Talk:e (mathematical constant)/Archive 6

I find this rather opinionated and these references may be outdated, but they do state e should be roman and so may warrant a discussion here.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)

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 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 ${\displaystyle \ln x=\int _{1}^{x}dt/t}$. This is not circular.

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,

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

-

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)

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)

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

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)

That exception to SELFPUB is typically for biographical articles (a better link is

talk

) 11:22, 8 July 2015 (UTC)

Slawomir: You say "is typically for", meaning the application of the rule sited is unclear to begin with. "This is not an article about you or your accomplishments" is true, but the lines I will add to the table are exactly about me and my accomplishments just as the other lines in the table are about individual accomplishments. Existing references already violate the logic you are trying to defend. For example, the reference for 1994 Nemiroff & Bonnell says "Email from Robert Nemiroff and Jerry Bonnell" (the 2 people claiming the accomplishment and therefore not a third party) and then refers to a web page (not an email) published by them. The 1999 Gourdon reference says "Email from Xavier Gourdon to Simon Plouffe", and again sites a web page (not an email) written by Gourdon claiming verification by Gourdon (not a third party). I could go on. Just look at what already exists in the table. But I think all those entries should stay. There is no doubt the people listed did what they say they did. Look at my documentation referenced in this discussion above. It is actually better than several already existing table references and I have emails that confirm third-party (not me) verification of my results. Or are you just saying that I can't update the article page but someone else can on my behalf? Could other knowledgeable Wikipedia authors please join in with an opinion? Rick314 (talk) 03:47, 9 July 2015 (UTC)

That is an incorrect reading of SELFPUB that probably should be clarified at the actual policy page. The exemption is only for articles about a person, their works, activities, etc. For example, in an article on a

the reliable sources noticeboard

, but I can basically guarantee that the response there will not be very different from my own.

As for the other entries in the table, I believe that at least some of them should be removed. Xavier Gourdon is a published expert, so this actually would fall under the exemption mentioned at

talk

) 11:43, 9 July 2015 (UTC)

Slawomir: Rather than continuing to discuss this with me, I see you concluded you are right and deleted much of the Known Digits table, substituting the incorrect statement "Since that time [1978], the proliferation of modern high-speed desktop computers has made it possible for amateurs to compute billions of digits of e." Let's continue this discussion at the math project page Wikipedia_talk:WikiProject_Mathematics#E_(mathematical_constant) where your knowledge of the subject and Wikipedia editing rights are brought into question. Then we can return here after that is resolved. Rick314 (talk) 04:53, 10 July 2015 (UTC)

See Wikipedia_talk:WikiProject_Mathematics#E_(mathematical_constant) and I think we are done here. In summary, the answer to my original How should I proceed? is that I first need to have my work published by the right person in the right place (details in the discussion) and then it could be added to the table. The same applies to others whose table entries were removed by Slawomir. Thanks to all who participated in this discussion. Rick314 (talk) 18:10, 11 July 2015 (UTC)

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 ${\displaystyle x^{1/x}}$ to the mathematical constant e, much less to commit

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) ${\displaystyle \lim _{x\to 0}f(x)=e}$ 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. "

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.

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.

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

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 ${\displaystyle \lim _{x\to 0}(1+x)^{1/x}=e}$, 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 ${\displaystyle \lim _{x\to 0}(1+x)^{1/x}}$.

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

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

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

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

the two functions are instances of

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

where x^(1/x) occurs at n = 0, and (1+x)^(1/x) occurs at n = 1. From n = 0 to n = 1, the

And your reference supporting this new proposed addition...?

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.

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 ${\displaystyle (n+x)^{1/x}}$ 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.

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 ${\displaystyle (n+x)^{1/x}}$ 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 ${\displaystyle \lim _{x\to 0}(1+x)^{1/x}=e}$ and the maximum of ${\displaystyle x^{1/x}}$ is at ${\displaystyle x=e}$, you cannot assert that these are related "because the family ${\displaystyle (n+x)^{1/x}}$". That's a classic non sequitur fallacy, and surely requires a source:

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

Yes, consensus is required, and I don't see consensus emerging from this discussion. Instead, you've asserted on the one hand that your one parameter family "explains" something (which it does not), and on the other that it does not (in which case it is irrelevant and so already covered in the article). Here you continue to defend your one-parameter family as being relevant, because you labor under the mistaken belief that this family is uniquely defined. In actual fact, any two real-valued continuous functions on an interval are

homotopic

. The homotopy is not uniquely defined, as you maintain. It is trivial to find many such examples for the pair of functions ${\displaystyle x^{1/x}}$ and ${\displaystyle (1+x)^{1/x}}$. For example, ${\displaystyle nx^{1/x}+(1-n)(1+x)^{1/x}}$, ${\displaystyle x^{n/x}(1+x)^{(1-n)/x}}$, etc... This is why, to mention your example in connection with the mathematical constant e, we need reliable sources that make the connection. If we eliminate the original research from your proposed change, all it contains is the following two pieces of encyclopedically useful information: ${\displaystyle \lim _{x\to 0}(1+x)^{1/x}=e}$ and that the maximum of ${\displaystyle x^{1/x}}$ occurs at ${\displaystyle x=e}$. Both of these facts are already covered in the article.

talk

) 11:38, 8 July 2015 (UTC)

You are correct concerning homotopy. I should have been more specific in using "single-parameter." I should have said, "the only single-parameter, single-term defined family...." As for the remainder of your points, our disagreements have already been stated. And yes, other editors' input would be required for a consensus to emerge. Bwisialo 01:37, 9 July 2015 (UTC)

You're still wrong. For example, ${\displaystyle (n+x+n(1-n)x^{2015})^{1/x}}$, ${\displaystyle ((1-n^{2})+x)^{1/x}}$,${\displaystyle (x+\sin(n\pi /2))^{1/x}}$, (etc...) are each one "term", or at least do not consist of more "terms" than your original example. I suggest that you drop the assertion that the family you've given is somehow special, that invoking it is an exception to

talk

) 13:16, 9 July 2015 (UTC)

I am here because Sławomir Biały pinged

Fair enough. I tried to advance as far as I could arguments in favor of the proposed change, but I understand the arguments against it. Thank you. Bwisialo — Preceding undated comment added 17:17, 9 July 2015 (UTC)

In reverting the proposed change below JohnBlackburne writes, "rv as consensus for change and clear opposition after lengthy discussion." The change below is entirely different than the previously proposed change and it is verifiable with a cited source. There has been no discussion of the change below, and the change below resolves the problem of original research with a cited source. Proposed change:

Several properties of exponential functions can be connected with the exponential inequality

${\displaystyle e^{t}\geq 1+t\,}$

where e^t = (1 + t) only at t = 0.^[1] Based on this inequality expression, e^1/x ≥ 1 + 1/x and, hence, e ≥ (1 + 1/x)^x, such that

${\displaystyle f(x)=\left(1+{\frac {1}{x}}\right)^{x}}$

global maximum

of ${\displaystyle {\sqrt[{x}]{x}}}$ occurs at x = e.

is an increasing function with a horizontal asymptote at y = e. More generally,

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

Additionally, the above exponential inequality can be used solve Steiner's Problem. The expression

${\displaystyle e^{\frac {x-e}{e}}\geq 1+{\frac {x-e}{e}}\,}$

can be reduced to ${\displaystyle {\sqrt[{e}]{e}}\geq {\sqrt[{x}]{x}}\,}$, yielding the solution that the

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

occurs at x = e. — Preceding unsigned comment added by Bwisialo (talk • contribs)

We can use some of this. I have added the inequalities to a new section on properties, to do with inequalities. Arguably these are more fundamental than the connection with Steiner's problem. I have included the proof of Steiner's problem, with edits.

talk

) 12:01, 13 July 2015 (UTC)

Hi, I'm not a mathematician, I came here after reading the article on Conlon Nancarrow, who wrote a piece for player piano ([1]) in the ratio of e:π. From the ==History== section I followed the link to Bernoulli where a note in Jacob Bernoulli#References appears to say that his published solution of 1690 was the answer to his own self-posed problem, published in 1685. To clarify the chronology of this History section, I wonder if this could somehow be worked into the sentence, such as

"The discovery of the constant itself is credited to Jacob Bernoulli,[7][8] who in 1690 published an attempt[9]<ref>Note from the above Bernoulli article</ref> to find the value of the following expression (which is in fact e): ..." >MinorProphet (talk) 12:03, 3 August 2015 (UTC)

Hi, this article seems to suggest that both Euler and Bernoulli discovered the number, in consecutive paragraphs. Also, I think the mentions of compound interest should either be made clearer as to what the link is (in the intro para) Iamsorandom (talk) 22:45, 16 February 2016 (UTC)

Which consecutive paragraphs suggest that both Euler and Bernoulli discovered the number? It's clear that Bernoulli discovered the number, Leibniz used the letter "b" for it, and Euler much later used the letter "e". A detailed discussion of compound interest appears in the relevant section of the article.

Biały

00:08, 17 February 2016 (UTC)

Due to

Virtually no reliable sources have paid attention to the dictates of the ISO on this. Sławomir Biały (talk) 14:18, 2 August 2016 (UTC)

ISO does not dictate - it facilitates. On matters of style, WP is not required to follow reliable sources, though it often does. I see no reason myself not to follow ISO on this. Dondervogel 2 (talk) 09:31, 10 September 2016 (UTC)

Just by curiosity, I checked randomly in 10 maths books of my library. All 10 type the operator "e" slanted, and not upright (by the way, the operators are never in italics in books or articles, they are typed in a slanted font, which is not the same as italics). Personally, I always type the operator e as a letter symbol in a formula when I use TeX (which makes it slanted), and all the mathematicians I know do the same. I don't think ISO helps at all in these matters: it is too often totally disconnected with the reality of scientific publications. Sapphorain (talk) 10:38, 10 September 2016 (UTC)

Wikipedia is required to follow reliable sources. See

Biały

12:47, 10 September 2016 (UTC)

Not on matters of style. Also, i and e are neither variables nor operators - they are mathematical constants, which is precisely why they should be upright. Dondervogel 2 (talk) 13:31, 10 September 2016 (UTC)

Euler's number, the subject of this article, is explicitly mentioned in the guideline. Also, it's just false that we don't follow reliable sources in matters of style. We do follow reliable sources, precisely for the reason that not following those sources would be assigning

WP:UNDUE weight to minority views. In this case, on the matter of how "e" is usually typeset. Sławomir Biały (talk

) 13:40, 10 September 2016 (UTC)

The rule is about single-letter variables and operators. The imaginary unit and Euler's number are just examples, and they are also incorrect examples. As stated, the rule does not apply to mathematical constants and therefore applies to neither e nor i. Dondervogel 2 (talk) 22:25, 10 September 2016 (UTC)

Nope, wrong. It explicitly is about italicization if e and i. This came out of precisely this sort of perennial discussion. There was very strong consensus against this proposal when you made it at

WT:MSM. This apparent denial of the consensus (and black-letter word of the guideline) appears to be tendentious. I consider this discussion closed. The matter was already settled more than a year ago. The appropriate avenue to lobby for change is now an RfC, not to make an end-run around past consensus in this strange way. Sławomir Biały (talk

) 00:44, 11 September 2016 (UTC)

The fact is that mathematical constants are usually typed with a slanted font - as are also all unspecified constants in mathematical formulae. This is easy to check by browsing randomly on math books and papers. It is not the role of wikipedia contributors to decide how mathematical texts should be printed, and certainly not to adopt a style which is used almost nowhere else. Sapphorain (talk) 15:33, 10 September 2016 (UTC)

The purpose of

WP's style guide is "to make using Wikipedia easier and more intuitive by promoting clarity and cohesion, while helping editors write articles with consistent and precise language, layout, and formatting". In other words it really is the role of WP contributors to decide how mathematical texts should be appear in Wikipedia. The criteria are clarity and cohesion, both of which would be well served by use of an upright e if this were followed uniformly throughout the project. Reliable sources are not relevant here unless they promote clarity. Dondervogel 2 (talk

) 22:37, 10 September 2016 (UTC)

Let me add my voice to the consensus that the ISO does not determine usage on WP, the ISO standard is non-standard, and we shouldn't adopt it. --

talk

) 23:17, 10 September 2016 (UTC)

Right. I agree that we're not constrained per se by "sources" on matters of style, but we should generally follow the style used in the mathematical community. What style is used in the community for the number e couldn't be clearer-cut. ISO is often useful, but in this case they screwed up big time. It's an idiotic recommendation, and we should not only not follow it, but we should make it clear that we're deliberately rejecting it because it's nonsense. --Trovatore (talk) 23:37, 10 September 2016 (UTC)

To point out the obvious: adopting a guideline that nobody uses promotes neither clarity nor cohesion. It would be much better to use an italic e throughout the project. Would that be an acceptable compromise, in the name of clarity and cohesion? Sławomir Biały (talk) 00:36, 11 September 2016 (UTC)

I agree the MOS trumps ISO. That is its purpose. It would help clarity and cohesion if you were to listen to my argument for why e and i are out of scope of that particular guideline, because that guideline contradicts itself. If I were to say "All traffic lights are green except the red ones" would you conclude that amber lights were green? I think it is more likely that you would (rightly) ignore the statement because of the self-contradiction. To answer your question directly: if the mos made a clear (ie, non-contradictory) ruling on italicization of mathematical constants, based on consensus, it would help clarity and cohesion to implement that statement project wide. Dondervogel 2 (talk) 08:23, 11 September 2016 (UTC)

The last paper I wrote with a typewriter was in 1987 or 88. Since the 1990s mathematical journal are only accepting manuscripts typed with some brand of TeX. Most of them now even specify in which brand you should submit. And wikipedia also does use a TeX variant. As a result, in all mathematical papers and books (and in wikipedia) all isolated roman letter symbols one types in a formula will invariably appear slanted, whatever they represent. Unless of course one takes the trouble of typing {\rm… }, which nobody does (if an author did such an implausible thing it would most likely be suppressed at copy-proof). If ISO's recommendations, or anyone's recommendations, are not compatible with this simple observation, then they are disconnected from the real world, do not belong to a reliable source, and should not be invoked. In fact nothing at all needs to be invoked in this particular matter. As we don't need any "source" informing us that an apple is a fruit, we don't need any "manual of style" instructing us how to type mathematical constants. Sapphorain (talk) 11:07, 11 September 2016 (UTC)

I do not accept that it is implausible that an author might use correct italicization of variables (italics) and constants (upright). It is my experience that journal copy editors treat all single letter symbols as if they are variables (except unit symbols), which leads to many characters incorrectly appearing in italics. When I point out this error at proof stage, they are in nearly all cases willing to correct it. Dondervogel 2 (talk) 18:30, 11 September 2016 (UTC)

It is your opinion that this is an error. That opinion is not shared by the rest of the world, notably Wikipedia. Sławomir Biały (talk) 18:53, 11 September 2016 (UTC)

That the use of italics for a mathematical constant is in my opinion an error is one thing we can agree on. That the WP guideline is self-contradictory is not an opinion, but a demonstrable fact. Dondervogel 2 (talk) 19:34, 11 September 2016 (UTC)

Yes, you already pointed that. But as also already mentioned, the WP guideline is not needed, and not called for in that matter. The TeX version of WP will type your math formulae correctly - that is, as the majority of professional mathematicians do. Just use <math>... </math>, and save your time and energy by not leading a rearguard action. Sapphorain (talk) 20:09, 11 September 2016 (UTC)

It is not self-contradictory. Since, however, the plain English written there seems to be too difficult for certain editors to parse, I have gone ahead and improved the wording to reflect the established consensus in this matter. I therefore assume that this matter is completely satisfactorily resolved. Sławomir Biały (talk) 20:25, 11 September 2016 (UTC)

I don't really see this edit as an improvement. If a reader is already familiar with logarithms, then the base of the natural logarithm does not need further explication, and if a reader is not familiar with logarithms, then telling them in a confusing way that e is the x-coordinate of a point on that graph also does not seem very clarifying. The first paragraph does say (later) that this means that e is the unique number whose natural logarithm is one. Sławomir Biały (talk) 01:39, 24 November 2016 (UTC)

Please examine v:Calculus I#Natural logarithm and exponential function. It seems to me that the natural logarithm came first and the ${\displaystyle e}$ came a little later. Am I mistaken?--Samantha9798 (talk) 01:46, 24 November 2016 (UTC)

But what does this have to do with the graph of the function? We already say that e is the base of the natural logarithm. Saying that the point ${\displaystyle (e,1)}$ is a point on the curve ${\displaystyle y=\ln x}$ is just an obfuscated way of saying the same thing! Sławomir Biały (talk) 01:49, 24 November 2016 (UTC)

I admit that saying it was the point (${\displaystyle e}$,1) was unwise. I did switch to (${\displaystyle x}$,1), but you undid that as well. As a matter of pedagogy, it makes sense to me that the natural logarithm should be taught first. ${\displaystyle e}$ follows logically from the natural logarithm. The dates are 1618 for ${\displaystyle e}$ and 1619 for some notion of the natural logarithm. You are going to confuse student for ever and ever just because of a few years priority? This is an important matter of pedagogy. Which is easier to learn? Some number theory formula or a graph with a point on it?--Samantha9798 (talk) 02:01, 24 November 2016 (UTC)

You added my idea back into the first sentence in your words. I am comfortable with your wording. I like the new top diagram you added.--Samantha9798 (talk) 02:16, 24 November 2016 (UTC)

I doubt that Napier referred to e as the x-coordinate of a point on a curve in 1618 (or 1619). That requires a good source if we're going to say that. Sławomir Biały (talk) 12:18, 24 November 2016 (UTC)

Could someone add a sentence on how to pronounce this constant ? — Preceding unsigned comment added by Lobianco (talk • contribs) 09:49, 6 November 2016 (UTC)

...It's the letter e. Wouldn't say it's that hard to pronounce. -- numbermaniac (talk) 07:36, 19 March 2017 (UTC)

One suspects that he means the pronunciation of "Euler" 71.84.210.136 (talk) 21:04, 20 May 2017 (UTC)

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