that the constant is lower-case e rather than upper-case E? If so, this seems to be a bit of a flaw in the way that wikipedia displays page names..

JMWt (talk) 14:37, 14 March 2015 (UTC)

And it appears as such at the top of the page (and this one). It’s a technical limitation of Mediawiki, that it normally doesn’t distinguish between upper and lower case for the first letter of a page name and displays it as upper case. So Cat and cat are the same article (but CAT isn't). Mostly this doesn’t matter, as most names are capitalised and common words like 'cat' are normally capitalised when used as a heading. For exceptions such as this which only make sense as lower case the magic word {{DISPLAYTITLE}} can be used (that's actually a template, but it does the same thing and is how it looks when editing).--JohnBlackburne^words_deeds 14:49, 14 March 2015 (UTC)

I am sorry, perhaps I am being imprecise, but the url for this page is https://en.wikipedia.org/wiki/E_%28mathematical_constant%29 - perhaps it is just my machine, but for me that displays as a capital-E JMWt (talk) 15:10, 14 March 2015 (UTC)

Also that the very top of this talkpage, the wikiproject templates call it "E (mathematical constant)" JMWt (talk) 15:14, 14 March 2015 (UTC)

This is not normally anything we'd worry about as editors. The talk pages and their templates do not have the same degree of format fine-tuning as do the main pages. The purpose of the talk page is to talk about the topic, not to present the topic, and historically editors have seen the wiki markup codes directly when editing. Since the names of articles are case-insensitive to the first letter, and one needs to be aware of that as an editor, one tends to not even notice this. I don't think that it would make sense to change the URL to have a lower-case 'e' in the address bar. With the move to a more WYSIWYG editing interface, perhaps someone might consider tuning the talk page templates, but I would not bet on it. —Quondum 15:48, 14 March 2015 (UTC)

The reference to April 1 1994 is simply wrong...

When one actually reads the reference to the 1,000,000 record of April, 1994, it says that the computation was done to 10,000,000 NOT the claimed record of just 1,000,000!!!! Correct this error please!!!! — Preceding unsigned comment added by 24.110.98.78 (talk) 21:50, 6 April 2015‎

Only the first million were checked, so the reference is correct. -- [[User:Edokter]] {{talk}} 22:15, 6 April 2015 (UTC)

I independently verified Nemiroff & Bonnell's 1994 5,000,000 digits and sent an email to Nemiroff 2015-03-15 telling him that he could remove "(currently unchecked)" from that page. He just hasn't done so. Is there any reason I shouldn't change their 1994 April 1 record from 1,000,000 to 5,000,000? I also asked Nemiroff for their 10,000,000 digit 1994 results (mentioned on the page linked to, but not provided). He replied that he couldn't find those files. Rick314 (talk) 21:25, 28 June 2015 (UTC)

The reason is that everything on Wikipedia needs to be verifiable through third party sources. We cannot accept self-published assertions. -- [[User:Edokter]] {{talk}} 21:35, 28 June 2015 (UTC)

I don't understand -- I am saying I provided 3rd-party verification for Nemiroff, to Nemiroff, 3 months ago. Are you saying the 1994 results can now be updated by Nemiroff but not me, or what? Rick314 (talk) 21:51, 28 June 2015 (UTC)

Erwin (Edokter) please reply. I see you are a Wikipedia administrator and so can provide clarification regarding Wikipedia processes. I provided a link above to Nemiroff & Bonnell's 1994-05-01 5,000,000 digits. I verified their results by comparing all 5,000,000 digits against the output of my own program, and they agree. I told them so in an email 3 months ago. What more has to be done before extending their 1994 milestone from 1,000,000 to 5,000,000 digits? Rick314 (talk) 16:59, 29 June 2015 (UTC)

circular definition

The article says "The number e is an important mathematical constant that is the base of the natural logarithm", and the article on natural logarithm says "The natural logarithm of a number is its logarithm to the base e". This is a circular definition. any ideas on how to fix it? 24.246.91.183 (talk) 05:00, 23 April 2015 (UTC)

The "definition" in the first sentence of the article is not a definition in a mathematical sense. It's just a way for a reader to look at an article and find out quickly what the article is about.

So really I don't see anything that needs to be fixed here. I'm not saying the lead sentences of the two articles are perfect, or necessarily what I would have written, but it's not a problem per se that taken together they give a circular definition, not if you can get genuine mathematical definitions from the articles themselves. Which, I believe, you can. --Trovatore (talk) 05:56, 23 April 2015 (UTC)

The natural logarithm is formally defined as the integral $\ln x=\int _{1}^{x}dt/t$. This is not circular. Sławomir Biały (talk)

talk, Trovatore, and Sławomir Biały, something still needs to be fixed: a math article should have a mathematical tone and mathematical jargon. And Sławomir Biały, where did you get that "formal" definition of e? Dandtiks69 (talk) 23:49, 25 May 2015 (UTC)

Dandtiks69 you need to be more specific about what you think the problem or problems with the article are, and how they can be addressed. I agree with Trovatore, and do not see anything seriously wrong that needs fixing with the lead or definition.--JohnBlackburne^words_deeds 00:36, 26 May 2015 (UTC)

From the lead:

The natural logarithm of a positive number k can also be defined directly as the area under the curve y = 1/x between x = 1 and x = k, in which case, e is the number whose natural logarithm is 1.

From the article natural logarithm, the first line of the "Definition" section reads:

Formally, ln(a) may be defined as the integral,

$\ln(a)=\int _{1}^{a}{\frac {1}{x}}\,dx.$

-Sławomir Biały (talk) 11:26, 26 May 2015 (UTC)

I was confusing the formal definition of e instead of ln (x), sorry. The one for e is the limit, as x approaches infinity, is (1+1/x)^x. — Preceding unsigned comment added by Dandtiks69 (talk • contribs) 21:10, 26 May 2015 (UTC)

Exponential-like functions

I just undid this change to the Exponential-like functions section. Although it made mathematical sense, in that there were no errors that I could see, as a whole it turned a concise and clear section into a mess, which seemed to be trying to do far too much and draw on too many things to be easily understood, touching on an covering material already covered elsewhere, in the main theory sections of the article. As such it was excessive and out of place in this article.--JohnBlackburne^words_deeds 02:22, 5 June 2015 (UTC)

I accept much of your criticism and have re-edited my change to be much shorter and more streamlined. Now using only one example, what the change adds to the article is a connection between an exponential function property and a main theoretical representation of e as a limit. True, the change does reference a main theoretical point covered elsewhere, but only as much as is needed to illustrate the connection. I hope you find the re-edited change satisfactory for the article. Bwisialo — Preceding undated comment added 07:31, 5 June 2015 (UTC)

I also did not find this an improvement. We don't need to illustrate a "connection" of the extrema of functions like $x^{1/x}$ to the mathematical constant e, much less to commit original research in doing so. The global maximum already is at x = e. That's the connection, without any need for embellishment. This is a famous mathematical problem. What's written is already quite standard and clear, without the need to inject our own interpretations. Sławomir Biały (talk) 11:28, 5 June 2015 (UTC)

Let f(x) = (1+x)^(1/x); g(x) = x^(1/x); and h(x) = (n+x)^(1/x)

On the level of explanation and illustration, there is a clear difference between the two following statements that relate a limit property to an extrema property:

1) $\lim _{x\to 0}f(x)=e$ and the global maximum of g(x) occurs at x = e. = e is the connection or shared functional property between these two expressions, and as such does not need to be stated.

2) f(x) and g(x) are instances of h(x), and the extrema of h(x) for 0 ≤ n < 1 form a continuous curve from the global maximum of g(x) until they approach the limit coordinates (0, e) where h(x) = f(x).

(1) treats f(x) and g(x) as two discrete / isolated functions, with the exception of = e. (2) illustrates a continuity of functional properties between the limit property of f(x) and the extrema property of g(x).

On the issue of original research: (2) is is not a synthesis that states a new thesis but is an explanation of sourced statements in a different way. "SYNTH is when two or more reliably-sourced statements are combined to produce a new thesis that isn't verifiable from the sources. If you're just explaining the same material in a different way, there's no new thesis." The statement that f(x) and g(x) are instances of h(x) is a simple and verifiable one. The remainder of the explanation in (2) derives from routine calculations of plugging in values for n, and "Routine calculations do not count as original research."

As such, I argue that my proposed change or something equivalent be added to the section.Bwisialo

I disagree. This does not actually explain anything in the article. One is still left the task of verifying through some method that the maximum of x^{1/x} occurs at x=e. Sławomir Biały (talk) 08:28, 17 June 2015 (UTC)

That the maximum of x^{1/x} occurs at x=e is already stated and verified in the article, prior to my proposed change. Any additional verification of this maximum seems unnecessary and, second, would be a change other than the one I am proposing.Bwisialo

So why is the section enhanced by your proposed revision? It seems like we agree that it's a red herring. Sławomir Biały (talk) 17:24, 17 June 2015 (UTC)

My comments here are admittedly long, but I am trying to address potential misunderstandings, clarify potential confusions, and answer your question.

I feel that you are not extending the principle of charity when interpreting my comments. Obviously, I do not agree that it is a red herring. If I did, I would not propose the change. For my part, if I understand your comments correctly, the stated reasons behind your objection seem to change in an inconsistent way. Your first comment suggests that the change suggests that the change introduces into the article a new thesis and personal interpretation based on original research. As a response to my most recent comment, your last comment seems to suggest that the change is redundant to the verified statements lim_x→0 (1+x)^(1/x) = e and f(x)_max {{x^(1/x) = e. Perhaps there is mutual misunderstanding on these topics.

Your comments do consistently pose the question: what does the proposed change add to the section? I have stated an answer to this, and rather than restate those comments, I will provisionally phrase the answer somewhat differently.

The change neither advances a new thesis nor is it redundant. As I suggest in my previous comments, it explains the expressions lim_x→0 (1+x)^(1/x) = e and f(x)_max {{x^(1/x) = e in a different way than what is presently in the article -– specifically, in a way that illustrates a connection between these two expressions.

Illustrating a connection is different than the statement that both of the these expressions = e. In effect, such a statement merely lists the two as discrete expressions that both fall under a category of expressions that = e.

Exponential-like functions –- including (1+x)^(1/x) –- express central functional properties of e, and the relationships / connections between these functional properties are worth indicating or stating briefly. The relationship / connection between lim_x→0 (1+x)^(1/x) = e and lim_x→∞ (1+1/x)^(x) = e is obvious and is indicated in the article by the use of the word “similarly.” The same applies to the relationship / connection between f(x)_max x^(1/x) = e andf(x)_min x^x = 1/e.

What is less obvious and merits a brief illustration is the relationship / connection between, for example, lim_x→0 (1+x)^(1/x) = e and f(x)_max x^(1/x) = e -- a limit property and an extrema property. The change I am proposing, and what it adds to and enhances in the section / article, is a brief illustration of the relationship / connection between the relevant functional properties of (1+x)^(1/x) and x^(1/x) as they relate to e. I am not suggesting that it is necessary to illustrate every relationship / connection between extrema properties and limit properties of exponential-like functions: the example serves the purpose of illustrating that the connections are there and illustrating one example of such connections. Again, illustrating such relationships / connections consists of something other and more than listing the functions as discrete expressions under a category of expressions of e. Bwisialo

I think I have been charitable in even responding, trying to get you to see the mathematical error involved in trying to say that the existence of this one-parameter family of functions somehow links up the value of a limit with a critical point, to see for yourself thst the entire point is pure mysticism. I see now that further discusdion is a waste of time, since it's clear you plan to continue this pointless discussion regardless. So I'l just be gery clear. The material in question does not belong on Wikipedia. It is WP:OR. If you cannot find a reliable source that clearly and explicitly says that the limit of the function (1+x)^{1/x} at x=0 is related to the critical point of x^{1/x} because of the existence of the one parameter family of functions that you cooked up, it doesn't belong here. It is a novel synthesis, not appearing in published reliable sources. That's not allowed here. Sławomir Biały (talk) 23:20, 18 June 2015 (UTC)

Whether it is original research is debatable, and needs additional editors' comments to achieve consensus. As I have stated and argued above, the verified sources are lim_x→0 (1+x)^(1/x) = e and f(x)_max x^(1/x) = e. The one parameter family of functions is straightforwardly verifiable from the sources, and -- per "SYNTH is when two or more reliably-sourced statements are combined to produce a new thesis that isn't verifiable from the sources. If you're just explaining the same material in a different way, there's no new thesis." -- the proposed change is precisely explaining the sourced material in a different way and is not a novel synthesis. Bwisialo

I beg to differ: "The properties of the two functions can be shown to be continuous with one another via the function (n+x)^(1/x)." This certainly qualifies as a "new thesis". This includes a statement that "properties" are "continuous", whatever that might mean. Supposing that we remove this statement, all that remains is a statement that $\lim _{x\to 0}(1+x)^{1/x}=e$, which already appears elsewhere in the article in context. The section under discussion is a short, precise, and clear discussion of the Steiner problem and Euler's theorem on the infinite tetration. It does not need a red herring about the limit $\lim _{x\to 0}(1+x)^{1/x}$. Sławomir Biały (talk) 01:00, 19 June 2015 (UTC)

What you have quoted is a provisional draft statement that may commit SYNTH unintentionally due to wording. Another provisional, revisable statement could be the following:

The global maximum for the function

$f(x)={\sqrt[{x}]{x}}=x^{\frac {1}{x}}$

occurs at x = e. This functional property of x^(1/x) is related to the limit property of the function

$\lim _{x\to 0}\left(1+x\right)^{\frac {1}{x}}=e:$

the two functions are instances of

$f(x)=(n+x)^{\frac {1}{x}},$

where x^(1/x) occurs at n = 0, and (1+x)^(1/x) occurs at n = 1. From n = 0 to n = 1, the minima and maxima of (n + x)^(1/x) form a continuous curve from the global maximum of x^(1/x) until converging on the limit coordinates of (0, e).Bwisialo

And your reference supporting this new proposed addition...? Sławomir Biały (talk) 02:37, 19 June 2015 (UTC)

The first two functional properties are verified; that the two functions are instances of (n + x)^(1/x) is straightforwardly verifiable; and the final sentence is merely a description of the graphic representation of plugging in different values for n. Bwisialo

Not what I'm asking for. What's the source that the limit of the function (1+x)^{1/x} has anything to do with the extrema of x^{1/x}. Your saying these functional properties are related. If there's no source for this strong claim, you'll need to publish this elsewhere first. We don't accept original arguments. Sławomir Biały (talk) 04:01, 19 June 2015 (UTC)

In using the word "related," I don't intend to mean anything other than what is stated in the subsequent statement in the remainder of the passage. That can be reworded. — Preceding unsigned comment added by Bwisialo (talk • contribs) 04:33, 19 June 2015 (UTC)

The section under discussion is about how e arises as an extremum of x^{1/x}. If what you propose to add is not connected with this after all, then it is redundant with material in the article already. If the family of functions $(n+x)^{1/x}$ is a notable family and published reliable sources have discussed its connection with the mathematical constant e, then we can include some discussion of it in the article. The appropriate context for discussing this family of functions would be determined by how the sources in question make that connection. But this is all hypothetical, because you've been asked to present sources several times, yet haven't done so. Sławomir Biały (talk) 14:04, 19 June 2015 (UTC)

It is more correct to say that the section under discussion is about e: Properties: Exponential-like Functions. The proposed addition is about the functional properties of x^(1/x) and (1+x)^(1/x) considered as instances of (n+x)^(1/x), which exhibits maxima and minima converging toward a limit. As such, e: Properties: Exponential-like Functions is the appropriate context for the proposed change. Again, the proposed change is intended to be a way of "explaining the same material in a different way," which is neither redundant nor asserts a new thesis. It is intended to be a way of explaining the functional properties of x^(1/x) (a maxima property) and (1+x)^(1/x) (a limit property) in relationship to one another.

The issue under dispute seems to come down to this: Does (n+x)^(1/x) need to be in published works in order to be used in the section/article as way of explaining functional properties of exponential-like equations? Does (n+x)^(1/x) count as original research? Ultimately, I consider (n+x)^(1/x) as an explanatory way of representing a set of routine calculations, which fall within the guidelines of acceptable use and which do not constitute original research -- the calculations being: using various values between 0 and 1 and graphing the results.Bwisialo

No, this is not the issue. You need to find sources that relate $(n+x)^{1/x}$ to the mathematical constant e, which is the subject of this article, and that do so in a way that connects the limits of one function to the extrema of another, using this family of functions. As far as I can tell, none of what you have said here actually does what you say it does, namely "explaining the functional properties of x^(1/x) (a maxima property) and (1+x)^(1/x) (a limit property) in relationship to one another." Indeed, all you have shown is that there is a one parameter family of functions between two given functions. That's true for any pair of functions at all, so it cannot be used to "explain" how a property of one is related to a different property of another. Although it's true that $\lim _{x\to 0}(1+x)^{1/x}=e$ and the maximum of $x^{1/x}$ is at $x=e$, you cannot assert that these are related "because the family $(n+x)^{1/x}$". That's a classic non sequitur fallacy, and surely requires a source: WP:SYN. And no, WP:CALC is absolutely not about this. And regardless of how we read "routine" there, a precondition of WP:CALC is that you get consensus, which you clearly do not have. If you disagree, go ahead and ask for clarification at WP:OR/N, but you're not likely to get a very different response there. Sławomir Biały (talk) 14:33, 5 July 2015 (UTC)

As I stated in my previous post, and as I read your post, the issues under dispute concern questions of sources and original research. You are certainly correct that consensus is required. To clarify a few other points, however, I would add the following. First, "explaining" does not necessarily mean, and does not in this case mean, anything more than "describing" something in a particular way. Your addition of "because of" states a new thesis and goes beyond what the proposed change is intended to state. Second, for what it's worth, it is not true that "any pair of functions" can be related to one another as instances of a single-parameter family; and the only single-parameter family that relates x^(1/x) and (1+x)^(1/x) is the one used in the proposed change, though this is not the point of the proposed change.Bwisialo

287,000 digits in 1988 and 1,000,000 in 1992

I (Richard Nungester) found e to 1,000,010 decimal places 1992-02-12. This precedes the currently listed table entry of 1994-04-01 by Nemiroff & Bonnell. I posted the program header in a Usenet post 1993-10-05 that includes the program date, algorithm (with 4 enhancements of my own), execution platform, execution timing, and results (first 20 and last 20 digits). I still have the Turbo Pascal 6.0 1-file 1360-line program with "Modified Date" tag of 1992-02-12. It can still be run. I am a novice at Wikipedia editing and submission guidelines. How should I proceed? Rick314 (talk) 20:54, 21 June 2015 (UTC)

I also recovered my 1988-05-24 files resulting in 287,187 decimal places, done on an HP-150 with 8 MHz 8088 CPU in 30.8 hours. This was over twice Wozniak's prior results with algorithmic improvements explained in the source code file header, so seems worthy of another table row entry. I am planning on uploading the key files to Wikipedia Commons and referencing them as proof. Please let me know if anything else would be expected before I go forward with this. Rick314 (talk) 00:00, 26 June 2015 (UTC)

I don't think it's a good idea to add these. The one beating Wozniak, since it was actually "published" online, seems like it would be more appropriate that the other, unpublished one. Most of the entries of the table are really problematic, falling on the wrong side of sources like WP:SELFPUB, WP:RS. This post is just one indication of why Wikipedia keeping "it's own" list of numerical records is a bad idea: such a list is inherently problematic. I think the table should be reduced, including only those records that are notable in the sense of having been published in reliable sources, or possibly those self-published by experts with a proven publication record. (The latter would be the most generous interpretation of WP:SELFPUB.) Sławomir Biały (talk) 23:13, 28 June 2015 (UTC)

Thank you Slawomir. Your comments and links were very helpful, but I do hope to continue with my table changes. Regarding WP:SELFPUB keep in mind that changes to the table are only a statement about what I did and seem to fit the exceptions given there. WP:RS seems to apply to whole Wikipedia articles not just lines in an existing table in an article. My reference to the 1993 Usenet post (above) and references to the 1988 source code and 1992 source code show dates, algorithms and results (first and last digits, in the file headers). These programs can still be compiled and executed, and I have have 3rd-party verification of my complete output. The 1988 program is clearly the predecessor of the Usenet-described 1992 program (with only its 2 additional algorithm enhancements). So I think I am ready to proceed with the table updates, but further comments are welcome. Rick314 (talk) 02:54, 8 July 2015 (UTC)