Talk: $e$ (mathematical constant)

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
Date	Process	Result
June 19, 2007	Good article nominee	Not listed
June 21, 2007	Peer review	Reviewed
July 18, 2007	Good article nominee	Listed
August 31, 2007	Good article reassessment	Kept
January 25, 2024	Good article reassessment	Kept
Current status: Good article

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This article has previously been nominated to be moved. Please review the prior discussions if you are considering re-nomination.

Discussions:

E (mathematical constant) → Euler's number, Moved, then reverted, 30 June 2005, discussion
RM, E (mathematical constant) → E (Euler's number), Not moved, 15 May 2011, discussion
RM, E (mathematical constant) → e (number), Withdrawn, 17 August 2017, discussion
RM, E (mathematical constant) → E (number), Not moved, 14 February 2023, discussion
RM, E (mathematical constant) → Euler's number, Withdrawn, 18 March 2023, discussion

Archives

Index 1, 2, 3, 4, 5, 6, 7, 8, 9

This page has archives. Sections older than 90 days may be automatically archived by when more than 5 sections are present.

GA Reassessment

E (mathematical constant)

The following discussion 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.

Article (edit | visual edit | history) · Article talk (edit | history) · Watch • Watch article reassessment page • GAN review not found

Result: Kept. ~~ AirshipJungleman29 (talk) 19:31, 25 January 2024 (UTC)[reply]

This 2007 listing contains significant uncited material, including whole subsections, which means that GA criterion 2, which requires nearly everything to be cited, is not met.

I also think that this article could perhaps use more detail (Pi is around 4x the length) but I am not an expert. ~~ AirshipJungleman29 (talk) 19:13, 30 December 2023 (UTC)[reply]

Three things.

1) The assertion that "whole subsections" are uncited is untrue: every subsection has at least one citation inline.

2) On the question of detail, the article seems to cover "the main points" as required by GACR; comparison with the massive topic of Pi is inapposite, in the manner of OTHERSTUFFEXISTS.

3) On the unstated issue of whether and how "math" – mathematical working – should be cited, I think that before every single math GA is dragged to GAR, the question should be fully aired on an appropriate Wikipedia forum to obtain consensus. The principle that we don't cite "The sky is blue.[21][22][23]" should hold here, as "all the truths of mathematics are tautologies" (Wittgenstein). In other words, any piece of correct mathematical working, like "6/3 = 2" says no more than "A = A", which is about as "the sky is blue" as it is possible to get. If working is to be disallowed, then it is hard to see how even the best, clearest, and most helpful all-math articles can meeat the criteria. Chiswick Chap (talk) 11:42, 21 January 2024 (UTC)[reply]

Chiswick Chap, the reason every subsection has an inline citation is that substantial work has been done over the past few weeks by editors such as XOR'easter and Dedhert.Jr; this was the article's previous state. WP:CALC, which I am fully in agreement with, covers your third point.

I wholly believe maths articles can meet the criteria—see Derivative, which the same two users recently kept at GA standard. The subsection "computational example" there is uncited and allowed per WP:CALC. ~~ AirshipJungleman29 (talk) 18:03, 21 January 2024 (UTC)[reply]

Thanks for letting me know. Good to know there's been progress, and it certainly seems to be a good thing for that to be documented on this page also. Chiswick Chap (talk) 19:03, 21 January 2024 (UTC)[reply]

I think this is good enough to keep. Many thanks to XOR, now very sadly retired, and to Dedhert.Jr. ~~ AirshipJungleman29 (talk) 19:31, 25 January 2024 (UTC)[reply]

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.

Clarification of derivative characterization

In the introduction section, the function $e x$ is characterized as the unique function $f$ that equals its own derivative and satisfies the equation $f (0) = 1$ . The latter part of this is unnecessary information, and implies that there are other, unrelated functions that equal their own derivatives but do not equal 1 at x=0, which is not true. The only other functions that equal their own derivatives are trivial variants of $e x$ like $2 e x$ and $e x +1$ . This would be better rephrased as such, as the fact that the function satisfies $f (0) = 1$ is not inherently useful to a reader. Pradyung (talk) 20:26, 13 March 2024 (UTC)[reply]

which is not true of course it's true: saying "those are just trivial variants" doesn't make them cease to exist. It also has the virtue of being precise and meaningful, whereas your version leaves completely unclear what the phrase "trivial variants" is supposed to mean. --JBL (talk) 20:36, 13 March 2024 (UTC)[reply]

Also, the function

2 e x

is a function that equals its own derivative and is not the function

e x

. This can be verified by looking at the values at 0, which are 1 in one case and 2 in the other one. So, the condition

f (0) = 1

is fundamental, since without it the assertion is wrong. D.Lazard (talk) 21:14, 13 March 2024 (UTC)[reply]

As I mentioned, the condition f(0) = 1 is not a meaningful or unique property, and the only functions other than e^x that equal their own derivatives are e^x multiplied by a real coefficient, which can hardly be considered distant enough from e^x. Pradyung (talk) 01:36, 14 March 2024 (UTC)[reply]

I understand where you're coming from; it's not the interesting part of the specification of the function, and the other solutions, except the constantly zero function, also "involve e" in some sense.

But you need to say something to make the claim well-specified, and your attempt doesn't seem to be an improvement for readability or accuracy. --Trovatore (talk) 01:57, 14 March 2024 (UTC)[reply]

How about "The (natural) exponential function

f (x) = e x

is the unique non-constant function

f

that equals its own derivative (excluding vertical dilations of

e x

); hence one can also define

e

as

f (1)

."? I think this makes it much more clear. Pradyung (talk) 03:40, 14 March 2024 (UTC)[reply]

Again,

x\mapsto e^{x}

and

x\mapsto 2e^{x}

are not the same function, although they are strongly related and their graphs look similar. Moreover, if you consider functions up to a vertical dilatation, the value

f(1)

is not defined, since it changes when applying a vertical dilatation. D.Lazard (talk) 10:10, 14 March 2024 (UTC)[reply]

I'm kind of mystified as to why you think that "makes it more clear". I get that it's possible to read the existing text and be confused about why it's important that f(0) be equal to 1. But I don't think your proposal helps at all. --Trovatore (talk) 00:27, 15 March 2024 (UTC)[reply]

Many readers of this article would have no idea what "vertical dilation" means. XOR'easter (talk) 16:50, 15 March 2024 (UTC)[reply]

Lede dwells too much on details of definition

It strikes me that the lede of the article spends too much space on various definitions of the constant. Currently two whole paragraphs of the lede are devoted to various representations and definitions. I think these could be summarized in plain English, and details deferred to a later section. (WP:MTAA warns against too many equations in the lede, and I think they are certainly excessive here.) Thoughts? Tito Omburo (talk) 23:18, 17 March 2024 (UTC)[reply]

"E (mathematical constant" listed at Redirects for discussion

The redirect E (mathematical constant has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at Wikipedia:Redirects for discussion/Log/2024 April 9 § E (mathematical constant until a consensus is reached. Utopes _{(talk / cont)} 01:48, 9 April 2024 (UTC)[reply]

Base of Natural Logs

We had a discussion as to whether we should say, "e is the base of natural logarithms" or "e is the base of the natural logarithm" and no clear consensus was reached. Nevertheless I thought the agreement was to use the normal English phrase, "base of natural logs." Now this has been reopened by an editor who not only used the phrase, "base of the natural logarithm" but reverted the correction without an explanation. Can we settle this please?

There are an infinity of natural logarithms but they all have one thing in common. If they are natural logarithms then their base is e. If you say 'the natural logarithm' then you mean just one value. But which value? For example the natural logarithm of 2 is 0.69314718…. Its base is e but why choose ln(2) rather than ln(3)? It is much easier to indicate that the base of all natural logarithms is e. Of course the base of the natural logarithm function is e but why use stilted English instead of the common phrase. OrewaTel (talk) 09:59, 15 April 2024 (UTC)[reply]

There are an infinity of natural logarithms (one for each real number), and "the natural logarithms" means all of them considered together. So "e is the base of the natural logarithms" is a mathematically correct formulation, but in "e is the base of the natural logarithm", the singular is definitively wrong. If considered formally, the sentence "e is the base of natural logarithms" means "the base of some natural logarithms". So it is mathematically less accurate, although often used coloqually. As I am not a native English speaker, I am not able to decide whether this colloqual usage is dominant or not. D.Lazard (talk) 10:35, 15 April 2024 (UTC)[reply]

You might want to note, as you did in the archive, that "logarithm" also refers to a unary operation, not just a single number, at least in its common use in English. Tito Omburo (talk) 10:40, 15 April 2024 (UTC)[reply]

In fact, the archive had pretty solid consensus for "base of the natural logarithm", but I would add that this is very standard in introductions on the subject. (For example, Stewart's Calculus.) The article itself uses the phrase "the natural logarithm" numerous times. I would argue also that "base of natural logarithms" is incorrect or misleading, because there is only one natural logarithm, namely ln. Tito Omburo (talk) 10:37, 15 April 2024 (UTC)[reply]

The phrase "the natural logarithm" is used 6 times in the article. The first time is the one we are discussing. The second time it is clearly talking about the natural logarithm function. ("The natural logarithm is the inverse of the exponential function.") Third and fourth times are specific instances. (Which is correct as we can talk about the natural logarithm of a particular number.) The fifth and sixth time explicitly talk about the natural logarithm function. So I ask again, for which specific number are we taking the natural logarithm?

In Stewart's Calculus the chapter heading on page 64 is "Natural Logarithms". The following clause is in the introduction, "The logarithm with base e is called the natural logarithm". Thereafter whenever Stewart talks in general he uses 'natural logarithms' but when talking about a logarithm of a specific number he uses 'the natural logarithm'. Sometimes he writes 'the natural logarithm function'. OrewaTel (talk) 11:24, 15 April 2024 (UTC)[reply]

In common parlance, "the natural logarithm" refers to the function. This is not controversial. Remarks like "for which specific number are we taking the natural logarithm" suggests that you fail to understand this. I suggest that you make an attempt. You can compare ghits if you like [1] versus [2], which show that the former is far more common. Tito Omburo (talk) 11:51, 15 April 2024 (UTC)[reply]

In common parlance, one talk of "the natural logarithm of 3" not of "the value at 3 of the logarithm". For making correct (but rather pedantic) the last formulation, one must add "function" at the end. Also, for centuries, everybody used "table of logarithms", not "table of the logarithm". "Logarithm" is often used as an abbreviation for "logarithm function", but this is an abbreviation that should be used only when it is clear from the context that one talks of the function.

Nevertheless, the counts of ghits are not WP:reliable sources. Similar counts at Scholar Google are more significant, and provide a slight advantage on plural vs. singular.

In any case, you changed (without any discussion) on the 18 March a previous formulation that was previously discussed. I have thus restored the previous formulation.

By doing this, I remarked that on the 23 March, you removed from the lead all definitions of the constant. This may be confusing for many readers, as they should be able to recognize immediatly the common definitions that thy may have already encountered. So, I have restored the most basic definitions in the lead, leaving in § Definitions the definitions that are more technical or require external concepts, such as integrals, inverse functions, etc. I cleaned up also § Definitions for giving the definitions at the beginning of each paragraph, and removing explanations of auxiliary concepts that are linked to. D.Lazard (talk) 14:53, 15 April 2024 (UTC)[reply]

I don't really object to the new lede, less the lengthy second paragraph from the old version. I still feel that "base of the natural logarithm" is more than adequately supported, both by regular usage in mathematics and by sources. And scholar actually shows a slight preference for the singular rather than plural: [3], [4]. This, together with the massively more prevalent ghits of the singular, strongly suggests that the singular is typical in English usage. (See also, in complex variables, one talks of the "branch of the logarithm", also, one talks of the "sine and cosine", the "natural exponential", and so forth, it being understood that one is referring to functions.) Note that at one point, you had agreed that "natural logarithm" refers to the unary operator, rather than just its values. As had User:Trovatore, User:Ancheta Wis, and User:Quantling. So, I'm hardly out on a limb: the earlier consensus clearly favored the singular. As Trovatore wrote:

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.

The direct answer to "which specific logarithm" is "the 'natural' one", as opposed to, say, the "common" one or the "binary" one. Here the "ones" are not numbers, but rather logarithms, a logarithm in this context being the logarithm function to a particular base, rather than a value of that function.

It's difficult to disagree with this. Do you agree? If so, I will restore the old consensus singular. Tito Omburo (talk) 15:21, 15 April 2024 (UTC)[reply]

I've put back in the "previous version that was discussed", per your post, since the outcome of the discussion was "the base of the natural logarithm", with the singular, which was also the earlier consensus version prior to this edit you had made, contrary to any clear consensus I was able to find, with one other editor User:Sapphorain making good arguments for the status quo. Tito Omburo (talk) 15:32, 15 April 2024 (UTC)[reply]

Note also the singular in the French article fr:Logarithme népérien. Tito Omburo (talk) 15:48, 15 April 2024 (UTC)[reply]

"Base of the natural logarithms" sounds better to me, but it turns out that both phrases are used these days in reputable sources,[5][6] and the singular seems to be more popular now. So either one is fine. --Macrakis (talk) 14:53, 15 April 2024 (UTC)[reply]

The French article on e starts, "Le nombre e est la base des logarithmes naturels." There is an article "Logarithme népérien" but that makes it very clear that it is talking about the 'fonction'. So to obviate a silly edit war, I'm adding the word function to the lead. It is stilted English but at least it is correct and makes sense. OrewaTel (talk) 02:51, 17 April 2024 (UTC)[reply]

The lede of that article also includes the sentence: "Ce nombre est défini à la fin du XVIIe siècle, dans une correspondance entre Leibniz et Christian Huygens, comme étant la base du logarithme naturel." Thus clearly, even in French, this usage is quite standard. Tito Omburo (talk) 20:59, 17 April 2024 (UTC)[reply]

The problem with "the base of the natural logarithm function" is that functions generally do not have a base. Logarithms, on the other hand, have a base. For example the natural logarithm is one logarithm, which has base e, and the common logarithm is a different logarithm, which has base 10. Logarithms, in this sense, are not numbers. --Trovatore (talk) 19:00, 17 April 2024 (UTC)[reply]

When you say logarithm you either mean a number or a function. A logarithm (singular) that is a number is a specific instance of the logarithm function. So when you make a statement about 'the natural logarithm' then either you are talking about the function or just one number. If you are talking about a single instance then we must ask, "Which instance and why not all the others?" Consequently the statement that e is the base of the natural logarithm is only meaningful if it refers to the natural logarithm function. OrewaTel (talk) 19:30, 17 April 2024 (UTC)[reply]

Basically "the natural logarithm" in isolation always refers to the function, unless there is an understood value to take the logarithm of. So the sentence "e is the base of the natural logarithm" is clear and unambiguous. The problem with "e is the base of the natural logarithm function" is that the object of the preposition has "function" as the head, and "the base of a function" makes no sense. --Trovatore (talk) 19:34, 17 April 2024 (UTC)[reply]

Agree. Basically, a logarithm is its own thing. Numbers don't have bases; functions don't have bases. Logarithms have bases. Tito Omburo (talk) 20:56, 17 April 2024 (UTC)[reply]

year of discovery

In https://wiki.riteme.site/wiki/Jacob_Bernoulli the year of discovery is stated as 1683 while in this article it is 1685. Can someone clarify the discrepancy. Thanks 192.56.200.12 (talk) 13:34, 18 April 2024 (UTC)[reply]

I think Bernoulli's work wasn't published until 1685. Both articles give 1683 as the discovery date. Tito Omburo (talk) 15:31, 18 April 2024 (UTC)[reply]