# Talk:e (mathematical constant)

Page contents not supported in other languages. E (mathematical constant) has been listed as one of the Mathematics good articles under the good article criteria. If you can improve it further, please do so. If it no longer meets these criteria, you can reassess it.
Article milestones
DateProcessResult
June 19, 2007Good article nomineeNot listed
June 21, 2007Peer reviewReviewed
July 18, 2007Good article nomineeListed
August 31, 2007Good article reassessmentKept
Current status: Good article

## Math Expressions

I don't think the global maximum of the graph x^(x^x) for positive x always occurs at x = 1/e for any n < 0. 2601:182:D81:74F0:CDE6:F67F:9965:4ED1 (talk) 01:40, 24 January 2023 (UTC)

You have copied the function incorrectly. The function $f(x)=x^{x^{x}}$ has no maximum. It becomes infinitely large as x grows.
The function in the article is $f(x)=x^{x^{n}}$ where $n<0$ .
This can be rewritten as $f(x)=x^{1/{x^{m}}}$ where $m>0$ .
That certainly has a maximum value. That it occurs at $x-1/e$ is plausible but that fact should be referenced. OrewaTel (talk) 05:08, 24 January 2023 (UTC)
I've removed it as crufty, likely OR, and unlikely to meet WP:DUE. --JBL (talk) 18:39, 24 January 2023 (UTC)

## Non-trivial

The following statement was made in e (mathematical constant)#Properties

ex is important in part because it is the unique non-trivial function that is its own derivative (up to multiplication by a constant)


The adjective 'non-trivial' was removed as being unnecessary. But there is a trivial function that meets the criterion.

Let $f(x)=0$ then
${\frac {d}{dx}}f(x)=0=f(x)$ An argument has been made that the zero function is not trivial because it is ex multiplied by a constant, namely zero. Of course the zero function is the product of any function by zero but that doesn't stop it from being a trivial case. Consensus please.

Is the zero function a separate trivial case or is it a part of the $e^{x}$ family? OrewaTel (talk) 02:01, 25 January 2023 (UTC)

Well, trivial or not, it is still true that the only functions that are their own derivative are $e^{x}$ times a constant, so I agree that "non-trivial" is not strictly necessary. The statement would also be true with "non-trivial", of course. As to whether "non-trivial" adds enough to the statement to be worth the extra verbiage -- I guess I don't really care one way or the other. --Trovatore (talk) 04:13, 25 January 2023 (UTC)
"the unique function that is its own derivative (up to multiplication by a constant)" is true as stated. There are no functions other than Kex for constant K that are their own derivative. If instead we insert "non-trivial" then aren't we suggesting that there are also trivial functions that are equal to their own derivative but are not of this form?; that's not actually true, right? —Quantling (talk | contribs) 14:51, 25 January 2023 (UTC)
The description «  $e^{x}$ is the unique function that is its own derivative (up to multiplication by a constant) » is exactly correct and doesn’t need any amendment. --Sapphorain (talk) 16:56, 25 January 2023 (UTC)
FWIW I think $e^{2x}$ is equal to its own derivative up to multiplication by a constant; the true statement is that $e^{x}$ is the unique (up to multiplication by constants) function that is equal to its own derivative. It might be better to say something that avoids the jargon-y phrase "unique up to". (I agree that "nontrivial" doesn't add anything and so probably shouldn't be there.) --JBL (talk) 18:15, 25 January 2023 (UTC)

Would it help to make the multiplicative constant explicit, as in:

As in the motivation, the exponential function ex is important in part because it is the unique function (up to multiplication by a constant K) that is equal to its own derivative:
${\frac {d}{dx}}Ke^{x}=Ke^{x}$ and therefore its own antiderivative as well:
$\int Ke^{x}\,dx=Ke^{x}+C.$ Quantling (talk | contribs) 18:46, 25 January 2023 (UTC)

I made the "K" edit to the article (WP:BRD). Thoughts? Please respond there or here. —Quantling (talk | contribs) 16:22, 26 January 2023 (UTC)

## Requested move 14 February 2023

The following is a closed discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a move review after discussing it on the closer's talk page. No further edits should be made to this discussion.

The result of the move request was: not moved. (non-admin closure) 21:16, 21 February 2023 (UTC)

E (mathematical constant)E (number) – Simpler and more recognizable for non-mathematicians; as per lead: "The number e..." Also consistent with other instances at Category:Mathematical_constants. Related terms are already disambiguated at E number (disambiguation). fgnievinski (talk) 19:05, 14 February 2023 (UTC)

Oppose This mostly seems unnecessary, as there is a redirect here, and given that the disambiguation page mentions other numbers associated with E, the title here seems more appropriate. Thenub314 (talk) 21:47, 15 February 2023 (UTC)
Oppose Not only is this unnecessary but, for Europeans, E Numbers have a specific meaning (approved food additive chemicals). e (mathematical constant) is obvious and unambiguous making it more obvious for mathematicians and non-mathematicians alike. A look at Category:Mathematical_constants shows that "name (number)" is an unusual title. OrewaTel (talk) 22:01, 14 February 2023 (UTC)
Comment E Number already exists. Having two separate pages called E Number and E (number) seems silly. OrewaTel (talk) 02:55, 16 February 2023 (UTC)
Of all the parenthetical disambiguations in Category:Mathematical constants, the suffix "(number)" is the most common one, e.g., Category:0 (number), Category:1 (number). fgnievinski (talk) 02:57, 17 February 2023 (UTC)
Of the 85 pages in Category:Mathematical constants only 3 take the form "<name> Number" and these are pages such as Plastic number - only one has an actual number namely 6174 (number). There are 2 sub-categories, Category:0 (number) and Category:1 (number). Meanwhile there are 52 pages of the form "<name> constant". OrewaTel (talk) 01:20, 19 February 2023 (UTC)
• Support. Disambiguation parentheticals are support to be the least specific that they need to be, so this is an improvement based on our titling policies. Rreagan007 (talk) 17:36, 15 February 2023 (UTC)
• Support. Some Wikipedia editors have an odd notion of "mathematical constant", as though the word "constant" meant that the number was important in some way. It doesn't; the word "constant" in mathematics means only that it always has the same value. The current title seems to reinforce this misconception, which we should avoid doing. --Trovatore (talk) 18:53, 15 February 2023 (UTC)
• Support. Correct and simpler. --Sapphorain (talk) 21:35, 15 February 2023 (UTC)
• Strongly oppose, per WP:LEAST: the title E (number) may be confusing for most readers, who may suppose that the article is about common uses of numbers, or about elementary arithmetic. So, E (number) is ambiguous, while E (mathematical constant) is not. If the move would be done, we would be faced to an WP:incomplete disambiguation which would require hatnotes in several articles. D.Lazard (talk) 17:12, 16 February 2023 (UTC)
What are these several ambiguous articles, that are not currently linked from E number (disambiguation)? fgnievinski (talk) 02:53, 17 February 2023 (UTC)
• Oppose. Having both E number and E (number) is too confusing. The notion that "mathematical constant" might be confused with meaning "important" doesn't bother me; it doesn't matter to me whether the user reads the article because they were looking for an unchanging number called e or an important number called e. —Quantling (talk | contribs) 17:50, 16 February 2023 (UTC)
If I understand correctly, an E number is not a "number called e" (or a number called "E") (regardless of whether the number could be changing or not), so I don't see confusion with that topic. —⁠ ⁠BarrelProof (talk) 21:58, 16 February 2023 (UTC)
There is no possible confusion for people who know what is in the articles, but you cannot ask readers to understand this subtle distinction before searching Wikipedia. WP:LEAST applies here. D.Lazard (talk) 09:23, 17 February 2023 (UTC)
• Support. Under WP:SMALLDETAILS, it's perfectly acceptable for E (number) and E Number to exist as separate articles. The proposed title also improves on multiple of the WP:CRITERIA for article titles: it's self-evidently more WP:CONCISE, it's been shown earlier in the discussion to be more WP:CONSISTENT with similar articles, and it's more WP:RECOGNIZABLE to a general audience. ModernDayTrilobite (talkcontribs) 16:50, 17 February 2023 (UTC)
• Oppose, for the same reasons mentioned by D.Lazard. The current title clearly indicates that the topic is about something nontrivial in mathematics. The proposed replacement makes one think of something having to do with elementary arithmetic or some general thing about numbers, which is confusing and seems ambiguous. The current title is non-ambiguous and clear. I see no reason to change it. PatrickR2 (talk) 02:58, 20 February 2023 (UTC)
• Oppose. Category:Mathematical constants does not show any form with "(number)" except after a literal number, such as 6174 (number). And as already said, E (number) is potentially ambiguous. — Vincent Lefèvre (talk) 22:59, 20 February 2023 (UTC)
• Oppose. This seems like a waste of effort, and "number" seems ambiguous. If giving this a more explicit name than e the most common in the literature is base of the natural logarithm which is quite a mouthful followed by Euler's number, which is unfortunately a proper name rather than a descriptive one (this is a fundamental constant that predates Euler and is pervasive in all areas of mathematics and science). (Euler's constant is less common as a name for e because it refers to a different constant.) e (mathematical constant) seems like a fine compromise. –jacobolus (t) 06:29, 21 February 2023 (UTC)
• Oppose – As pointed out eloquently and repeatedly above, e is not "just a number". "Mathematical constant" has both a separate article and a category for a reason. Also, WP:IAR (and WP:SMALLDETAILS be damned), E number vs. E (number) is just too confusing. Favonian (talk) 14:41, 21 February 2023 (UTC)
The thing is that mathematical constant is a horrible article. It flat-out says things that are not true. --Trovatore (talk) 18:25, 21 February 2023 (UTC)
The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

## Cis(θ)

$\cos \theta +i\sin \theta =e^{i\theta }$ is arguably the most beautiful equation in mathematics. This is so fundamental that the expression cos θ + i sin θ has its own function, namely, cis(θ). Euler's formula can be written more concisely as cis(θ) = e and several good faith edits have added it to the section that defines cis(θ).

These have been reverted because this is the section.

Because this series is convergent for every complex value of x, it is commonly used to extend the definition of ex to the complex numbers. This, with the Taylor series for sin and cos x, allows one to derive Euler's formula:
$e^{ix}=\cos x+i\sin x,$ which holds for every complex x. The special case with x = π is Euler's identity:
$e^{i\pi }+1=0,$ from which it follows that, in the principal branch of the logarithm,
$\ln(-1)=i\pi .$ Furthermore, using the laws for exponentiation,
$(\cos x+i\sin x)^{n}=\left(e^{ix}\right)^{n}=e^{inx}=\cos(nx)+i\sin(nx),$ which is de Moivre's formula.
The expression
$\cos x+i\sin x$ is sometimes referred to as cis(x).

We only need to write Euler's formula once in each section OrewaTel (talk) 21:19, 10 March 2023 (UTC)

Is it your argument that it is the expression cos(x) +isin(x) that is represented by cis(x) rather than the function cos(x) +isin(x); along the lines of 1 + 2 + 3 is a summation but 6 is not (except in a degenerate sense, I suppose)? If it is the expression then I suppose it is okay have cis(x) and eix separated as the article currently does. On the other hand, if it is the function then I'd like to combine the two places where cos(x) +isin(x) appears; so that its equality to eix is given AND that, whether written as cos(x) +isin(x) or eix, it is sometimes called cis(x). — Preceding unsigned comment added by Quantling (talkcontribs) 21:36, 10 March 2023 (UTC)
I'm having trouble following the above discussion, but let me throw in my two cents: The $\operatorname {cis} (\theta )$ notation is not very common, and for good reason, as it's entirely redundant with either the notation $e^{i\theta }$ or the notation $\cos \theta +i\sin \theta$ , depending on which aspect you want to emphasize. For most purposes the expression $e^{i\theta }$ is preferable, as it is shorter and less complicated than $\cos \theta +i\sin \theta$ ; when you want to manipulate it, take derivatives or integrals or whatever, it's usually more convenient. When you want to make the connection with the trig functions, fine, you know $e^{i\theta }=\cos \theta +i\sin \theta$ , and you just apply that fact. Having the abbreviation $\operatorname {cis} (\theta )$ doesn't really seem to accomplish anything — it's not particularly shorter than $e^{i\theta }$ ; it's less convenient to manipulate; and if you want to make the connection with the trig functions, well, they're kind of hidden. The abbreviation doesn't really call them out.
So I think we should mention $\operatorname {cis} (\theta )$ briefly, probably just once, to acknowledge the fact that some sources do use it, but after that there's no particular value in referring to it. --Trovatore (talk) 21:55, 10 March 2023 (UTC)
Update: I took a look at the edit under discussion, and I'm even more confused about how this discussion is supposed to relate to it. Is the debate really over whether the formula should be inline or displayed? Or maybe whether the equality with $e^{ix}$ should be called out explicitly? I suppose I'd come down in favor of "inline", as the displayed takes up a lot of screen real estate for not very much. I'd probably leave out the "equivalently" bit, as it's clear from the preceding text. --Trovatore (talk) 23:21, 10 March 2023 (UTC)
The problem was an edit that added a line that explicitly stated
$cis(x)=e^{ix}$ immediately after, "is sometimes referred to as cis(x)."
I thought that restating the formula just 10 lines after
$e^{ix}=\cos x+i\sin x,$ was a bit soon.
As regards the notation 'cis(x)', it is not particularly useful. I've never used it for real, although I had to use it at school. It should be mentioned because it exists. OrewaTel (talk) 09:23, 11 March 2023 (UTC)
I agree with Trovatore. Therefore, I have moved the disputed sentence to the end of the paragraph, and I have replaced “referred to” with “abbreviated”. After all, the only advantage of 'cis(x)' is to be a mnemonic for Euler’s formula (cis may be read as an initialism of “cos plus i sin”). D.Lazard (talk) 09:47, 11 March 2023 (UTC)

## Requested move 18 March 2023

The following is a closed discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a move review after discussing it on the closer's talk page. No further edits should be made to this discussion.

The result of the move request was: Withdrawn by nominator. Rreagan007 (talk) 22:35, 18 March 2023 (UTC)

E (mathematical constant)Euler's number – Per WP:NATURALDISAMBIGUATION. Rreagan007 (talk) 04:23, 18 March 2023 (UTC)

• Oppose. It's usually known as e, not Euler's number. A great many sources in the literature call it e and nothing else, a decent number say both "Euler's number" and e, while hardly any call it "Euler's number" but not e – with the result that e is the most WP:RECOGNIZABLE. Also, when discoveries are misleadingly named after a person other than the actual discoverer, it's undoubtedly justified to follow the misleading name when it is the clear WP:COMMONNAME, but more dubiously justified when it is just for the sake of disambiguation. Adumbrativus (talk) 07:22, 18 March 2023 (UTC)
• Oppose per 'e' is the clear WP:COMMONNAMEblindlynx 14:57, 18 March 2023 (UTC)
• Oppose per Adumbrativus. 〜 16:50, 18 March 2023 (UTC)
• Oppose. It is good enough as is. —Quantling (talk | contribs) 17:50, 18 March 2023 (UTC)
• Oppose. I would really like to get rid of "mathematical constant" (and indeed the rather awful article by that name) but this isn't the way. --Trovatore (talk) 18:00, 18 March 2023 (UTC)
• Oppose Not only for the above reasons but because of the possible confusion with Euler's constant.
The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

## Base "of the natural logarithm" vs. "of natural logarithms" vs. ...

@D.Lazard: @OrewaTel: and everyone else: WP:PLURAL gives a whole lot of reasons we go with singular over plural, and several of them seem to apply to the present case. Perhaps this is a multi-way decision in disguise?:

1. It is the base of natural logarithms -- sure it is the base of each logarithm in a set, but because this a property of each element of the set, why talk about the set in the first place, rather than a single element of it?
2. It is the base of the natural logarithm -- this doesn't introduce a set. However, natural logarithm can refer to a function or the result of applying that function to a value, and if it is interpreted as the latter then it may leave the reader wondering which value it is being applied to.
3. It is the base of the natural logarithm function -- this clarifies that it is the function we mean, not an application of the function to some value. But maybe it is now too wordy / pedantic.

Additional possibilities? Favorites? —Quantling (talk | contribs) 14:28, 22 March 2023 (UTC)

Off-topic ramble: There was an episode of The X-Files in which Mulder and Scully needed a five-digit code to unlock something, and they settled on 27828 (sic) because, as Scully said, "Euler's number is the basis of all natural logarithms". You know, not just some of them. Almost as good as the one where they found a corpse and were trying to decide if the cause was cyanide poisoning person had drowned in sea water, and she listed "bradycardia" among the symptoms.
I think "base of the natural logarithm" makes the most sense to me. The problem with "base of...function" is that functions in general don't have a base. --Trovatore (talk) 16:58, 22 March 2023 (UTC)
Update: I find on a web search that she actually called it "Napier's constant". --Trovatore (talk) 17:02, 22 March 2023 (UTC)
"Natural logarithm" is both a unary operation (or function), and the result of the operation (similar distinction as between “addition”—the operation, and “sum”—the result). So, “base of natural logarithms” is the correct choice if one wants to emphasize on the results. If one wants to emphasize on the operation, singular is better. General rules for actions and operations suggest to not have an article ("base of natural logarithm”), but, reading Natural logarithm, it seems that these rules are not commonly used in this case, as “the natural logarithm” appears many times. I am not sufficiently fluent in English for choosing between these two versions, and I feel both as unnatural. This is why I prefer the plural without article (side advantage for the short description: it is slightly shorter that the singular with article) D.Lazard (talk) 17:37, 22 March 2023 (UTC)
There are a number of side issues. It is true that 'Logarithm' may be a function but the phrase "The base of a logarithm function is ..." is not used. You can use the phrase, "This function finds the logarithm to base e of a specific number." Equally well you could say that 0.6931... is the logarithm of 2 with base e. But again this is the logarithm of a specific (named) number.
In standard English the correct general sentence is e is the base of natural logs. OrewaTel (talk) 23:27, 22 March 2023 (UTC)
Natural logarithms aren't objects that have a common "base". Natural logarithms, as objects, are just numbers. On the other hand the natural logarithm, in context, is a type of logarithm, and different types of logarithms have different bases. So I disagree with your claim about the "correct general sentence"; the best short solution is in fact "base of the natural logarithm", with no plural. --Trovatore (talk) 02:02, 23 March 2023 (UTC)
Next step? We might have a majority opinion here, but perhaps not a consensus. —Quantling (talk | contribs) 18:56, 23 March 2023 (UTC)