Talk:e (mathematical constant)

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)

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 $\mathbb {R}$)   $\ \Longrightarrow \ [f(1)/f(0)]^{f'(0)/f(0)}\ \,=\ \,e$
-- 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#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)

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: [1]. I can't think of any good reason not to mention it somewhere in the page. Sapphorain (talk) 21:02, 29 November 2017 (UTC)

(edit conflict) 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 • 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)

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

S%C5%82awomir_Bia%C5%82y, 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#Mean lifetime 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:

If f(t) is an exponential function, then the quantity $f(t)/f'(t)$ is a constant, sometimes called the time constant (it is the reciprocal of the exponential growth constant or decay constant). The time constant is the time it takes for the exponential function to increase by a factor of e: $f(t+\tau )=ef(t)$.

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 f(t) ) . "
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

If f(t) is an exponential function, then $f(t)/f'(t)$ is a constant 'for all t'. quoted amendment dedicated to Hesselp 10:02, 30 April 2018 (UTC) When f describes a physical process and t is associated with time, this constant is often called the time constant $\tau$ of this process, and the reciprocal is called its exponential growth rate (>0) or decay rate (<0).

The number e is the factor by which all exponential functions change during the elapse of one time constant:

$f(t+\tau )=e\cdot f(t)$.

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 t'. 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?

6a.  If f(t) is an exponential function, then $f(t)/f'(t)$ is independent of t; sometimes this value is called time constant of f(t), symbol $\tau$.  (It is the reciprocal of the exponential growth constant or decay constant.)  The time constant is the time it takes for the exponential function to increase by a factor of e. So for all t:

$e=f(t+\tau )/f(t).$

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?

6b.  If $f(t)$ is any solution of the differential equation $y'=y/\tau$,  (an exponential function with time constant or e-folding $\tau$), then for all $t$:

$e={\frac {f(t+\tau )}{f(t)}}.$

-- 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)

Known digits edit -- first million digits of E

I edited the table "Number of known decimal digits of e" to add the calculation for the first million digits of e; this information had been on this page some time ago (not added by me, BTW), and I noticed today it was gone so added it back.

This was undone with the reason "entries more recent than 1978" are "rather ridiculous." (Why are they "rather ridiculous"?) I added the information again, and noted that "This is a significant increase over the previous calculation and 1,000,000 is a notable number."

Undone again, with the (partial) comment "Already linked in external links. Secondary source needed for mention here." The external link referred to doesn't assign credit to those who computed it nor when this was done. I added the entry back with another reference as a secondary source.

Undone again, with the comment "the added source isn't particularly reliable; it's just a listing in a table on some web page, and it disagreed about the number of digits by a factor of 10 -- also, 1 million isn't a "notable number", it's just a round number."

The added source is a website maintained by two French mathematicians.

The number one million is notable enough to have its own Wikipedia page, and the one million digits of e that were calculated were used in research. Obviously, I think this is a worthwhile entry, so I added the entry again, removing the reference to the French maths site and adding references to two research articles.

Undone again, with the comment "Nothing about this entry makes it notable given the current state of affairs. Use the talk page to make your case if you must."

The current state of affairs is not relevant -- the state of affairs in 1994 is, and the calculation of e to one million digits (by two PhD astrophysicists at NASA) at that time is indeed notable; it's a significant increase over the previous result. Further, these are results that have been used in research (more recently, however, the 2-million digits of e have been used).

I hope the most recent removal of the info I attempted to add (rather, restore) is undone; the information is useful, assigns credit, and the only thing controversial about it is that a number of editors appear to want the table to end with Wozniak. Nice to be reminded (again) of why I hate editing Wikipedia pages and do it so rarely. Ciao. Owlice1 (talk) 05:26, 5 July 2018 (UTC)

A few things. First of all, the fact there's an article for 1 million is completely irrelevant. That has no bearing whatsoever on whether or not this entry should be in the list. Also, you're neglecting to mention that this entry that you want to restore is just one of a whole mess that got removed (the list has expanded and shrunk at various points). Presumably we want to draw the line somewhere. I honestly think this wouldn't be terrible if it were added back in, but these become of increasingly less historical significance as we go on. Finally, why are you so eager to restore this entry but not any of the others? –Deacon Vorbis (carbon • videos) 14:45, 5 July 2018 (UTC)

The fact that there is an article for 1 million demonstrates the number itself is indeed notable, and I posted that in response to the complaint that the number isn't notable. (If it's not a notable number, delete the page for it.) Yes, you want to draw the line somewhere, and that somewhere is at Wozniak. That's been made very clear. Nevertheless, this one additional entry is useful for the reasons I've already mentioned: it's a significant increase over the previous result, it is a notable number, it is used in research. This result came 16 years after the previous one. Other results of two million, etc., followed closely on the heels of this one, too closely, I would say, to be noted, as with the ever increasing capability in computing power and the speed at which the increases were (and are) being made, that will then always be the case: greater numbers will always be found. I've answered every criticism of the edit. As to why I didn't try to restore any others, well, we've seen how well it worked when I tried to restore just one with so much going for it! Why on earth would I bother with any others, here or anywhere else? I'm done. Owlice1 (talk) 16:06, 5 July 2018 (UTC)

It's been asserted that the 1994 calculation is notable. Notability is established by secondary sources. If this calculation is indeed noted in reliable sources, it can be restored. Sławomir Biały (talk) 16:44, 5 July 2018 (UTC)

I provided several sources, as one can see from looking at my edits. If you find none of them reliable, then please let me know what is a reliable source, and let me know, too, if you would, why this addition needs more reliable secondary sources than others listed in the table, each of which has one source, at least one of which, which is a link to a website (deemed unacceptable for my addition), doesn't work. Thanks. Owlice1 (talk) 17:21, 5 July 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.

---

Number of known decimal digits of e

Date	Decimal digits	Computation performed by	Reason for notability
1690	1	Jacob Bernoulli	First value
1714	13	Roger Cotes	First reasonable precision
1748	23	Leonhard Euler	Euler's number crunching professionality
1853	137	William Shanks	World known number cruncher
1871	205	William Shanks	... doing it again
1884	346	J. Marcus Boorman	Last known effort by rote human calculation
1949	2,010	John von Neumann (on the ENIAC)	First computerized result getting public attention
1961	100,265	Daniel Shanks and John Wrench	Hereditary deficiencies?
1978	116,000	Steve Wozniak on the Apple II	Egalitarian approach to e —The End

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.

References

---

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

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. Template:Z175 –Deacon Vorbis (carbon • 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 • videos) 15:20, 5 July 2018 (UTC)