Talk:E (mathematical constant)/Archive 7

Requested move 17 August 2017

 * 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. No further edits should be made to this section. 

The result of the move request was: Withdrawn. No such user (talk) 07:29, 18 August 2017 (UTC)

E (mathematical constant) → e (number) – ...or perhaps e (constant). Phrase "(mathematical constant)" is unwieldy, and as far as I can tell this is the only article using it; disambiguator "(number)". is well-attested in e.g. Category:Integers. "(mathematical constant)" does not add anything particularly specific that "(number)" does not, and the article itself starts with The number e is... WP:NCDAB recommends that If there are several possible choices for parenthetical disambiguation, use the same disambiguating phrase already commonly used for other topics within the same class and context, if any. Otherwise, choose whichever is simpler. For example, use "(mythology)" rather than "(mythological figure)". No such user (talk) 09:45, 17 August 2017 (UTC)


 * e (number) is easily confused with e numbers. e (constant) might also refer to the elementary charge.   Sławomir Biały  (talk) 10:54, 17 August 2017 (UTC)


 * Oppose per Sławomir Biały. That, and I'm familiar with the confusion that happens with the title "E (number)" as the creator of R from E number. (In fact, I think E (number) may need to be retargeted to E (disambiguation) as a R from incomplete disambiguation, but that can be a discussion for another day.) Steel1943  (talk) 14:06, 17 August 2017 (UTC)


 * Oppose. Both other suggested possibilities have been there for over 10 years without seeming to be an issue.  But I don't think that you can compare this to, for example, 42 (number).  There, the "42" always means the same number, but the article with "(number)" is about the number itself, while any others just happen to include that number in some way.  Here, the symbol $e$ can apparently refer to different numbers, so something more specific would seem to be needed.  --Deacon Vorbis (talk) 14:36, 17 August 2017 (UTC)
 * oppose. put it well – both alternatives can easily be interpreted in other ways. The current name works well as most people would describe it as a mathematical constant, not in any other way, and not find it too unwieldy.-- JohnBlackburne wordsdeeds 15:40, 17 August 2017 (UTC)
 * Oppose as per above Power~enwiki (talk) 22:58, 17 August 2017 (UTC)


 * The above discussion is preserved as an archive of a requested move. Please do not modify it. Subsequent comments should be made in a new section on this talk page or in a move review. No further edits should be made to this section.

Perceived circularity in "definition"
On 26 October 2017 Radagasty (1 contribution here) perceived a logical circularity in the "definition" of e, manifest in the first sentence of the current lede. I fully agree to his argumentation and repeated his citations of the addressed phrases in the relevant articles. His perspective (and mine) were brushed aside essentially by - there being more definitions, all being equivalent, offered to select from, - the first sentence being non-technical/descriptive to the lay reader, - favouring one definition via defining ln by integrating its derivative.

To start with why I put quotes around "definition", I'd like to introduce e as a specific real number, which turns up a lot in mathematics in many contexts, not immediately seen as intimately connected. I do think, there is a certain similarity to &pi; or &phi;, turning up also not only near circles, or ratios of lengths. Maybe, one should not overemphasize this, but definitely, e is a real number that deserves an identifier, and it is slightly questionable to me, if this "baptizing" is a full blown "definition".

There being many occasions in math where e turns up, is no excuse to lead the addressed lay reader in the first sentence along the circle, established by wiki-links, "e is the base of ln", and "ln is the log with base e". The claim that the ln is defined in other terms below, is imho of no help to the lay reader, who, highly probable, will have some notions about a log being the inverse to an exponential.

From my superficial knowledge of real analysis I like it to see e nailed down by its absolutely convergent power series. I reason this by all real numbers being accessible by equivalence classes of these, and by them being extremely handy in dealing with the mentioned limits, which allow to fix the appearance of e in the specific exponential, invariant wrt derivation, and the related log, establishing thereby the prominent role of these two specific functions. This all is, imho, no obligatory content for the lede, but mentioning the usefulness of the "defining" power series might be appropriate.

Please, see below my suggestion for a modified lede, which contains the modification I consider to be immediate improvements. I tried to keep the changes as small as possible, and of course, everybody is free to ignore or modify it.

The number $e$ is a mathematical constant, approximately equal to 2.71828. Among many other guises it is the limit of $e$ as $n$ approaches infinity, an expression that immediately arises in the study of compound interest. The constant was discovered by the Swiss mathematician Jacob Bernoulli while studying this topic. It can also be calculated as the sum of the infinite series


 * $$e = \displaystyle\sum\limits_{n = 0}^{ \infty} \dfrac{1}{n!} = \frac{1}{1} + \frac{1}{1} + \frac{1}{1\cdot 2} + \frac{1}{1\cdot 2\cdot 3} + \cdots$$

Prominent examples of the many remarkable appearances of this constant are: The function $(1 + 1/n)^{n}$ is called the (natural) exponential function, and is the unique exponential function equal to its own derivative, so $e$ can be spotted as the unique positive number $a$ such that the graph of the function $f(x) = e^{x}$ has unit slope at $y = a^{x}$. The natural logarithm, or logarithm to base $e$, is the inverse function to the natural exponential function. The natural logarithm of a positive number $x = 0$ can also be defined directly as the area under the curve $k$ between $y = 1/x$ and $x = 1$, in which case $e$ is the value of k for which this area equals one (see image). In this view $e$ is the unique number whose natural logarithm is equal to one.

Also called Euler's number after the Swiss mathematician Leonhard Euler, $e$ is not to be confused with $x = k$, the Euler–Mascheroni constant, sometimes called simply Euler's constant. Occasionally the number $e$ is termed Napier's constant, but Euler's choice of the symbol $e$ is said to have been retained in his honor.

The number $e$ is of eminent importance in mathematics, alongside 0, 1, $\pi$ and $i$. All five of these numbers play important and recurring roles across mathematics, and are the five constants appearing in one formulation of Euler's identity. Like the constant π, $e$ is irrational: it is not a ratio of integers. Also like π, $e$ is transcendental: it is not a root of any non-zero polynomial with rational coefficients. The numerical value of $e$ truncated to 50 decimal places is

Best regards, Purgy (talk) 11:29, 24 November 2017 (UTC)


 * @Purgy.  One detail, concerning your  "is the unique exponential function equal to its own derivation". You intend to say, I suppose: "is the unique antilogarithmic function equal to its own derivation".   For there are infinitely many exponential functions with this property (type  a·ex). -- Hesselp (talk) 15:55, 24 November 2017 (UTC)


 * See exponential function. Antilogarthmic function appears to be a neologism.   Sławomir Biały  (talk) 16:00, 24 November 2017 (UTC)


 * Support proposal. Looks good to me.  I imagine others will want to copy edit it.   Sławomir Biały  (talk) 12:58, 24 November 2017 (UTC)
 * Support although I don't care for the word "guises" in the second sentence. Also (I tried adding this in the past and it was reverted) if displaying to 50 places, it would be nice to include the formatting 2.7 1828 1828 45 90 45 as an easy pattern to remember e to 15 decimal places (this was from an old math textbook I had). There is a natural break that matches with the 50-place representation, which can continue with "23536 02874...." ~Anachronist (talk) 21:12, 25 November 2017 (UTC)

I changed the lede, hopefully, sufficiently respecting the remarks about uniqueness (still avoiding what I call a paleo-logism), guises, and the mnemotechnics. Purgy (talk) 09:08, 26 November 2017 (UTC)

No mnemonics in WP?
@Anachronist, there is obviously adamant opposition to have your mnemonic suggestion show up in WP. :( Sorry, I do not know a solid reason for this, and I tried, at least. Perhaps I can get a reason for this, please? Purgy (talk) 16:37, 29 November 2017 (UTC)
 * It is apparently a known trick in order to remember the first 15 decimals of e, and it is not difficult to find several mentions of this online. Here is a detailed version: . I can't think of any good reason not to mention it somewhere in the page. Sapphorain (talk) 21:02, 29 November 2017 (UTC)
 * That source doesn't seem particularly reliable. But even with one, why include it?  It's not particularly interesting or relevant; it's not even really much of a mnemonic, just a grouping of digits that anyone may or may not find easier than not using.  --–Deacon Vorbis (carbon &bull; videos) 22:04, 29 November 2017 (UTC)
 * I don't think the mnemonic should appear in the lead, since that strikes me as undue weight, not especially helpful in that context, and on the wrong side of WP:NOTTEXTBOOK. We should follow MOS:DIGITS.  If mnemonics are included elsewhere, they should be cited to reliable sources, not blogs, to establish due weight.   Sławomir Biały  (talk) 21:58, 29 November 2017 (UTC)
 * I don't like calling it a "mnemonic", but I see nothing wrong with a grouping of digits that aids memory, particularly in this case when the grouping aligns with the 5-digit grouping. I'll remind everyone that MOS:DIGITS isn't a hard requirement, it's a recommendation from which we can deviate in instances where an alternative grouping has value, as it does here. I see no problem in retaining the easy-to-remember grouping without explanation (which is how I originally saw it displayed in a textbook, no explanation, just a display, and it was obvious why it was displayed that way). ~Anachronist (talk) 02:23, 30 November 2017 (UTC)

I honestly do not care much if the formatting is either way, but my slight preference is to keep the mnemonic and the hint to it (without referring to presidents!). Here are some more thoughts. - Given the sourced 50 decimals I would not need any "reliable source" for making evident the repeating patterns in the first 15 decimals. Personally, I favour grouping in 3 digits (like 2,718,281.828 45 $\approx 1 \text{M}e$), which would suggest to give 48 or 51 decimals ... - The use of the word "mnemonic" may strongly depend on the personal thesaurus of the involved people, but I like to call anything that is capable of plausibly aiding in memorizing a mnemonic device. I claim repetitive patterns are a mnemonic device. - The weight of the specific formatting (three additional gaps) plus giving a hint to the mnemonic is manifest in below 50 characters (well below one average line), so I have a hard time to consider this 1% as undue within the ~40 lines lede. - The effort does not dig into specific techniques how to memorize the patterns, but just makes them evident. Really, I never noticed that pattern before, but I am not much in memorizing any naked facts, and would have been thankful for this in my youth. I belong to those, who do not see the pattern spontaneously.

As said, I do not care, but I am surprised about the harshness of rejection. Sorry. Purgy (talk) 09:33, 30 November 2017 (UTC)

Circular Definition
This article begins with the definition:

"The number e is a mathematical constant that is the base of the natural logarithm: the unique number whose natural logarithm is equal to one."

And, referring to the article on the natural logarithm, we find:

"The natural logarithm of a number is its logarithm to the base of the mathematical constant e, where e is an irrational and transcendental number approximately equal to 2.718281828459."

These two definitions are circular, and, without the numerical approximation of e in the latter, could (save for the description of e as irrational and transcendental), could apply to the common logarithm, or any other base.

In other words, it really only describes without defining. It would perhaps be useful to resolve the circularity. — Preceding unsigned comment added by Radagasty (talk • contribs)


 * I'm not concerned. There are multiple definitions of both terms given in the opening paragraphs.  The opening sentence at Natural logarithm uses a non-technical definition to be more descriptive to the lay reader. power~enwiki ( π,  ν ) 04:13, 26 October 2017 (UTC
 * Both articles give multiple, independent definitions that do not rely on each other. This should hopefully be clear here; two more definitions are given in the next two sentences, and there is further explanation in the body of the article.  --Deacon Vorbis (talk) 04:16, 26 October 2017 (UTC)
 * There is no circularity in the definition given in this article. The natural logarithm is defined here independently of the number e as the indefinite integral of 1/x.   Sławomir Biały  (talk) 10:31, 26 October 2017 (UTC)


 * In any case, the first sentence of the article is not meant to be a formal definition, and it is a mistake to treat the first sentence as such. The first sentence is a general description of the topic for the intended reader. If we want to have a formal definition, it should be lower in the article, such as in the "Definitions" section of natural logarithm. &mdash; Carl (CBM · talk) 14:33, 20 November 2017 (UTC)


 * @CBM. "The first sentence is a general description of the topic for the intended reader.". Sounds very reasonable. So, instead of a first sentence with a link to an article with in its first sentence a link back to $e$,  I propose: "The number $e$ (2.718...) is closely connected with any exponential curve, just as $$\pi$$ (3.141...) with any circle and $$\phi$$ (1.618...) with any golden rectangle."   Together with the basic picture visualizing this connection: curve, asymptote, tangent, and vertical segments of length 1 (at the tangent point), $e$ and 1/$e$ . -- Hesselp (talk) 17:44, 20 November 2017 (UTC)


 * No, because "just as" is deeply misleading when used in this way. --JBL (talk) 20:23, 20 November 2017 (UTC)


 * That is less clear than what we have now. The articles currently say the key facts: e is the base of the natural logarithm, and the natural logarithm is the logarithm with base e. The actual definitions are not circular, but because the natural log and e are closely related, the first sentences of the articles may well refer to each other. &mdash; Carl (CBM · talk) 21:40, 20 November 2017 (UTC)


 * The starting sentence "The number $e$ is the base of the natural logarithm" doesn't make clear at all why the number is somewhere between 2 and 3.  Whereas you can see this clearly by comparing two ordinates in the picture of a (arbitrary) exponential curve and a (arbitrary) tangent.   "Just as" you can estimate $$\phi$$ by comparing length and width in the picture of a golden rectangle.  What could be 'misleading' in this? How to get an exponential curve?  Mark in a grid the points (0, 0.5) (3, 1) (6, 2) (9, 4) (12, 8) and draw a smooth curve by hand.   This curve can be seen as being the graph of the natural logarithmic or exponential function by choosing appropriate scales along axes. (The subtangent of the curve has to be seen as having length 1, etc.)


 * About what is cited as "key facts". The mutual linking can be avoided as well by starting with: "The number $e$ is the base of the exponential function identical to its derivative." Isn't this even more a 'key fact'?   -- Hesselp (talk) 20:54, 21 November 2017 (UTC)

How $e$ is related with every exponential process
Radagasty (26 October 2017) correctly states the circularity in the starting sentences of e (mathematical constant) and Natural logarithm. And three commentators correctly state that there are alternative characterizations later on. As an alternative I propose to start the article with the following rewording of the well known story on continuous compounding (see section Compound interest): "The number $e$ (2,718...) shows up when exponential/organic growth (constant growth rates on equal time intervals) is compared with lineair/anorganic growth (constant increments on equal time intervals) starting at the same moment with the same value and the same rate.  At the moment the lineair proces has doubled the start value, the exponential process reaches 2,718... = $e$ times this value." Visualized in a picture with: an exponential curve and asymptote (horizontal), a tangent, vertical line segments at the tangent point and at the point where lineair growth has doubled the start value, and "1", "2" and "$e$". Support? -- Hesselp (talk) 19:51, 19 November 2017 (UTC)


 * No. That's pretty incomprehensible.  --Deacon Vorbis (talk) 20:15, 19 November 2017 (UTC)
 * No also. Very cryptic and not of any help. Sapphorain (talk) 20:55, 19 November 2017 (UTC)
 * Nope. Also, none of the definitions in the lead is circular.  In particular, the natural logarithm is defined in the first paragraph of the lead independently of the subject of the article.  Also, what you wrote is mathematically wrong.   Sławomir Biały  (talk) 01:15, 20 November 2017 (UTC)
 * Obviously not, it's incomprehensible. The current version, by contrast, is good.  --JBL (talk) 02:44, 20 November 2017 (UTC)
 * No. The proposed change is mathematically wrong also, unless those "equal time intervals" are infinitesimally small. I prefer a simpler definition like "the area under the curve y=1/x is 1.0 in the range x=1 to e." ~Anachronist (talk) 05:53, 20 November 2017 (UTC)


 * @Anachronist. 1. Mathematically wrong? Please clarify this. For an exponential process ('function' in mathematics) can be described by the condition:  $$f(t_1+\Delta t) / f(t_1) = f(t_2+\Delta t) / f(t_2)$$ for all $$t_1$$ and  $$t_2$$, and for all finite $$\Delta t$$.   Yes?


 * 2. The visualizations of both the arbitrary-exponential-curve-with-arbitrary-tangent definition  and  the 1/x-curve-with-square-equals-the-sofa-shaped-region definition, show $e$ as a line segment compared with a unit segment. In my opinion the first construction is more comprehensible and more general ('simpler') than the second. Which argument(s) do you have for choosing the second (with the not at all trivial process of equalizing the sofa-area to the square-area)? -- Hesselp (talk) 14:16, 20 November 2017 (UTC)
 * You seem confused. Exponential functions, like $$f (t)=2^t $$ are the subject of a different article.   Sławomir Biały  (talk) 16:39, 20 November 2017 (UTC)


 * Doesn't 'natural logarithm'  denotes a function as well?  What is the connection between your remark and my two questions to Anachronist? -- Hesselp (talk) 17:44, 20 November 2017 (UTC)


 * You said that every exponential process has the property that "At the moment the lineair proces has doubled the start value, the exponential process reaches 2,718..." That's either wrong or not even wrong.  But in any case, consensus is pretty clearly against this proposal.  Time to move on.   Sławomir Biały  (talk) 18:36, 20 November 2017 (UTC)


 * Agreed. --JBL (talk) 20:23, 20 November 2017 (UTC)


 * Can be said here, what is seen as 'wrong' in the sentence cited by Sławomir Biały? Notice that the first illustration in the section In calculus (with tangents in (0, 1) to the dotted and dashed curves as well), shows three times that doubling the ordinate of the tangent, coincides with a 2.7-fold ordinate of the exponential curve.


 * And this. Can we maintain the following order of discussion? 1. Is the proposed alternative (mathematically) correct? 2. Then, if yes, what are its (dis-)advantages? 3. Then, voting on the desirability of the alternative. -- Hesselp (talk) 20:57, 21 November 2017 (UTC)


 * Here is a really simple thing about Wikipedia that you should learn as soon as possible: if you propose an edit to an article and immediately a half-dozen people express clear and unambiguous disagreement, the likelihood that you will get what you want is 0. This is true regardless of how much effort you spend arguing about it.  There are all sorts of ways that people get changes made to Wikipedia, but this is absolutely not one of them.  Moreover, you should at this point have collected enough data points to understand that the plausible outcome of continuing to behave in this way is that your freedom to edit becomes increasingly restricted until you are banned entirely.  I would prefer that you instead learn to accept when consensus is against you and stop the tedious arguing about edits that are never going to happen. --JBL (talk) 21:20, 21 November 2017 (UTC)
 * Nothing wrong with honest inquiry, as long as we don't get into WP:DEADHORSE territory, which it's approaching, I'll admit. The caption in the graph defines e quite succinctly and more simply than this talk page proposal: $$e$$ is the value of $$a$$ such that the slope of $$f(x) = ax$$ at $$x = 0$$ equals 1; and indeed, this is one of several definitions already present in the lead section. The proposed definition doesn't work for any arbitrary "equal time intervals", so in that sense it is mathematically wrong. The lead section is fine as it is, offering a variety of simple ways to define the constant. ~Anachronist (talk) 00:56, 22 November 2017 (UTC)
 * Addendum: I hadn't realized until now that Hesselp has been topic-banned from articles and talk pages related to mathematical series since November 7. Hesselp: You have violated that ban by starting this talk page conversation. I advise you to disregard this conversation and not reply here or anywhere. You need to find other topics of interest. Had I noticed your ban, I would have removed this thread and blocked your account rather than replied. ~Anachronist (talk) 17:50, 22 November 2017 (UTC)


 * @Anachronist. I did not  "start this talk page conversation";   I was number six who came in. And this discussion is not about series(sequences) or sequences(series), but about possibilities to improve the first sentence in $e$(mathematical constant).  Yes, you can use series-representation and series-notation to express number e.  But you can use that representation for any number (and any function), so in your interpretation my current topic ban should regard almost all mathematics.  That’s not what admin TomStar81 wrote me on 7 November 2017.  You (almost) said that you are going to block my account. If you think that is fair in the present situation, and the best for Wikipedia – I cannot stop you.  From my side, I thought (and think) that my successive proposals for the starting sentence of this article, and my attempts to explain them, could contribute to an improvement. -- Hesselp (talk) 22:43, 22 November 2017 (UTC)


 * "Can be said here, what is seen as 'wrong' in the sentence cited by Sławomir Biały?" This is not a classroom debate.  It is not the role of Wikipedia editors to point out your mathematical mistakes (especially not as hints have already been given, like the exponential function $$f(t)=2^t$$).  If you wish to discuss the errors in your mathematics, you can email me.  I charge a standard consulting fee of $500(US), payable as a bitcoin escrow, for my services, should you wish to employ them.   Sławomir Biały  (talk) 13:21, 22 November 2017 (UTC)

More discussion on three proposals for the start of the article
Proposal 19 November 2017  "The number $e$ (2,718...) shows up when exponential/organic growth (constant growth rates on equal time intervals)  is compared with lineair/anorganic growth (constant increments on equal time intervals) starting at the same moment with the same value and the same rate.  At the moment the lineair proces has doubled the start value, the exponential process reaches 2,718... = $e$ times this value." Visualized in a picture with: an exponential curve and asymptote (horizontal), a tangent, vertical line segments at the tangent point and at the point where lineair growth has doubled the start value, and "1", "2" and "$e$".

Five negative reactions: 'pretty incomprehensible',  'Very cryptic and not of any help',   'mathematically wrong',   'incomprehensible',   'mathematically wrong also, unless those "equal time intervals" are infinitesimally small'. As far as I understand, the 'mathematically wrong' by Sławomir Biały is based on his conception that the number $e$ doesn't has to do with exponential curves and exponential functions (or anyway less than with the natural logaritmic function). My argument against this opinion I mentioned here, second sentence. And on the 'mathematically wrong also' by Anachronist: I don't think you can mention one exponential function not satisfying "constant growth rates on equal - finite - time intervals". And not one non-exponential function satisfying it on all interval-pairs. Apart from this, the two short descriptions in parentheses are not essential in this proposal. -- Hesselp (talk) 22:43, 22 November 2017 (UTC)

Proposal 20 November 2017  "The number $e$ (2.718...) is closely connected with any exponential curve, just as $$\pi$$ (3.141...) with any circle and $$\phi$$ (1.618...) with any golden rectangle." Together with the basic picture visualizing this connection: curve, asymptote, tangent, and vertical segments of length 1 (at the tangent point), $e$ and 1/$e$.

With a general description of the topic instead of a complete formal definition, as asked for by CBM. Two negative reactions, only motivated by: - 'deeply misleading' (without any explanation) - 'less clear than what we have now'  ('closely connected with any exponential curve' versus 'base of the natural logaritm'). -- Hesselp (talk) 22:43, 22 November 2017 (UTC)

Proposal 21 November 2017  "The number $e$ is the base of the exponential function identical to its derivative." No reactions on this proposal (close to the present version, without the mutual linking signaled by Radagasty, 26 October 2017) until now. -- Hesselp (talk) 22:43, 22 November 2017 (UTC)


 * No change. Regarding the first proposal, which I am astonished Hesselp is still pushing: Presumably Hesselp believes that the "exponential process" $$f(t)=2^t$$ should grow by a factor of e when the "linear process" doubles its start value.  This requires a very strong source to be believable.  The second proposal is based on a false analogy: the numbers e, π, and &phi; are defined in very different ways, and seems to contain the fallacy of the first proposal albeit less explicitly.  The remaining one is already discussed in the first paragraph of the lead in a much clearer and more explicit way, though Hesselp apparently hasn't read the first paragraph of the article yet because he denies that it does this.   Sławomir Biały  (talk) 12:26, 23 November 2017 (UTC)


 * @Sławomir Biały. You ask for 'a very strong source' for my first Proposal 19 November 2017. My answer: see the subsection Compound interest.  Concentrate in this twenty lines on: $1.00,  $2.00,   $2.71828... and 'continuous compounding' (read this as: 'exponential process'). For a visual analogon: draw a tangent to the exponential curve (at the right of the text), find the point on this tangent with ordinate double the ordinate of the 'starting point' (the point of tangency), and estimate the surplus of the continuous-compounding proces at the lineair-doubling moment. This Bernoulli-source is strong enough?


 * Based on the discussions until now, my favorite opening of the lead should be: Proposal 23 November 2017 :   "The number $e$ (2,718...) decribes the surplus of exponential (continuous compounding, cumulative) growth, over lineair growth with the same value and growth-rate at the start.  At the moment the lineair process has doubled the start value, the exponential process reaches 2,718... = $e$ times this value." With as main arguments that this makes visible: (1) The number shows up not only in abstract mathematics (special 'nice' exponential and logarithmic functions) but as well in every (ideal) organic process. In the case of bacteria (with parent generation P and new generations F1, F2, ...):  (P+ F1+F2+F3+... all new generations) = e·P  when (P+F1) = 2·P.   (2) The value is somewhere over 2. -- Hesselp (talk) 21:29, 23 November 2017 (UTC)


 * Hesselp, I am familiar with compound interest. But that is not what the sentence you wrote conveys.  It says that any exponential process satisfies a certain specific scaling law that involves e.  That is quite simply not true, as the exponential function $$f(t)=2^t$$ clearly illustrates.  So, a source for that statement, as you phrased it, would be required.   Sławomir Biały  (talk) 22:50, 23 November 2017 (UTC)


 * @Sławomir Biały.  The tangent in arbitrary point (u, 2u) on the graph of your function f  (with equation y(t) = 2u + (t-u)·2uln2 )  grows to double value at moment v (with 2·2u = 2u + (v-u)·2uln2  or  v = u + 1/ln2 ). So  f(v) / f(u) = 2u+1/ln2 / 2u = ..... e.   Same result as Bernoulli. -- Hesselp (talk) 09:45, 24 November 2017 (UTC)


 * This is clearer than what you wrote above, but not appropriate for the first sentence of the article. It can be added elsewhere to the body of the article, with a source.   Sławomir Biały  (talk) 12:55, 24 November 2017 (UTC)


 * Additional remarks, on your (paraphrased): It's quite simply not true that any exponential process satisfies a certain specific scaling law that involves e. 1.  I don't understand what you mean by 'a scaling law'.  For by 'an exponential growth/process'  I mean a special way a certain quantity varies in the course of (mostly) time, not depending on any 'scaling' &thinsp;whatever. 2.  The connection of any exponential function with the number e shows up already in the fact that the  e-logarithm is needed to describe its slope.  This leads to  one more variant for the first sentence of the lead:  Proposal 24 November 2017   "The number $e$ is, for every exponential function f, the constant value of  (base of f)&thinsp;^&thinsp;(f'&thinsp;/&thinsp;f)  ."  -- Hesselp (talk) 15:55, 24 November 2017 (UTC)
 * Without the additional clarification, I read your proposal as saying that, if the value is doubled from $$t=1$$ (say), to $$t=2$$, then the exponential process will multiply by a factor of e. That is, $$f(2)/f(1)=e$$.  But this is clearly not true for the function $$f(t)=2^t$$.  One cannot avoid a discussion of the instantaneous growth rate in any case, not the growth rate on equal intervals.   Sławomir Biały  (talk) 16:07, 24 November 2017 (UTC)


 * Improved formulation: - $$f$$ is an exponentional relation (unscaled)  $$\Longrightarrow \ f(t+f(u)/f'(u))\ /\ f(t)\ \,=\ \,e $$ - $$f$$ is an exponential function (domain $$\R$$) &thinsp; $$\ \Longrightarrow \ [f(1)/f(0)]^{f'(0)/f(0)}\ \,=\ \,e $$ -- Hesselp (talk) 12:16, 25 November 2017 (UTC)


 * Further improved(?): For every exponential process $$f$$ (with subtangent $$\mathrm{s}_f$$, the constant value of $$f/f'$$) and for all $$t$$, $$f(t+\mathrm{s}_f)\,/\,f(t)$$ equals  $$e$$ . I'm still trying to find back my source(s). Someone else?   -- Hesselp (talk) 17:31, 30 November 2017 (UTC)


 * Still no change. The new proposed wording is now even more incomprehensible than the earlier proposal, and possibly also even more wrong.  And the equation "$$(P+F1+F2+F3+...) = eP$$" quite literally is a violation of the topic ban concerning series.  The newest formulation has the virtue of being correct, but seems like original research unless sources can be found that introduce e this way.   Sławomir Biały  (talk) 12:56, 25 November 2017 (UTC)

User:Hesselp violation of topic ban
See WP:ANI. Sławomir Biały (talk) 23:11, 22 November 2017 (UTC)
 * In my opinion Hesselp is not in violation of his topic ban with his above contributions. In no way I want to dispute that on several other occasions Hesselp definitely acted in an extremely hard to digest, if not totally unacceptable manner. To me, the above exchange shows a rather biased consensus, not to consider any of his, imho partly sensible and constructive, reservations to the status quo, an ex cathedra declaration of the status quo being better without regarding any other idea, and an even quite remarkable level of threat and retaliation for engaging with some content, evidently considered as own. Purgy (talk) 09:25, 23 November 2017 (UTC)
 * His latest post completely misunderstands what others have already said of his rejected proposals (see his summary of my own objection, Carl's remarks, and JBL's, completely missing the point of each of them). This is exactly the same behavior that lead to his banning.   Sławomir Biały  (talk) 12:28, 23 November 2017 (UTC)

More on topic
- The phrase of the first sentence of the lede "the unique number whose natural logarithm is equal to one" is quite poor in mathematical context, since any logarithm of its base yields "one", and renders this first sentence rubbish, imho. BTW, this has already been mentioned by Hesselp and disregarded above.

- I do would like to see more prominently (not only way below) that the exponential function, with e as basis, is the unique one reproducing itself under derivation, not just the related, but quite local condition "unit slope at x = 0".

In stark contrast to Hesselp, I hopefully, sense early enough, when WP is at the boundaries of its capabilities to accept even sensible changes. Purgy (talk) 09:25, 23 November 2017 (UTC)


 * The phrase in question is what it means to be the base of the logarithm. It is provided as a definition of the base, because not all readers will know what that means.  I've added a mention of the derivative to the lead.  But Hesselp's proposed changes above were not remotely sensible.  Declaring that "At the moment the lineair proces has doubled the start value, the exponential process reaches 2,718... = e times this value" is unacceptable is not "ex cathedra".  That's just a totally unacceptable proposal.  Any reasonable fragments of his other proposals are already discussed in the lead.   Sławomir Biały  (talk) 12:30, 23 November 2017 (UTC)


 * I enjoy your edit about the invariant exponential under derivation, and dare to suggest that the local assertion about the slope could be mentioned as immediate consequence after this global statement.


 * However, I can't help but reading the sentence

The number e is a mathematical constant that is the base of the natural logarithm: the unique number whose natural logarithm is equal to one.


 * as "The number e is ... the unique number whose natural (= base e!) logarithm is equal to one." I do not want to read this in the lede about e. The interpretation of the above sentence that e is thus defined as the base of the natural logarithm seems awkward to me, and "what it means to be the base of the logarithm" is nothing I expect in introducing e.


 * May I suggest to completely omit the text about logarithm in the first sentence and start the lede with the numerical value?

The number $e$ is a mathematical constant that is approximately equal to 2.71828, ...


 * The facts involving the ln would naturally fit below, together with the inverse of the exponential. Purgy (talk) 17:02, 23 November 2017 (UTC)


 * One of the simplest ways to define e is as the base of the natural logarithm, so I don't see why that's a problem.  Sławomir Biały  (talk) 17:17, 23 November 2017 (UTC)
 * Yes, certainly, ... running in circles is big, simple fun:

The natural logarithm of a number is its logarithm to the base of the mathematical constant e, ...


 * More seriously, the introduction via compound interest and even via -hush-hush-Hesselp- via series seems to be more amenable to a majority of readers than the "ex cathedra" simplest way along a circle, touching the concept of logarithms, but, OMG!, that's WP, at its usual best. :D Purgy (talk) 18:08, 23 November 2017 (UTC)
 * Well, the natural logarithm is defined as the integral $$\ln x = \int_1^x \frac{dt}{t}$$, as described here. It's not circular, and we can easily draw a picture of it.   Sławomir Biały  (talk) 21:24, 23 November 2017 (UTC)


 * I think that the proposal to begin with the numerical value is completely reasonable; in that case, what do you think the second half of the first sentence should be? --JBL (talk) 02:24, 24 November 2017 (UTC)
 * By the way,, I think you do your own discussion a disservice by tying it to the obviously terrible proposals by Hesselp. I suggest making this its own section.  --JBL (talk) 02:28, 24 November 2017 (UTC)

Ongoing dissent about mnemonic format
Since there is obviously no accepted, not even a tolerated consent on having a cheap mnemonic in the formatting of a series of decimal digits of e or not, I hope to reopen an explicit discussion on this.

I found two other discussions on mnemonics for e in the archives, but they do not refer to this simple formatting of just the first groups of 5 within a lengthy report of e's decimals. As I perceive it, a grouping of decimal digits in three is widely agreed upon standard in engineering. Even the ISO-normed prefixes adhere to this. Imho, giving more than 4 digits without any grouping is inconsiderate, if not a recklessness at all. The grouping in 5 is, of course, a viable alternative, but foregoing this strict grouping, just at the very beginning, and synchronizing with it at the third group, could, imho, be a tolerable exception, when this exception yields a mnemonic, which is only objected to by a part of the editors. Additionally, I believe that a hint to the mnemonic were necssary in case of its implementation.

I do not care very much about this, but I miss the reasons for the strong aversion to the mnemonic, considering its minimal-invasive looks. Purgy (talk) 12:04, 27 January 2018 (UTC)


 * Based on the discussion in the section above, in which there was only mild objection to the grouping 2.7 1828 1828 45 90 45 for the first 15 digits, I went ahead and grouped it that way, and was immediately reverted for what seemed to be a rather subjective reason that it's "difficult to read". Given that this grouping appears in textbooks, I disagree that the grouping is "difficult to read"; obviously textbook authors, publishers, and students also disagree. I am also curious about the strong aversion. I have not yet seen a logical rationale for it. ~Anachronist (talk) 20:48, 27 January 2018 (UTC)


 * That seems like a very idiosyncratic way of grouping digits. Do other reference works really group the digits in this way?  I note that Donald Knuth's Art of computer programming includes a grouping of digits that agrees with thee one given in this article.   Sławomir Biały  (talk) 22:03, 27 January 2018 (UTC)
 * Noting an extraordinary idiosyncrasy for one side of the dispute, citing just one book using the other notation, is not very argumentative in the case of minimal layout differences. Purgy (talk) 09:26, 28 January 2018 (UTC)
 * The grouping of digits used in the article is also used in Abramowitz and Stegun. Currently 100% of the sources cited in this discussion use the convention currently adopted in the article, with 0% of those sources favoring the proposed change.  That seems like a very strong argument against this change to me.   Sławomir Biały  (talk) 14:28, 28 January 2018 (UTC)

Secretary problem
, the new section you've added contains a true theorem statement and is titled Secretary problem, but the theorem you've written there is not the one that actually goes by that name. They both involve the floor of n/e, so maybe they are related somehow, but certainly it is not standard to use that name for this problem. --JBL (talk) 14:20, 28 January 2018 (UTC)


 * The cited source mentions that it is related to the secretary problem.  Sławomir Biały  (talk) 14:25, 28 January 2018 (UTC)


 * That's fine, but it's not actually the same problem. --JBL (talk) 14:34, 28 January 2018 (UTC)


 * Nor do we say it is. Hopefully this is now clearer in the text.   Sławomir Biały  (talk) 14:42, 28 January 2018 (UTC)

Comments on 'alternative characterization 6'
- The value of the quotient $$f(x) / f'(x)$$ being independent of $$x$$ for an exponential function $$f$$, is mentioned in the third sentence of Exponential function. - And the property that equal absolute increments of the abscissa correspond with equal relative increments/decrements of the ordinate, is as fundamental for exponential functions. -- Hesselp (talk) 15:27, 27 April 2018 (UTC)


 * At least one reference that clearly and directly supports this characterization is required. Ideally, this reference should be a secondary source, showing that the characterization you gave is one that is widely used and accepted, like the others.   Sławomir Biały  (talk) 16:28, 27 April 2018 (UTC)


 * @Sławomir Biały and Joel B. Lewis ('uncited').  About references and sources: Secondary sources of the 'alternative-6' can be found in descriptions of exponential processes (e.g. radioactive decay). As in WP:Exponential decay sentence 5-6: "The exponential time constant (or mean life time or life time, in other contexts decay time or in geometry subtangent) [...] τ is the time at which the population of the assembly is reduced to 1/e times its initial value."   Putting e in front you get essentially:  "The number e shows up as constant growth/decay factor over the life time (f/f' ) of an arbitrary exponential process (f) ".


 * As more primary sources, focussing on the role of the number e in all exponential processes (continuous growth/decay), I mention three articles (in Dutch, in magazines on mathematics for teachers): - Euclides (Netherlands) 1998/99, vol. 74, no 6, p.197/8 - Wiskunde & Onderwijs ('Mathematics and teaching', Belgium) 2001, vol. 27, no 106, p. 322-325 - Euclides 2012/13, vol. 88, no 3, p. 127/8 . -- Hesselp (talk) 16:08, 28 April 2018 (UTC)


 * @D.Lazard. Interesting to see your modification of the first alt-6-version. Rewriting my text into your format, I get: If $$f(t)$$ is any solution of the differential equation $$y' = y/s$$,  then for all $$t$$:    $$ e = f(t+s) / f(t) $$. a. My choiche of t instead of x has to do with my mixed background in physics and mathematics. In my view, an exponential function is mostly a function of time, so t. But if there are better arguments for x,  excellent. The same for $$f(x)$$ instead of the sufficient (but still not everywhere usual?) $$f$$. b. Instead of 'for all t ' and 'for all s ' in my version, you have t = 0 and s = 1.   This leads to the question:  is the general case more or less difficult to grasp for a reader than the special case? (And in between there are the cases with only t=0 and with only s=1 as well.).  I don't comment on this question at the moment; only this: c. The solutions of your differential equation are of the type   a exp(x), not a very common type of exponential function, I think. -- Hesselp (talk) 16:08, 28 April 2018 (UTC)


 * I'm satisfied by the discussion at Exponential decay that something like this could be included as a characterization of e. However, I would still like to see a better source (in English!).  I think some effort should be made to tie it to the articles on exponential growth and decay.  I would rephrase the addition along the following lines to make that relationship clearer:
 * Thoughts? Sławomir Biały  (talk) 19:55, 28 April 2018 (UTC)
 * Thoughts? Sławomir Biały  (talk) 19:55, 28 April 2018 (UTC)


 * @Sławomir Biały. Some remarks on your proposal. i. On "..then the quantity $$f(t)/f'(t)$$..".  Why 'quantity'? why not 'quotient'? Even better: simply "..then $$f(t)/f'(t)$$.." . ii. On  "... $$f(t)/f'(t)$$ is a constant, sometimes called the time constant ..." . The real universal constant is  $$f(t+\tau) / f(t)$$, while $$f(t)/f'(t)$$ depends on $$f$$.  So I propose: "... $$f(t)/f'(t)$$ doesn't depend on $$t$$ (this value is sometimes called the time constant of $γ$ ) . " iii. On  "(it is the reciprocal of the exponential growth constant or decay constant)".   This interrupts the main message, maybe better in a footnote. Or leave it out, for 'the reciprocal of a time interval' I can't see as an elementary concept. iv. I understand that I've to wait until a sufficient number of reliable explicit secondary sources are found, for (maybe) consensus on the introduction or characterization of e as (something like)  "the stretching/shrinking factor of every exponential process (function) over any period equal to its time constant" . -- Hesselp (talk) 10:13, 29 April 2018 (UTC)


 * I think we should wait for native speakers of English to comment on the proposal. Some things about your critique strike me as misunderstanding idioms and grammar.   Sławomir Biały  (talk) 12:13, 29 April 2018 (UTC)


 * I like Slawomir's version. Unlike Hesselp's, it is actually possible to understand, is clearly written, and avoids obscurities.  Good job getting something usable out of this. --JBL (talk) 12:58, 29 April 2018 (UTC)


 * Even when not a native speaker, I want to join JBL's praise of Sławomir Biały's suggestion. However, since it's about charcterizing $e$ and not the time constant, I suggest to amend to




 * Honestly, I think this is mathematically obvious to a degree making additional math sources superfluous, and physics sources should abound. Purgy (talk) 15:28, 29 April 2018 (UTC)


 * Arguments against changing in Purgi's proposal "...is a constant. When ... this constant is often ..."   into   "...is independent of $t$. When ...this value is often ..." ?   To reduce the possibility of misunderstanding. (I know I had 'constant' as well in the first version of alternative 6.) -- Hesselp (talk) 08:36, 30 April 2018 (UTC)

Again: arguments against changing in Purgi's amended proposal: "...is a constant 'for all $f$'. When ... this constant is often called the time constant $$\tau$$ of this process, ..." into "...is independent of $t$. When ...this value is often called time constant  of the process (symbol $$\tau$$), ..." ? -- Hesselp (talk) 16:05, 30 April 2018 (UTC)


 * I prefer Sławomir Biały's version because it does not waste time getting to the connection with e. By comparison, Purgy's version emphasizes and expands on the parts that are least closely related to the topic of this article.  I suggest adding Sławomir Biały's version verbatim.  --JBL (talk) 22:10, 3 May 2018 (UTC)


 * @JBL. Please explain what you mean with "don't randomly break equations just for kicks." (Summary 3 May 2018) And your "More general is not better" isn't clear to me as well, for you advocate Slawomir's proposal using the most general situation. -- Hesselp (talk) 16:23, 4 May 2018 (UTC)


 * Seeing the bare entry now, the notable connection to time constant and decay/growth rate of exponential processes totally stripped off, I revert to D.Lazards longer standing "three"-version. Furthermore, I plead for a more explicit consensus before any other edits on this detail. Reversion already done by JBL. 06:04, 4 May 2018 (UTC) Purgy (talk) 06:00, 4 May 2018 (UTC)

Two proposals
Balancing the proposals, arguments and opinions shown on this talk page until now, could there be consensus on the following 'version 6a' ? Arguments? Ideas for improvement?

Or could there be consensus on the much shorter 'version 6b' ? A compromise of "this only uses the concept of derivative as prerequisites", "properties of exponential functions and terminology that is unrelated with the definition of e",  "emphasizes and expands on the parts that are least closely related to the topic of this article" and "the notable connection to time constant and decay/growth rate of exponential processes totally stripped off". Or the remark in parentheses better in a footnote? then also naming 'exponential growth constant/rate and exponential decay constant/rate? Arguments? Ideas for improvement?

-- Hesselp (talk) 16:23, 4 May 2018 (UTC)


 * Opinion: Positive consensus is required.  I will not be commenting on these specific proposals.  Proposals which already seem already to have positive consensus are in the previous section, and do not require Hesselp's "improvements".  Sławomir Biały  (talk) 11:02, 27 May 2018 (UTC)

Please, improve on the given reasons
I do not think that giving reasons why which entries are given in a table degrades a featured article; maybe reasons even help the more-digits researchers. I just concede that my reasons are a bit tongue-in-cheek. I also think that "trillions" of digits of e are inappropriate in a FA. The revert first - think later approach is often really annoying.

Since that time, the proliferation of modern high-speed desktop computers has made it possible for all those sufficiently interested and equipped with the right hardware, to compute digits of any representation of e up to the lifetime of this hardware.

Please, feel cordially invited. Purgy (talk) 09:26, 5 July 2018 (UTC)


 * [[Image:Information.svg|25px|alt=Information icon]] Welcome, and thank you for your attempt to lighten up Wikipedia. However, this is an encyclopedia and articles are intended to be serious, so please don't make joke edits. Readers looking for accurate information will not find them amusing. If you'd like to experiment with editing, please use the sandbox instead, where you are given a certain degree of freedom in what you write.  –Deacon Vorbis (carbon &bull; videos) 15:20, 5 July 2018 (UTC)


 * As for the bit about trillions, why not? It might not be the very best choice, but your proposed change is wordy, awkward, and gives no indication about the amount of digits that is reasonably attainable.  –Deacon Vorbis (carbon &bull; videos) 15:20, 5 July 2018 (UTC)


 * [[Image:Information.svg|25px|alt=Information icon]] I herewith withdraw all things cordial with respect to Deacon Vorbis. He is just entitled to spit on me his condescending qualification efforts, in the same way as any IP and even vandals are entitled to edit WP.


 * [[Image:Information.svg|25px|alt=Information icon]] To all those, capable to make their good faith perceiveable, I want to reinforce my cordial invitation for improvement of the suggestions I made in absolutely positive intentions. I will try to explicate these in a reply to Owlice1 above. Purgy (talk) 07:02, 6 July 2018 (UTC)

Sharp inequalities
By implication, you're asserting that the following is not a sharp inequality: User:Sławomir Biały, did you really mean that? —Quondum 00:05, 1 August 2018 (UTC)
 * the unique base of the exponential for which the inequality $f(t)$ holds for all $e$ ... is $f(t)$.
 * the unique base of the exponential for which the inequality $f(t)$ holds for all $f(t)$ ... is $t$.
 * the unique base of the exponential for which the inequality $e$ holds for all $t$ ... is $a^{x} &ge; 1 + x$.


 * I don't understand your objection to the property in question. I object to calling it a "mundane observation".  It's an important estimate, and it also uniquely characterizes the number e.  I understood your initial edit summary to be, wrongly, saying that $$a^x>1-x$$ is always true.  (Though, I inferred that you perhaps meant something like $$a^x>-1+x$$ instead, which is not a sharp inequality for any a.)  But, for the line $$y=1+x$$, with both y-intercept and slope equal to unity, the sharp inequality $$a^x>1+x$$ for all nonzero x holds, if and only if $$a=e$$.  How is that a "mundane observation" that therefore justifies removal?   Sławomir Biały  (talk) 00:58, 1 August 2018 (UTC)


 * Maybe you misunderstood my claim, and I fail to see the error in my edit summary ("[...] There are many similarly formed expressions, e.g. a^x > 1 − x, that give different constants"), other than the obvious slip of using ">" instead of the intended ">=". I restated this corrected in the above, but you still may have missed it.  (Your word order above, especially with your comma placement, is difficult to parse.)  The function $x$ is tangent to some graph $e$ for every base $a^{x} &ge; 1 − x$ at the point $x$.  Thus every base $e–1$ satisfies a sharp inequality of this type by choosing a suitable constant $ax$.  In effect, without motivating the particular choice of coefficients, the statement says that we can find a tangent to $1 + bx$, a statement which I consider to be worthy of the description "mundane".  Also, when looked at like this, it says nothing about the specialness of $a$.  I would say that few people are going to say that $(0, 1)$ stands out as clearly more special than say $a$.  —Quondum 12:08, 1 August 2018 (UTC)


 * Precisely the same unmotivated choice enters any definition of the number e. Why shouldn't e be the unique number such that $$d/dx(e^x) = -e^x$$?  Why is $$e=\lim_{n\to\infty} (1+1/n)^n$$ instead of $$\lim_{n\to\infty} (1-1/n)^n$$?  Why is $$e>1$$ instead of $$<1$$?  Perhaps all of these are "mundane observations" that the article is better without?  Ultimately, your objection seems to boil down to the contention that a logarithm can have many bases, so there's no reason to prefer e to any other base.  Hopefully you now understand why I find that silly in the present context.
 * Secondly, as you note, if $$a>0$$ then $$a^x > 1+(\ln a)x$$ for all nonzero x. Conversely, the number $$a=e^b$$ is the unique number such that $$a^x > 1 + bx$$ for all nonzero x.  Here we are simply taking the special case of $$b=1$$, just like all of the other characterizations.  I don't see what that has to do with it being any more mundane than other characterizations present in the article.  I think it's rather interesting that the number e can be completely characterized by sharp inequalities, without any reference at all to calculus.  Don't you?  Sławomir Biały  (talk) 20:31, 1 August 2018 (UTC)


 * I should not have to tell you that notability is a criterion in choosing what to include in an encyclopedia.
 * For this last "rather interesting" motivation to be valid, how do you propose to mathematically define $b$ for real $ax$ and $e$ such that this does not rely on calculus, limits, or something more abstruse, while not implying $1 + x$ as a particularly simple case? —Quondum 23:51, 1 August 2018 (UTC)
 * Lack of notability was not your stated reason for removing it. You removed it because you personally found it mundane.  It can be readily sourced to many standard textbooks, which a moment of research would have verified.  I'm not going to engage here in original research.
 * Obviously, any fact about real functions requires the real number system at some level. But this characterization does not use calculus in the way that others do, because they all involve limits very explicitly.  I had hoped that you might find it interesting that the convexity of the exponential function, an important fact in analysis, makes possible a characterization of e that does not explicitly use limits, derivatives, or integrals.  That is, what is commonly known of as "calculus".  Since you're apparently unable to see why it's interesting on your own, though, much more "mundane" arguments favoring the inclusion of the fact can be given, based on it being a standard thing about e that is covered in most modern calculus textbooks.  Your personal interest is not required, but it might lighten the mood a bit if you tried.  Sławomir Biały  (talk) 23:15, 2 August 2018 (UTC)
 * After some thought, the challenge I posed can be managed algebraically on a restricted domain: $1 − x$. To answer your final question, no, I do not find this interesting.  If I did, I would get more excited about similarly using a sharp inequality to claim that it is possible to differentiate without any reference to calculus.
 * You seem to be prepared to spend inordinate amounts of energy to avoid overtly conceding that you might have acted in a way that comes across as possibly thoughtless or inconsiderate. My motivation for spending so much energy I need to review: I do not find debate (or co-editing) with you rewarding.  —Quondum 19:03, 2 August 2018 (UTC)
 * Ok, fair enough. I strongly disagree with your removal of the material, and have stated my reasons.  Your agreement is not required.  The fact that it has been restored with a source is now sufficient.  If you have further doubts about the content, I can readily supply more sources.   Sławomir Biały  (talk) 23:15, 2 August 2018 (UTC)

Overspecified?
I find the description
 * The function $ax$ is called the (natural) exponential function, and is the unique exponential function of type $a$ equal to its own derivative ($x$). This is easily spotted at $exp(x)$, where $R>0 × Q → R$ for any value of $f(x) = e^{x}$, but $a^{x}$ is the only positive number such that the slope at $f(x) = f'(x) = e^{x}$ of the graph of the function $x = 0$ also equals $a^{x} = 1$.

clumsy, especially in that it overspecifies the premise. Were I to copyedit this, keeping in mind that this article is not primarily about the exponential function, it might become something like
 * The only real or complex function that is equal to its own derivative – i.e. such that $a$ – is $a = e$, or a constant multiple thereof. This leads naturally to the exponential function, which has as its base the number $x = 0$.

I see that it is very neatly stated in e (mathematical constant). —Quondum 02:34, 30 July 2018 (UTC)
 * I would write
 * The only real or complex function that is equal to its own derivative (that is, such that $y = a^{x}$), and is equal to 1 at 0 (that is $1$) is $f(x) = f'(x)$. This is the exponential function, which has as its base the number $f(x) = e^{x}$.
 * D.Lazard (talk) 07:52, 30 July 2018 (UTC)


 * While agreeing on the now current formulation being clumsy, I do like mentioning the whole class of functions $$x\mapsto a^x, \; a \in (\mathbb R,)\mathbb C$$, in which the instantiation with the specific value $$a=e$$ achieves an important property. Purgy (talk) 09:25, 30 July 2018 (UTC)


 * I agree with Purgy. This article is about the base e, not the natural exponential function per se.  So it makes more sense here to ask, what distinguishes e from other bases?  Sławomir Biały  (talk) 10:39, 30 July 2018 (UTC)


 * I have restored the old version of the lead, prior to the addition of discussion of the natural exponential function, to refocus it on the number e itself. There were also a number of questionable structural changes that took place which, on balance, were not good.   Sławomir Biały  (talk) 10:46, 30 July 2018 (UTC)


 * I agree that the focus should be in the number $e$. As such, even with Sławomir's revert, I still find the paragraph starting "The constant can be characterized in many different ways. For example, ..." to be an excessive diversion into the details of contexts in which $f(x) = f'(x)$ occurs for the lead of an article.  Since all the examples given are intimately related, I would find it sufficient to mention that it is the base of, say, the natural logarithm, and leave exploration of perspectives for the body of this or other articles.  But I see that this would effectively duplicate the first sentence of the lead, and which already seems to answer the question that Sławomir aptly posed; should this paragraph not simply be removed?  —Quondum 11:27, 30 July 2018 (UTC)


 * I faintly recall a consensus not to award predominance to the logarithmic approach, and to simply start with the numerical value. In so far I object to Sławomir Biały's revert. I am not aware in detail of the mentioned questionable structural changes.
 * My suggestion for avoiding excessive diversification within the lead would be to give just
 * an approximate value (~5 decimals),
 * the/a series expression(s) (no limits, no exponentials, no logs),
 * the historical beginning (Bernoulli?),
 * the fundamental status of transcendence, and
 * a remark on the ubiquity in math, and, perhaps, the relation with $f(0) = 1$.


 * Cheeky enough to revert Sławomir Biały. Purgy (talk) 12:26, 30 July 2018 (UTC)


 * I am unable to find any consensus for the new lead in the discussion page archive. I think the old lead was better, since it points out immediately that e is the base of the natural logarithm, which defines the number very early on and clarifies why it should be of interest.  The new lead, by contrast, puts rather peripheral matters first, like the series and limit definitions of the number, and ancillary historical details.  Unless there is affirmative consensus for the new revision, I think the old lead should be restored.   Sławomir Biały  (talk) 15:33, 30 July 2018 (UTC)


 * I do not think we should eschew all mention of the natural logarithm and natural exponential function. I also don't think that we should offer any general theorems about solutions of differential equations in the lead of the article.   Sławomir Biały  (talk) 15:23, 30 July 2018 (UTC)


 * I agree with the general perspective that (as I interpret it) seems to be given by both Purgy and Sławomir to keep it at an overview introduction level in the lead. Giving a brief historical context and prominent associations (such as the limit and its association with natural logarithms).  I am dubious about including an actual series expression in the lead.  The listing of the well-known constants in the Euler identity is a bit of a flourish that works well in a lead, and some properties such as being transcendental belong there.  I would move long decimal expansion into the body.  I agree with Sławomir that more technical aspects with the flavour of a theorem or dealing with differentiation, continuity etc. should be left out of the lead: this is an article that the average nonmathematical high-schooler should be able to read easily without losing the sense of what is being said.  So, for example, it is fine to mention that it is transcendental, but adding a definition thereof is possibly counterproductive.  As the lead is now, merging everything after "transcendental" into the body would work for me, possibly along with with the series expression.  —Quondum 17:45, 30 July 2018 (UTC)


 * I disagree in part with this proposal. If we cut the last paragraph, then the lead would then contain no mention, whatsoever, of the fact that e is the base of the natural logarithm.   Sławomir Biały  (talk) 21:27, 30 July 2018 (UTC)


 * Ah, sorry, I (incorrectly) remembered it being in the start, from the different version. As I indicated in my previous post, I feel that the prominent association with the natural logarithm (and/or similar, such as the natural exponential) should be mentioned in the lead, though this does not imply delving into the mathematical detail.  My bland description of cropping the lead would thus not be exactly what I'd meant to suggest.  —Quondum 22:39, 30 July 2018 (UTC)


 * This does point to my chief objection to the present version of the lead. The most important things, about the logarithm in particular, have been moved to the last paragraph, for questionable reasons.   Sławomir Biały  (talk) 10:47, 31 July 2018 (UTC)


 * As restored now, it does feel a bit more natural to me and the extreme artificiality that led to my original comment is diminished, though some of my earlier comments about excess detail in the lead remain. I am hesitant to get into the detail of the exact balance to strike, though, since past experience has shown that the sense of where a suitable balance lies varies between interested editors.  —Quondum 12:15, 31 July 2018 (UTC)

My suggestion is shaped by dangling Damokles' sword of how (relevant) math is perceived in the public. So my thoughts were: a number must primarily have a value (5 decimals), one should be able to calculate it a bit by oneself (one simple series), history is a general cultural highlight (Bernoulli), buzzwords are attractive (transcendency), and finally, respecting the ubiquity in math, rounded off perhaps with Euler.

Intentionally, I do not enumerate details of the ubiquity, but I oppose to bring the logs upfront and not on equal footing with exponentials. I think hyperbolic angles lost a bit of momentum, integrals are educationally after derivations. My gut feelings are that maybe continuous interest is closest to public interest (pun attempted), and understanding. How much of all this is in the lead is determined by its length.

To be honest, the unitary log as raison d'être for $e$ looks dramatically circular to me (we had this, I did not start it then). Purgy (talk) 12:49, 31 July 2018 (UTC)


 * The most important feature of the constant is that it is the base of the natural log/natural exponential. These matters should be discussed first.  Then history and nomenclature, followed by the number-theoretic properties.  A lead which pushes until the very last paragraph any relation to the natural log and exponential is unacceptable.  I would add that your objection to the consensus lead is nonsensical, since both the natural log and natural exponential are discussed in the first paragraph.  It just happens that the characterization in terms of the natural log (which is not circular, please read the article and discussion page archives) is much shorter.  The relation to the natural log is explained in the first paragraph, and the accompanying graphic.  It is also discussed in much more detail in the article itself.   Sławomir Biały  (talk) 01:15, 1 August 2018 (UTC)


 * Well, I certainly do not need that badly a certain version of the lead that I would throw around phrases like "most important", "unacceptable", "nonsensical", and not even "not circular" (in an expectable setting). I just ask to take my utterances as one possible way to weigh certain points in this article for best meeting the needs of a vaguely specified non-professional audience, and beg pardon, in case I bothered someone with my aspects. Since I am quite sure that I won't change my opinion on this that easily, I humbly beg to be allowed to disagree to the above ex cathedra, without being considered imbecile. Meanwhile, I get to know what to expect from an increasing number of editors, without feeling bothered myself too much. Purgy (talk) 09:01, 1 August 2018 (UTC)


 * The introduction suffers from non-key properties of the topic, just as Quondum commented here first. In addition, it creates a circular reference with the natural logarithm article because the latter defines its topic via the main article.  It's sad that building a tree of knowledge seems so difficult even in mathematics. --ilgiz (talk) 21:22, 27 December 2018 (UTC)

Math indentation
Please see the linked user essay in my edit summary. Using a colon to indent creates a definition list. These formulas are not definition lists. The math extension already includes its own standard way to indicate that the math is a block element rather than inline. This isn't just arbitrary, there are accessibility issues at play as well as semantics. Opencooper (talk) 18:17, 4 March 2019 (UTC)
 * Colons are used for displayed formulas in all mathematical articles. If you want to change that, a WP:Consensus is required. You may try to get one by starting a WP:RfC. But without a consensus that includes mathematics editors, any systematic change like your edit of this article would be WP:disruptive editing. D.Lazard (talk) 18:39, 4 March 2019 (UTC)
 * Wow, we're already at the stage of personal attacks and calling editors disruptive? What ever happened to assuming good faith? Manuals of style are guidelines, not death pacts that one must blindly follow without actually thinking about them. Also, this goes against the MoSes for lists and for accessibility, both which call out this specific misuse. The only reason I linked my personal essay is because I feel it breaks it down better for non-technical users and goes more in depth on the issue. Opencooper (talk) 18:45, 4 March 2019 (UTC)
 * I Have never written that you or your edit are disruptive. I have only written that repeating such an edit without a consensus would be disruptive. This is not a personal attack, only a warning. If you not intend to repeat this edit, there is absolutely no problem. Also, in the case of a contradiction between MOSes and the uses of a WP community, this cannot be solved by a single editor, without consensus of the community. D.Lazard (talk) 19:02, 4 March 2019 (UTC)
 * I haven't repeated it, hence your implication is unnecessary and presumes guilt (similar to "when did you stop being a wifebeater?"). If contradictions can't be solved without the community, how come you get to decide that the Math MoS trumps all then and shut down discussion? Isn't this the point of this discussion, for us, the community, to try to resolve issues? Dogmatically following every other article only makes us consistent in being worse, not addressing issues. Let's not all jump off bridges because the first guy decided to. Opencooper (talk) 19:13, 4 March 2019 (UTC)

What is e?
Recently an editor tried to replace introduction of e via ln by exp. I recall this happening time and again. Personally, I prefer slightly the ln (for integrals over derivatives), but I can imagine that a good deal of the readership might prefer the exp.

Maybe either an enlightening sentence, or reporting about different views, or a sentence just less apodictic than just currently, or whatever is appropriate. Purgy (talk) 08:01, 18 March 2019 (UTC)