Talk:E (mathematical constant)

ISO 80000-2

Due to ISO 80000-2 the operator "e" should be typed upright, not in italics. <spane class="autosigned">— Preceding unsigned comment added by 131.188.134.245 (talk • contribs)

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

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

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

Wikipedia is required to follow reliable sources. See WP:WEIGHT. In this case, following the ISO would give the views of a tiny minority undue weight, when the rest of the world uses the conventions adopted in this article. Also, per WP:MSM#Roman versus italic: "For single-letter variables and operators such as the differential, imaginary unit, and Euler's number, Wikipedia articles usually use an italic font." Sławomir
Biały 12:47, 10 September 2016 (UTC)

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

Euler's number, the subject of this article, is explicitly mentioned in the guideline. Also, it's just false that we don't follow reliable sources in matters of style. We do follow reliable sources, precisely for the reason that not following those sources would be assigning WP:UNDUE weight to minority views. In this case, on the matter of how "e" is usually typeset. Sławomir Biały (talk) 13:40, 10 September 2016 (UTC)

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

Nope, wrong. It explicitly is about italicization if e and i. This came out of precisely this sort of perennial discussion. There was very strong consensus against this proposal when you made it at WT:MSM. This apparent denial of the consensus (and black-letter word of the guideline) appears to be tendentious. I consider this discussion closed. The matter was already settled more than a year ago. The appropriate avenue to lobby for change is now an RfC, not to make an end-run around past consensus in this strange way. Sławomir Biały (talk) 00:44, 11 September 2016 (UTC)

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

The purpose of WP's style guide is "to make using Wikipedia easier and more intuitive by promoting clarity and cohesion, while helping editors write articles with consistent and precise language, layout, and formatting". In other words it really is the role of WP contributors to decide how mathematical texts should be appear in Wikipedia. The criteria are clarity and cohesion, both of which would be well served by use of an upright e if this were followed uniformly throughout the project. Reliable sources are not relevant here unless they promote clarity. Dondervogel 2 (talk) 22:37, 10 September 2016 (UTC)

Let me add my voice to the consensus that the ISO does not determine usage on WP, the ISO standard is non-standard, and we shouldn't adopt it. --JBL (talk) 23:17, 10 September 2016 (UTC)

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

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

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

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

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

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

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

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

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

Pronunciation

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

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

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

The graph of $y=\ln x$ is not magical

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

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

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

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

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

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

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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)

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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. Sławomir Biały 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^words_deeds 15:40, 17 August 2017 (UTC)
• Oppose as per above Power~enwiki (talk) 22:58, 17 August 2017 (UTC)

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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.

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

(edit conflict) 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. — 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. — 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 φ 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, 2^u) on the graph of your function f  (with equation
y(t) = 2^u + (t-u)·2^uln2 )  grows to double value at moment v (with 2·2^u = 2^u + (v-u)·2^uln2  or  v = u + 1/ln2 ).
So  f(v) / f(u) = 2^u+1/ln2 / 2^u = ..... 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'  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) ^ (f' / f)  ." -- Hesselp (talk) 15:55, 24 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. Sławomir Biały (talk) 22:56, 23 November 2017 (UTC)

User:Hesselp violation of topic ban

See WP:ANI#User:Hesselp violation of topic ban. 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, @Purgy Purgatorio:, 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)

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 π or φ, 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 (1 + 1/n)n 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 f(x) = ex 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 y = ax has unit slope at x = 0. The natural logarithm, or logarithm to base e, is the inverse function to the natural exponential function. The natural logarithm of a positive number k can also be defined directly as the area under the curve y = 1/x between x = 1 and x = k, in which case e is the 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 γ, 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, π 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

2.71828182845904523536028747135266249775724709369995... (sequence A001113 in the OEIS).

References

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·e^x). -- Hesselp (talk) 15:55, 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)