Talk:e (mathematical constant)

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Good articleE (mathematical constant) has been listed as one of the Mathematics good articles under the good article criteria. If you can improve it further, please do so. If it no longer meets these criteria, you can reassess it.
Article milestones
June 19, 2007Good article nomineeNot listed
June 21, 2007Peer reviewReviewed
July 18, 2007Good article nomineeListed
August 31, 2007Good article reassessmentKept
Current status: Good article

Comments on 'alternative characterization 6'[edit]

- The value of the quotient  being independent of for an exponential function , 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 is any solution of the differential equation  ,  then for all :    .
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 instead of the sufficient (but still not everywhere usual?) .
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 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: .

Thoughts? Sławomir Biały (talk) 19:55, 28 April 2018 (UTC)
@Sławomir Biały.  Some remarks on your proposal.
i. On "..then the quantity ..".  Why 'quantity'? why not 'quotient'? Even better: simply "..then .." .
ii. On  "... is a constant, sometimes called the time constant ..." .
The real universal constant is , while depends on .  So I propose:
"... doesn't depend on (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 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 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:


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  " a constant. When ... this constant is often ..."   into   " 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:
" a constant 'for all t'. When ... this constant is often called the time constant of this process, ..."   into
" independent of t. When ...this value is often called time constant of the process (symbol ), ..." ?
-- 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[edit]

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 is independent of t; sometimes this value is called time constant of f(t), symbol .  (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:

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 is any solution of the differential equation ,  (an exponential function with time constant or e-folding ), then for all :

-- 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[edit]

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)
While I do not feel particularly strongly on this and I am not disagreeing with Sławomir, I do believe that Purgie is making a valid point, even if a bit too flippantly. Entries in a table like this need to be more than just notable, they need to be interesting, specifically historically interesting. The table's function is to illustrate the growth of the number of known digits. It can not be complete, nor would we want it to be. It has to stop growing at some point and I think that it should stop when the next entry is no longer interesting. For years I did a corresponding lecture on the digits of π. As each new record was set I was forced to remove some items, even though they were notable at the time I started to talk about them, since they had stopped being interesting (what I could say about them I could easily have said the same about some newer entries). At this point, with over 500 trillion (I didn't really count the zeroes in Ye's table, but the number is up there) digits known, the fleeting record of the millionth digit calculation has lost all interest, at least for me. --Bill Cherowitzo (talk) 17:32, 5 July 2018 (UTC)
Yes, it has to stop growing at some point -- I do not disagree with that. I would ask, then, that those who have undone my edit defend stopping the table at 116,000 rather than 1,000,000, a number not achieved until 16 years after Wozniak's. It was after 1 million digits was reached that new significant records (2 million, 5 million, and so on) were set only months, or maybe even weeks, apart, not years. Owlice1 (talk) 17:52, 5 July 2018 (UTC)

The latest revision, with the added sources, looks reasonable to me. I say we let the addition stand. Sławomir Biały (talk) 18:06, 5 July 2018 (UTC)

I disagree with Sławomir Biały's and Owlice1's opinion that the entry under reversion should be included in the addressed table. I try to give answers to questions raised by Owlice1 and to explicate my reasons for objecting and also for my suggestion of an expanded table.
  • I admit calling new records in rote computing "rather ridiculous" is quite harsh. I can only mention the "rather" as mitigating: sorry. I do believe, however, that the achieved numbers have no profound importance.
  • I do not doubt the factuality of the intended entry and it being reliably sourced, but the notability of in math topics is to me just as big as any decade, perhaps slightly bigger as a multiple of , favoured in technical contexts. However, I do not see any notability wrt e itself, and not even wrt a number of digits in its representation in positional number systems. There being a WP article on the number is largely irrelevant in any other article. Therefore, there is no reason, stemming from digits themselves, to appear in the article about e.
  • The v. Neumann entry is relevant for the reason I tried to give: first automatically computed value. I think, v. Neumann is irrelevant in this context, the ENIAC is the relevant information, one of the first floor-sized computers, unavailable to the public. The 1961-entry might be skipped, its importance being perhaps only the increase of available digits in orders of magnitude. The 1978-entry is not important for S. Wozniak, but for the fact that then a publicly available device empowered the almost average Joe to calculate digits of almost any desired math constant to a degree, for which I have no tolerated verbiage. "Trillions" is not to my liking, because of Moore's Law and other reasons I consider obvious to most in good faith. The number of calculated digits is limited just by boredom.
  • I do not deny the existence of research values in the ongoing calculations, but their nexus to e are, at least to my knowledge, confined to the application of specific algorithms, possibly exchangeable to those for other constants, which are often considered barely as useful test samples. I conjecture that even a newly discovered quantum algorithm for calculating digits of e would not justify a new entry, but only an article in WP on its own.
  • I still believe that adding to the table a reason-for-notability column, containing good reasons, enhances the article. Since I was aware that my specific knowledge and active fluency in English would not be sufficient to supply really good entries there, I just did a sketchy draft, and explicitly asked for kind improvement in the edit summary.
I did not expect the qualification "flippant" (I am fully respectful!), and other reactions I consider not de rigeur. Purgy (talk) 10:15, 6 July 2018 (UTC)
The achieved result of 1,000,000 digits of e is notable. I've already pointed out a number of reasons why. I think you do not grasp this one, however: these digits were (are) available for download; this is actually useful. (Where are the 116,000 digits of e from the previous result? Published in BYTE. How useful were they? What could anyone do with them?) Generating the million digits and then publishing the result online, where the digits can be downloaded, makes them available for research. Here are three research articles that use this particular achieved result:
Ginsburg, N. and Lesner, C. (1999) "Some Conjectures about Random Numbers"
Shimojo, M., et al (2007) “A Note on Searching Digits of Circular Ratio and Napier's Number for Numerically Expressed Information on Ruminant Agriculture"
Lai, Dejian & Danca, Marius-F. (2008) "Fractal and statistical analysis on digits of irrational numbers"
Notice the last two articles mentioned were written more than a decade after these results were made available online for others to use, indicating that particular digit-set had some endurance for research (and still may, though I haven't looked for more papers using it; I have run across other papers using even the larger sets generated by Nemiroff & Bonnell, such as 2 million and 5 million digits of e, too).
Such research might not be something you'd want to do, but others clearly do. Calculating and then making these digits available online for anyone to use had not been done before. (Editing to add: at the million digit level.) Owlice1 (talk) 11:31, 6 July 2018 (UTC)
Yes, I do not grasp how often I have to ruminate that facts, typical of any irrational number, are very well notable at appropriate places, but are not notable within an article about e, which just happens to be irrational. Purgy (talk) 07:47, 7 July 2018 (UTC)

For some reason, the following comes to mind:

Who Where What
Colonel Mustard Library Wrench

I've restored Nemiroff & Bonnell to the table, with what I hope are enough references to satisfy all. Thank you for your patience. Owlice1 (talk) 11:47, 7 July 2018 (UTC)

It’s just not an interesting or remarkable result. As by 1978 it was already possible to generate over 100,000 digits on a 8-bit CPU, someone could have generated a million digits a few years later, and probably did long before 1994. There would be many such firsts that were not published as they are simply not interesting, no-one has noticed them. It really is not that interesting now anyone can download and run a program to generate digits.--JohnBlackburnewordsdeeds 12:26, 7 July 2018 (UTC)
"not reliably sourced." I provided primary and secondary sources. Which of these did you find not reliable? It is certainly not true that others did not notice these results. They were used in research, the Gutenberg project published them, and the results are even available through Amazon! (The reviews are rather amusing.) The research articles using them are not about algorithms for generating e, but about how to use the generated digits. Can they, for example, be used as a random number generator, or in cryptology? That is what some of this research is, and it's clear that others must not have wanted to generate these digits themselves, as they used (and cite) these results. Speaking of "not reliably sourced," I note this from your post: someone could have generated a million digits a few years later, and probably did long before 1994. (Emphasis mine.) That's not sourced at all. I'm restoring the entry. Owlice1 (talk) 17:32, 7 July 2018 (UTC)
The computation of one million digit has not been the object of a regularly published paper. This shows that, even at the time of this computation, this was not considered by the mathematic community as a significant result. Nevertheless, the list of these digits is useful and has been used for other research. This could be mentioned elsewhere in the article, but does not belongs to this section. This list of digits is also listed in the "See also" section, which is its natural place. I have reverted your edit for these reasons.
On the other hand, when you disagree with other editors, edit-warring is the worst way for dispute resolution, as you may be blocked for editing because of the WP:3RR rule. D.Lazard (talk) 18:01, 7 July 2018 (UTC)
I rarely edit Wikipedia pages; it wasn't until this discussion that I learned the phrase "edit-warring." I brought the discussion to the talk page when it was suggested I do so, and I've answered every criticism of the edit. There was some agreement/acquiescence that the addition could stand, which is why I put it back.
BTW, I missed the "regularly published paper" for the Wozniak result that shows it is considered significant by the maths community. Where is that, please? What I see given as a source for that is a BYTE magazine article (which I probably have in the stash of old mags in the basement; I may have to go look). At least one other entry in the table has a link to a website as the source, rather than to a journal article. I have asked before why my addition, which has multiple reliable refereed secondary sources that indicate the value of this result/addition, is unacceptable while others with only one source not as robust still stand; I never got an answer to that. The goal of some of the editors appears to be to end the table at Wozniak, no matter what. Owlice1 (talk) 18:29, 7 July 2018 (UTC)
May I point you to the fact that you keep refusing to recognize the statements about the disconnectedness to this article of the sources you mention, backing the non-notability wrt this article of the sourced fact you want included. The research work of these papers is in no way specifically connected to e, but to an assumed structure of randomness in its digits. Any other irrational number would do the same trick. You also seem to ignore in your comparison the given reason for the Apple II entry. I repeat: It is not primarily about Wozniak, but about the public availability of equipment, capable of calculating more or less arbitrarily many digits of any computable number, rendering any new "records" as of no relevance. Please, do not strive to make WP a Guinness Book of Records. Purgy (talk) 19:22, 7 July 2018 (UTC)
I'm not disputing the Wozniak entry, at all, although the different standards for sources for its inclusion wrt my addition are noted. You cannot assume randomness in the digits of e; indeed, that's one point of research, one that requires a million or more digits to accomplish. The Nemiroff/Bonnell results are notable for a number of reasons, including enabling that and other research (in computer science, math, and even apparently in ruminant agriculture) where other results at that time did not. I'm not "striv[ing] to make WP a Guinness Book of Records." I'm trying to add a useful/notable addition to that table. That's it.
Two additional papers for my own reference (and possibly future others'):
BiEntropy - The Approximate Entropy of a Finite Binary String,
A New Method for Symbolic Sequences Analysis. An Application to Long Sequences,
Owlice1 (talk) 20:01, 7 July 2018 (UTC)

Please, improve on the given reasons[edit]

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[1] First value
1714 13 Roger Cotes[2] First reasonable precision
1748 23 Leonhard Euler[3] Euler's number crunching professionality
1853 137 William Shanks[4] World known number cruncher
1871 205 William Shanks[5] ... doing it again
1884 346 J. Marcus Boorman[6] 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[7] Hereditary deficiencies?
1978 116,000 Steve Wozniak on the Apple II[8] 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.[9]


  1. ^ Cite error: The named reference Bernoulli, 1690 was invoked but never defined (see the help page).
  2. ^ Roger Cotes (1714) "Logometria," Philosophical Transactions of the Royal Society of London, 29 (338) : 5-45; see especially the bottom of page 10. From page 10: "Porro eadem ratio est inter 2,718281828459 &c et 1, … " (Furthermore, by the same means, the ratio is between 2.718281828459… and 1, … )
  3. ^ Leonhard Euler, Introductio in Analysin Infinitorum (Lausanne, Switzerland: Marc Michel Bousquet & Co., 1748), volume 1, page 90.
  4. ^ William Shanks, Contributions to Mathematics, … (London, England: G. Bell, 1853), page 89.
  5. ^ William Shanks (1871) "On the numerical values of e, loge 2, loge 3, loge 5, and loge 10, also on the numerical value of M the modulus of the common system of logarithms, all to 205 decimals," Proceedings of the Royal Society of London, 20 : 27-29.
  6. ^ J. Marcus Boorman (October 1884) "Computation of the Naperian base," Mathematical Magazine, 1 (12) : 204-205.
  7. ^ Daniel Shanks and John W Wrench (1962). "Calculation of Pi to 100,000 Decimals" (PDF). Mathematics of Computation. 16 (77): 76–99 (78). doi:10.2307/2003813. We have computed e on a 7090 to 100,265D by the obvious program
  8. ^ Wozniak, Steve (June 1981). "The Impossible Dream: Computing e to 116,000 Places with a Personal Computer". BYTE. p. 392. Retrieved 18 October 2013.
  9. ^ Alexander Yee. "e".

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

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 • 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)
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.
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)


I find the description

The function f(x) = ex is called the (natural) exponential function, and is the unique exponential function of type ax equal to its own derivative (f(x) = f(x) = ex). This is easily spotted at x = 0, where ax = 1 for any value of a, but a = e is the only positive number such that the slope at x = 0 of the graph of the function y = ax also equals 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 f(x) = f(x) – is f(x) = ex, or a constant multiple thereof. This leads naturally to the exponential function, which has as its base the number e.

I see that it is very neatly stated in e (mathematical constant)#Calculus. —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 f(x) = f(x)), and is equal to 1 at 0 (that is f(0) = 1) is f(x) = ex. This is the exponential function, which has as its base the number e.
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 , in which the instantiation with the specific value 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 e 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 0, 1, π.
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)

Sharp inequalities[edit]

"The point is, this inequality is sharp. Undid revision 852851072 by Quondum (talk)":

  • the unique base of the exponential for which the inequality ax ≥ 1 + x holds for all x ... is e.

By implication, you're asserting that the following is not a sharp inequality:

  • the unique base of the exponential for which the inequality ax ≥ 1 − x holds for all x ... is e–1.

User:Sławomir Biały, did you really mean that? —Quondum 00:05, 1 August 2018 (UTC)

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 is always true. (Though, I inferred that you perhaps meant something like instead, which is not a sharp inequality for any a.) But, for the line , with both y-intercept and slope equal to unity, the sharp inequality for all nonzero x holds, if and only if . 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 ax is tangent to some graph 1 + bx for every base a at the point (0, 1). Thus every base a satisfies a sharp inequality of this type by choosing a suitable constant b. In effect, without motivating the particular choice of coefficients, the statement says that we can find a tangent to ax, 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 e. I would say that few people are going to say that 1 + x stands out as clearly more special than say 1 − x. —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 ? Why is instead of ? Why is instead of ? 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 then for all nonzero x. Conversely, the number is the unique number such that for all nonzero x. Here we are simply taking the special case of , 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 ax for real a and x such that this does not rely on calculus, limits, or something more abstruse, while not implying exp(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: R>0 × QR. 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)