Talk:E (mathematical constant)/Archive 5

Making e more accessible
The story of compounding interest playing a part in Bernoulli's discovery of e is compelling and conceptually accessible. I've created a depiction of this here. I've avoided text labels to make it language-independent. Would this be a useful addition to the article? -- Scray (talk) 21:05, 20 April 2011 (UTC)
 * I went ahead along the bold route. -- Scray (talk) 03:03, 21 April 2011 (UTC)
 * The plot was quite cryptic, and the caption not that helpful. The key missing points were that the interest rate was 100% per year and that the time scale was in days, and logarithmic.  The 100% per year things is an arbitrary hack; the point is really to find whatever time period makes the simple interest come to 100%.  Did Bernoulli really use 100% and 1 year?  I don't know; but if he did, we should say so, and if he didn't we should probably not focus on that arbitrary case.  What would be more clear would be a linear time scale going out to 1.0 times the time that it takes the simple interest to reach 100%, which could also be years in the simple example if we want to stick with that.  Dicklyon (talk) 03:38, 21 April 2011 (UTC)
 * It was just an attempt, and thanks for the feedback. You're right - I inadvertently left out the interest rate, which is very important, and the numbers chosen are illustrative examples; editing the caption would fix all of that.  As far as the log scale - try plotting this on a linear scale - it's very hard to see the convergence.  At 1.0 times the time it takes to reach 100%, the value is still pretty far from $e$.  Maybe I misunderstand, but I tried a few permutations on the figure before generating the one I posted.  In my opinion, a visually simple figure that illustrates convergence to $e$ might motivate the more casual reader to dig in a bit more to understand the details.  -- Scray (talk) 03:53, 21 April 2011 (UTC)

Here's another attempt. If I get time tomorrow I'll try to guess what you are suggesting above; however,as I understand this article, Bernoulli was interested in the effect of reducing the compounding interval, so I don't think it would be accurate to plot with an increasing amount of investment time. -- Scray (talk) 04:15, 21 April 2011 (UTC)
 * A new version is depicted at right, and I hope it's an improvement (at 100 compoundings per unit time, the value is only 2.714567482, but I could reduce the number of partitions if you think that would look better). -- Scray (talk) 00:29, 22 April 2011 (UTC)
 * Better. But the parenthetical "(1 on the x axis)" suggests that 1 is the unit of time, and that the x axis is units of time, as opposed to compounding frequency.  If you do it by compounding time instead of frequency, from 0 to 1 on the x axis, where 1 is the time for which simple interest is 100%, you get a nearly straight line approaching a limit of e at 0.  Is that better?  I'm not sure.  But consider it.  Dicklyon (talk) 01:10, 22 April 2011 (UTC)
 * Sorry I'm so dense. Your suggestion is excellent - see new version at right (the downside is that it no longer has the appearance of an asymptotic relationship - but it is accurate).  I've attempted to write a clear caption, but welcome further tweaks.  -- Scray (talk) 01:54, 22 April 2011 (UTC)

I made one, too. This is perhaps too "vulgar"? Dicklyon (talk) 02:40, 22 April 2011 (UTC)
 * I like that - very clear. Mine have been somewhat Spartan because I gathered that it's preferable to avoid using English in WP figures, all things being equal.  Clarity is very important, though, and this concept is not entirely intuitive.  -- Scray (talk) 02:44, 22 April 2011 (UTC)
 * I'll also comment that I avoided any assumption of specific interval, in part because (as you said) we don't know that Bernoulli used and the annual case is arbitrary. Oh - might be good to add an explicit indication of $e$.  -- Scray (talk) 02:46, 22 April 2011 (UTC)
 * Right, that's what I meant by vulgar; not abstract or clean, but tied to things that non-mathematicians might relate to better. I have no idea what Bernoulli did, but as a mathematician, he probably abstracted it pretty quickly.  Good point about the English; and I uploaded to commons.  However, since it's a svg, it may be easier to translate than a pixmap would be.  I can add the e if you (and others) think this image is worth the trouble. Dicklyon (talk) 02:51, 22 April 2011 (UTC)
 * I think it's worth the trouble - we can see if others weigh in. I'd also consider dropping "0.5", "1.5", and "2.5" from the ordinate axis to reduce clutter.  Using quarter-intervals (0.25, 0.5, etc) on the abscissa might be more relevant to the points you've chosen as well.  -- Scray (talk) 03:16, 22 April 2011 (UTC)

Diagram in "Exponential-like functions"
Does anyone else think the red 'e' in the diagram in section "Exponential-like functions" should be moved to below the x axis? At first I took it to mean the height of the red dashed line. —Preceding unsigned comment added by 68.199.134.93 (talk) 02:13, 22 April 2011 (UTC)
 * I agree that would look better, also more consistent with other figures in the article. I tried (unsuccessfully) to edit starting with that svg file - I think I'd have to start from scratch to get it to look right.  -- Scray (talk) 02:41, 22 April 2011 (UTC)
 * Done - I hope others agree that this is an improvement. -- Scray (talk) 17:09, 22 April 2011 (UTC)

U+2107
I'm searching for a source which explains why U+2107 is called "Euler Constant". Specifically, do mathematicians ever use epsilon to refere to Euler's number, or any other numbers named for Euler? The only source I have found so far states:
 * The Euler number (pronounced 'oiler') goes by many names such as; epsilon, the exponential number, and Napiers number to name a few, but the name I shall refer to it as throughout this essay is also its most common name and is simply... e. -http://www.gizapyramid.com/ricks-e-proportion/rick-howards-research.html

But this source is not verifiable. Can anybody help me? --beefyt (talk) 22:19, 26 April 2011 (UTC)

Requested move

 * The following discussion is an archived discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. No further edits should be made to this section. 

The result of the move request was: Not moved. Jafeluv (talk) 23:52, 15 May 2011 (UTC)

E (mathematical constant) → E (Euler's number) – It's both shorter and more specific. What's not to like? In Category:Mathematical constants, this is the only constant with the phrase "mathematical constant" (or anything similar) as part of the article title. So it would seem to be a one-off disambiguator of arbitrary character. Euler's number is currently a redirect to this article. Capitalizing the first letter in the formal title is a technical requirement. DISPLAYTITLE can be used to make it appear lower-cased, italicized, or change the font. My plan is to get a set of matching titles for the Euler equation constants, something like "I (imaginary unit)" and "$\pi$ (pi)". Kauffner (talk) 09:30, 9 May 2011 (UTC)


 * See Talk:E_(mathematical_constant)/Archive_1 (and probably others) for a discussion, and rejection, of a similar proposal. Andrew Kepert (talk) 09:58, 9 May 2011 (UTC)


 * That was nearly six years ago, not the same proposal, and there was no formal vote. Kauffner (talk) 11:14, 9 May 2011 (UTC)


 * Oppose. Not the only other name; confusion with Euler number, genus (mathematics), etc; makes it look like e stands for Euler which it does not. And the suggestion of π (pi) borders on trolling. Xanthoxyl  &lt; 15:00, 9 May 2011 (UTC)
 * Euler presumably assigned this constant an $e$ just because that was the next available letter in the alphabet. But the usage of this letter by others is nonetheless understood to honor him. Certainly everyone seems to know the "$x$ is for Euler" story, so at this point it is an entrenched part of the mythology of the number. As for the Euler's number/Euler number issue, that will no doubt continue to confuse the unwary whatever title is given to this article. Kauffner (talk) 04:37, 10 May 2011 (UTC)
 * No, not really (on the last point), because the locution Euler's number is hardly ever used, except when explaining what e stands for (whether the story is correct or not). So no particular confusion, in the overwhelming majority of cases, when the phrase is simply not used.  That's also the (or at least a sufficient) reason that the move is not advisable. --Trovatore (talk) 04:46, 10 May 2011 (UTC)
 * Readers presumably want to know who came up with this number and why it is called $y$. You don't think we should explain what it stands for because that might confuse them? Kauffner (talk) 05:35, 10 May 2011 (UTC)
 * No, perhaps I didn't explain well enough. I have no objection to mentioning the phrase Euler's number very very briefly (but I mean very very very briefly) somewhere in the text.  My point was that the extent to which that phrase occurs is so entirely negligible that the unwary will have little chance to become confused.  Precisely because the phrase is so very rarely used, I do object to putting it in the title. --Trovatore (talk) 07:26, 10 May 2011 (UTC)
 * But it doesn't stand for Euler, as we just explained. Please go and work on a subject you actually understand. Xanthoxyl  &lt; 07:10, 10 May 2011 (UTC)


 * Oppose. Euler's number is not the (or even, really, a) common name for the constant.  It mostly survives as a way to explain why we call it e. --Trovatore (talk) 18:43, 9 May 2011 (UTC)
 * Oppose since it's not generally known as Euler's number, and there's scant evidence for Euler being related to why we call it e. Dicklyon (talk) 05:39, 10 May 2011 (UTC)
 * Oppose: Euler's number will be unfamiliar to most people and make them think maybe it is something different than what they are looking for. –CWenger ( ^ •  @ ) 17:21, 10 May 2011 (UTC)
 * Oppose. When people use the term "Euler's number", in my experience they usually mean the Euler-Mascheroni constant, not e.  There is no standard term for the number e.  Sometimes it's called "Napier's constant", but that's a pretty rare term that I think we should avoid giving our stamp of approval.    Sławomir Biały  (talk) 11:42, 13 May 2011 (UTC)
 * The above discussion is preserved as an archive of a requested move. Please do not modify it. Subsequent comments should be made in a new section on this talk page. No further edits should be made to this section.

This article is incomprehensible to laypeople
Which is fine for a mathematical subject, and the bulk of the article should not be simplified in anyway.

What's not fine is that an article about such an important number does not begin with an explanation comprehensible to a layperson, somebody who took college algebra 20 years ago and hasn't thought about math since. Having such a simplified explanation would not detract from the article in the slightest, because everything else could be maintained.

This is the current first sentence: "The mathematical constant e is the unique real number such that the value of the derivative (slope of the tangent line) of the function f(x) = ex is equal to f(x) at any x, and therefore is equal to 1 at the point x = 0." It is concise (and presumably correct) but the only people who will instantly understand it are people who are thoroughly familiar with e in the first place!

Even if someone insists on keeping that first sentence for the sake of specificity and accuracy - although I think it should be replaced - the first few sentences of this article should be written in laypeoples' terms. Here's the rest of the first paragraph:

"The function ex so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm to base e. The number e is also commonly defined as the base of the natural logarithm (using an integral to define the latter), as the limit of a certain sequence, or as the sum of a certain series (see the alternative characterizations, below)."

In this particular context, somebody without mathematical knowledge will have trouble understanding the following terms from the first paragraph: derivative, tangent line, function, exponential function, logarithm, natural logarithm, base of the natural logarithm, base e, integral, limit, sequence, and series.

That is just three sentences. The article should start with something along the lines of this (please note that I have no idea what I'm talking about and that this is almost certainly incorrect):

"The mathematical constant e is a unique real number that, when used as the basis of a function results in a tangent line when x=0 with a slope of 1. This is significant because, as a natural logarithm, e can be used to calculate blah blah blah."

All in human terms.

Just my suggestion

128.223.222.29 (talk) 20:33, 11 July 2011 (UTC)


 * Someone had obfuscated the first sentence from its original version. I have simplified it somewhat by restoring the original, which is similar in spirit to your suggestion.  Hopefully the other issues are also less severe now that the first sentence is at least more comprehensible (and agrees with the illustration).   Sławomir Biały  (talk) 13:21, 27 July 2011 (UTC)


 * I agree the program is of no help in understanding the mathematics. I'm not sure about the rest 128223.222.29's comments; I don't find them helpful, but I already know the material.  — Arthur Rubin  (talk) 07:44, 30 October 2011 (UTC)


 * The first paragraph is incomprehensible as it stands. It is far too technical. I think the opening paragraph should be re-written along the lines suggested by 128.223.222.29. Jayarava 10:00, 5 November 2011 (UTC) — Preceding unsigned comment added by Mahaabaala (talk • contribs)


 * MOS:LEAD implies that the probability of achieving "featured" status lies in a lead that is "clear, accessible style" (among other failings of the poverty of a mere "good" status, which this page really is). Some power to the IP's! (Some minor concessions, perhaps.)  GA status is imposed upon an article because it is not accessible to all readers, because it is only "nearly all readers". The next step (hinting A-class) is that "A non-expert in the subject matter would typically find nothing wanting."  See the talk page, at the top.  &mdash;  Cp i r al Cpiral  22:02, 3 November 2011 (UTC)

Use a different definition
I agree that the intro to this article is poorly suited for a general audience. Part of the problem is the chosen definition, which requires more concepts than necessary, including the idea of a tangent to a curve and raising a number to an arbitrary real number. Both are hard for laypersons to grasp and, indeed, the later concept is normally defined by first constructing the natural logarithm. I believe the area under 1/x definition and the infinite sum of 1/n! are both better for lay persons- the first only requires the high school level idea of graphing a function and the intuitive notion of area. The second has the advantage the formula is relativly simple and converges quickly so it can easily be tested on a calculator or spread sheet. I also think the lede should not begin with an explanation, before giving a definition. So here is a rough cut proposal:

"The number $e$ is an important mathematical constant that occurs in formulas in many branches of mathematics. Its value is approximately 2.71828. Mathematical calculations involving logarithms and exponential growth are often simpler when e is the base value.

Because e is so ubiquitous, there are many definitions for it, all equivalent. It is the unique number such that the region under a graph of the function y = 1/x between x = 1 and and x = e has an area of 1. It is also the sum of the series:


 * e = 2 + 1/2 + 1/(2x3) + 1/(2x3x4) + 1/(2x3x4x5) +1/(2x3x4x5x6) + ...,

which can be written more formally as
 * $$e = \sum_{n = 0}^\infty \frac{1}{n!}$$

where $x$ is the factorial of $e$."

We could even include a figure that show a spreadsheet summing, say, the first 10 terms.--agr (talk) 04:21, 8 November 2011 (UTC)


 * I don't think that works or's an improvement. My biggest concern is the way of using the graph is very unusual: it's not the area under the graph but the point on the x axis such that the area under the graph from 1 (why 1?) to that point is 1. It's obvious why this is true if you know that the integral of 1/x is ln(x), but if you don't know that it doesn't help you understand it. More generally reading from a graph does not seem a very formal definition. It's not really defined or explained by the series either, which is just a way to calculate it. A spreadsheet would just show that the series converges but that's pretty obvious as the terms get smaller very rapidly.
 * I don't know there is a way to make it much more accessible as it's a moderately advanced topic. It's something students might learn a year or two after they learn about e.g. π and trigonometry (at least in the education system here in the UK), and then only if they choose mathematics at an advanced level. It has many applications but only in more advanced calculus (answering questions like "what is y if y´= ky?"), statistics, complex numbers, etc. As such it's difficult to provide an elementary justification for it: it has to rely on or relate to at least some advanced concepts.-- JohnBlackburne wordsdeeds 15:52, 8 November 2011 (UTC)
 * The next layer of audience on this good pearl of an article is, ideally, the mathematically educated (say, an engineer), who are able to read the article for the first time, and in one sitting. Justification of existing positions (word-choice, concept-choice) of advanced articles is accomplished by hyperlink "assessment" (i.e. context evaluation). They should be able, in one sitting to get e, even if they "assess" many wikilinks.  The key is to make such assessment both available (i.e. A-class content), and, for "sitting time" purposes, single-depth. (If I have to venture to far away from e, say triple- and quadruple- depth "link assessments", I am over-challenged by the article (which is a good thing because actually, teaching is the purpose of such a set of volumes an encyclopedia is)).  This theoretical content criterion then determines the content (word and concept), and article size is, by this same means, also "automatically" adjusted. That is to say the articles words and concepts are well chosen. An example is DVdm's comment "Removal of relevant wikilinks = removal of content"  &mdash;  Cp i r al Cpiral  19:54, 8 November 2011 (UTC)


 * A definition of e is unlikely here, but we should present the canonical form (the mathematicians consensus) and mention the interesting forms and features. To me, a Wikipedia article on e is not a textbook case, or even a thorough definition of e, but if we link carefully (remember, links are both words and concepts) the reader can be self refreshed about e's place as declared in mathematics.  A definition of e is unlikely anywhere anyway.  Wolfram lists 94 formulas of e that are commonly used in math!  Each of these 94 is a form or definition of e.  To borrow computational terminology, we declare e, and it is "defined" elsewhere.  For example, per Derbyshire, Principia Mathematica takes 345 pages to define the number 1. (!) &mdash;  Cp i r al Cpiral  19:54, 8 November 2011 (UTC)

There is no reason this article's intro needs to be pitched to "the mathematically educated," nor should it be under Wikipedia policy. Per MOS:MATH: "The lead should as far as possible be accessible to a general reader, so specialized terminology and symbols should be avoided as much as possible." "The lead section should include, where appropriate: ... An informal introduction to the topic, without rigor, suitable for a general audience." All definitions of e are essentially equivalent and for the lead we can pick one or two most comprehensible to a general audience. The body of the article can and should go into more technical detail, of course. Also the fact that the 1/n! series converges rapidly is not obvious to a general audience and is a point worth illustrating. That this formulation of e can easily be computed by anyone with a calculator helps demystify the subject. --agr (talk) 13:51, 9 November 2011 (UTC)

To help understand the Stochastic representation
I propose the Stochastic representations section read as follows, in order to contain a more basic and more thorough understanding of the stochastic aspect of $e$.

< snipped> See below for final version.

Here is a C program for discussion's sake concerning the elementary method.

< snipped> C program confused the real issue I brought up. &mdash; Cp i r al Cpiral  22:04, 29 October 2011 (UTC)


 * definitely not. The text now there is concise and clear: the text above is a far less clear and much longer, while the source is even less accessible: even though I spent a decade programming C/C++ the above source does nothing to help me understand the mathematics.-- JohnBlackburne wordsdeeds 22:48, 29 October 2011 (UTC)
 * There are three sentences in the section. I think a section with three paragraphs would look stylish: 1)A basic laypersons description of "the miraculous facet of the gem" 2)A paragraph starting out "More formally..." including the uniform distribution concept, and 3)A mathematically worded description of the equation that does not include the formality.  But honestly, I flounder, and need some assistance to meet the many calls, in the other discussions, for a wider audience, for more comprehensibility, for Euler's insights. The C program is for those in this discussion who would like to compile and run it for the purposes of viewing its instructive output.  (I'm sorry the title was confusing.  I've changed it.) The Euler number is a terrible thing to hide in plain sight.  I read there what would might pass a mathematics masters exam, but might also easily fail to affect the spirit of even an Electrical Engineering graduate, such as myself.  I hope to read some more reasoning from others like you.  Thank you for your response.  &mdash;  Cp i r al Cpiral  05:10, 30 October 2011 (UTC)

Thanks to Mr. Blackburne's assessment, and the now-obvious reality of how busy we all are, I have taken the time myself, without further ado, to correct the logical errors in my first proposal, and to also make it more organically lucid. I am willing to struggle this addition into the section for this reason: formalities, although compact, are not easy to read. (Ask the computer programmer which of another's code they'd rather read, compact or lucid.) Formality in any guise is not basically understood (Why do we...?), although often memorized. Formality is compact. Genre-specific symbolic-experts only can unravel it. That's reality, and its OK with me. But consider also that experts must often skip-over the basics, and the precocious the elementary. That should be OK as well. &mdash; Cp i r al Cpiral  22:34, 1 November 2011 (UTC)
 * Please do not try to rework the above explanation or code sample with a view to adding it to the article because, while I agree that a sample program is great for a programmer who wants a good insight into details, it is simply not helpful for this article. Johnuniq (talk) 00:20, 2 November 2011 (UTC)
 * I've removed the C program and feel very sorry for the confusion. The already reworked proposed wording in the top box remains patiently waiting for comment.  Thank you all.  Lack of comments raises boldness. &mdash;  Cp i r al Cpiral  06:35, 2 November 2011 (UTC)

Reworked proposal. Happy editing. &mdash; Cp i r al Cpiral  21:52, 2 November 2011 (UTC)

Polished proposal. (See above). Thank you. &mdash; Cp i r al Cpiral  21:42, 5 November 2011 (UTC)
 * No, the text there is much better than your version above. Mostly yours is too long: why read so much text to get the same information that's there already? Adding a paragraph on [0, M] before switching to [0,1] just adds unnecessary complexity when either will do and [0, 1] is clearer. See WP:EDITORIAL for why words like 'easily' and 'bewildering' should never be used, and emphasis such as at the end of the first paragraph should also never be used in that way. Also although this doesn't affect the content your paragraphs above have far too many line breaks, making it difficult to read your text in the edit window and potentially difficult to edit if such text were used in an article.-- JohnBlackburne wordsdeeds 23:57, 5 November 2011 (UTC)
 * Right, I forgot about WP:EDITORIAL. I have de-emphasized this "amazing coincidence", for the elementary approach I use makes the mystery "2.7 because not 1, not 2, not 4, but, a sort of 3 that is e" clear to all readers (an A-class article characteristic). (No one has ever explained it?)


 * By the way, what now stands as "the full explanation" is now part of my proposed presentation of the section. Because of this inclusion, and because it was allowed to be corrected recently--the (0,1) was changed to the [0,1]--please permit me these further humble opinions, as a sort of penance that includes some kind of necessary, entire review: I believe this subsection to be unclear concerning:
 * the concept of E, expected value. (It is not mentioned in, or linked out of, the article.)
 * the operation of E. In the algorithm, there two operations of E, plus one operation of avg. It reads "E(U)=e", as if there was only one operation of E. (Granted, it does imply two runs where it says "sample averages".)  Should it say like avg( E(E(minU_1)) ) = avg(E(3)) = e?
 * the relation of e to n. It reads "U variables will approximate e". The random variable n is not e, although that statement begs the image Xe. The "U variables" appear to be n, but are not the U variables some N that is 1) random numbers N generated from 2) another set of random numbers n?  (Again I grant that it does imply some kind of N where it says "sample averages".)
 * the written but unspecified notations E and U should be clearly specified and therein wikilinked. If U is the continuous uniform distribution U(0,1), then let us not present it as it is now "U=min anything". U should be the standard usage, reserved for the Uniform Distribution U(0,1).  Its use here as E(U) is confusing.
 * closed form or open form: the spirit of the section and subsection is "a representation of e" (oh boy!), but the result "will approximate e" is anti-climactic.
 * Our short subsection reads like a personal genius' notebook, and is wikilink bare, and so I recall "Wikipedia is not a collection of information".


 * The way I see my proposal is this: We can either force the expert reader to skip over the elementary approach I propose, or force the average reader to skip over something that is not to be presented. Experts get used to skipping over the elementary. The average reader cannot get used to skipping over what is not there in either wikilink, or some explication.  Please note that the article is only 29 KB, and the recommended range is 30 to 50. Why have only one expression of e over the reals, when it works over discrete intervals as well, as I have shown? &mdash;  Cp i r al Cpiral  21:11, 6 November 2011 (UTC)

Would somebody please verify that this: random samples $e$, $e$, ..., $x$ of size $y = 1/x$ from a continuous uniform distribution on [0, 1] are used to approximate $x = 1$. If


 * $$U= \min { \left \{ n \mid X_1+X_2+\cdots+X_n > 1 \right \} },$$

then the expectation of $x = e$ is $e$: $n!$. Thus sample averages of $n$ variables will approximate $e$. means this (working backwards, deducing the meanings of the symbols):

E(U)=e, where (implied by the definition of "expected value"), and
 * E is a statistical operator that generates "the one selection that is the weighted mean from all possible values of U"

(implied by the set-builder notation used).
 * U is a function that generates one infinite sample: all possible values of n

The sample U represents is an (infinite sized) multiset, each member of which is an integer n in (the infinite range) [2,∞), generally varying around n=3.

Note how the reader covers concepts in statistics (expected value, sample, population), set theory (notation, multisets), mathematics (function, operator), and how infinity is mixed up in each, and how we might reduce this burden (on most readers) to focus more on $X_{1}$ itself by reducing those concepts as much as possible, as in the proposal at the top: 1) The infinite population of the real interval could instead be the simple, finite, whole number interval [0,1,2,3]. 2) The set notation could be eliminated (but left as a formality for expert mathematicians). 3)The ideas of a function and an operator can be eliminated. 4) The ideas of an infinite sample and an infinite population could be eliminated (see the above proposal) by being subtly implied at the same time as focusing on "why" n is usually 3, that "why" being a clear, fully inclusive logic that leaves the astounding result, "hanging in the air" from the one (absolutely necessary) remaining concept of "expected value". The fact is that it works very well over an easily imagined, uniformly discrete interval (correct me if I'm wrong, but the C program I had posted originally convinced me, graphically).

&mdash; Cp i r al Cpiral  21:35, 7 November 2011 (UTC) Who am I to Q.uestion why the poor (as in "desperate reaches from any masses") but necessary explanation of this I will do for English WP? A. contributor to a popular ("of the masses") encyclopedia.
 * I'm not going to wade through your lengthy interpretation and try and understand what you're getting at. I've taken a look at the section and tried improving it myself, adding a sentence which expresses it in simpler language, editing the rest so it uses simpler language, and adding a couple of wikilinks, which hopefully addresses some of your concerns. Anyone unfamiliar with the approach can now use the links to find out more. This is not a statistics article so it is not the place to go into a lot of detail on such techniques.-- JohnBlackburne wordsdeeds 00:21, 8 November 2011 (UTC)
 * Links are great, yea? Not sure which you are referring to with "this is not a statistics article". Here or there? In all respect, I watch for help with my proposal here from those who read it. (I find it odd that I ask for help and get told I can't teach.  Hmmm.) I highly recommend participants read the entire discussion page carefully and considerately as concerns the "good quality" issues going on.  Hopefully they don't really need to just go away. &mdash;  Cp i r al <u style="color:#2820F0;">Cpiral  10:43, 8 November 2011 (UTC)

I'm pretty sure that the stochastic aspects of $X_{2}$ can be opened up better than is currently available in the article. By "better" I mean an improved article rating, which means, in one real sense, a wider audience. One section of an article on $X_{n}$ might be made as accessible as possible to the widest, curious, audience possible by having subsections with increasing difficulty. The nature of curiosity of a mathematical underling is not capricious. Hilbert's gravestone says it of mathematicians even better: "We can know. We will know." Mathematics is a word coined by Pythagorus meaning 'that which is learned' (according to The Mystery of the Aleph by Amir Aczel p.14.)

In defense of the Elementary protocol section: Kronecker's finitism made him a forerunner of intuitionism in foundations of mathematics, and in is book On the Concept of Number attempted to banish from mathematics what he regarded as unnecessary levels of abstraction, "anything... that could not be derived from integers in a finite number of steps". "The computational method is easiest for non-mathematicians to understand."(John Derbyshire in Prime Obsession p. 185 and 198). In my proposal the Elementary protocol subsection (integer points) is tied by the Geometric protocol subsection (shapes) to the Formal protocol subsection (an algebra of real points and functions). As it is now the short, formal section Stochastic representations of $n$ keeps the article in 21st century math.

I showed above that this article is too short in size to be overly concerned about adding material. "Please note that the article is only 29 KB, and the recommended range is 30 to 50." In my proposal above I am trying to improve the article by using a computational method on whole numbers, and a "visual" geometric approach, in order to expand the audience that wants to understand the stochastic aspect of $e$. &mdash; Cp i r al <u style="color:#2820F0;">Cpiral  03:49, 14 November 2011 (UTC)
 * That is much worse than the first version you posted. This is not a statistics or statistical methods article, so it is not the place to explain the detail of the method. Even in such an article Wikipedia is not a textbook and so lengthy how-to sections do not belong anywhere. Finally it is poorly written in a chatty, unencyclopaedic style that gets in the way of any explanation, and with lots of advanced geometric concepts introduced without justification that just add to the knowledge needed to understand it.-- JohnBlackburne words<sub style="margin-left:-2.0ex;">deeds 04:12, 14 November 2011 (UTC)
 * The positive criticism, your worthy point that a chatty style interferes with explanation, is really a point on the style (what is chosen) of the article $U$ as it now stands. I reworked that stylistic problem in the above proposal (which concerns $e$'s stochastic nature). The negative criticism, that the content is "textbook", and "how-to" (what is banned) simplifies to the debate about ever using an heuristic argument (a debate about banning them) from Wikipedia mathematics articles in general. I find heuristics in featured math articles such as Euclidean_algorithm and there are possible others that use them. (So there. I make a counterpoint.) Your point that a geometric ("visual") progression has no justification, I contend lacks merit.  The Stochastic representations section lacks completion because there exist several "protocols", some with wider appeal.
 * Overall, I am not discouraged because I have passed more progress points (facts, arguments) by you (indicated by silence on those matters), than I have failed. I would love the honor of your attention in simple debate (but alas, expect silence) on one of the two worthy points raised about the above proposal:
 * Heuristic: refers to an experience-based technique for problem solving [textbook! how-to!], [but also for] learning [what is known], and discovery [of the known nature of a thing].
 * Encyclopedia: "a type of reference work... holding a summary of information...; the phrase literally translates as 'complete instruction' or 'complete knowledge'... Articles focus on factual information to cover the thing or concept... [focusing] [much more on the nature] of the things and concepts referred to by the [word or phrase] for which the article name stands", [$E(U) = e$]. The style of the articulation may vary from article to article, depending on the nature of the topic and the methods fit to convey the topic completely.
 * Is there room in this article for more than one protocol in the section Stochastic representations? Yes.
 * Should this article's style resort to a geometric heuristic to convey the stochastic nature of $U$? Yes, if wikilinked. &mdash;  Cp i r al <u style="color:#2820F0;">Cpiral  20:53, 14 November 2011 (UTC)

Arbitrary break
The policy I was referring to is here: WP:NOTTEXTBOOK: there is no need to explain and prove something unless it has encyclopaedic value. The geometric point is that discussing hypercubes (I think that's what you mean by supercube) etc. introduces very advanced geometry that simply confuses the issues. But the main issue is still this is not an article on stochastic methods, and a page-long digression into such has no place in this article.-- JohnBlackburne words<sub style="margin-left:-2.0ex;">deeds 21:34, 14 November 2011 (UTC)


 * You have indicated that you may find something in my efforts although Johnuniq warned. You directed my attention to proper Wikipedia content here.  I also learned at MOS:MATH that there is some leeway concerning proofs when they "really expose the meaning of some theorem". Like the calculus section, I think the sentences with all the n!'s, above, should be kept because they are part of sentences that also contain the sequence of ideas that are necessary to "obtain".  So it should be as obvious (if you have any interest at all) that any expansion of this section cannot be like the Derangements or Asymptotics sections and just give the formula.  The heuristic style is the only way to inform of such a stepwise thing as a random selection that then produces large numbers of trials of resulting random numbers that are then processed with probability theory to proceed to e.


 * The popular article on pi is 88KB. The popular article on the golden ratio is 76KB. The way to this possible addition to the popular article on e necessitated at least several multi-purpose sentences for each idea, divided into a few paragraphs.  Additionally, I'm sure you'll agree to the wider appeal that comes with the various approaches to such a remarkable attribute of e, and how articles like this one and pi and the golden ratio, should be somewhat soft on the readership where possible. Let us all then simply agree on something the right size to contain the sketches above. I hope I have produced that in my work, now drastically changed, above.  (Also please remember my constant appeals to article WP:CLASSES, and what it says it takes to get to the next level.) Thank you very much for your inputs. Happy editing! &mdash;  Cp i r al <u style="color:#2820F0;">Cpiral  03:10, 16 November 2011 (UTC)
 * Because this article is not too long does not mean anything no matter how appropriate can be included, and how long other articles are is irrelevant. As for MOS:MATH it says as a rule of thumb, include proofs when they expose or illuminate the concept or idea; don't include them when they serve only to establish the correctness of a result. The above does not illuminate e, it just shows how to calculate the result. It is perhaps a useful example for an article on stochastic processes but not here.-- JohnBlackburne words<sub style="margin-left:-2.0ex;">deeds 03:48, 16 November 2011 (UTC)
 * Yes and perhaps section 2.1 would fit also in an article on Bernoulli trials. And taken in isolation the current version of Stochastic representations would fit in the list article List of representations of e. This is a very clean and tight article, and 2.1 talks about bank accounts, and 2.2 talks about gambling casinos, as they should.  For encyclopedic fullness we add the first paragraphical explication to illuminate the stochastic process, and the second the statistical process.  Together these show physically (a "deed") what the current version only shows mentally (a "knowing"). My aim is to illuminate how the entire process in its fullness plays the same role as the bank accounts and gambling casinos do in the other sections that proceed to ascertain the meaning of e (and by no means, thanks to JohnBlackburne, to prove the "correctness of the math") in such a way that might appeal to the engineers and budding mathematicians.  To repeat, the current version packs analysis, advanced algebra, and set notation into 1 equation in 1 sentence; the proposed version adds geometry and probability in 3 formulas and 1 equation in 11 sentences which (if I, now become biased by induction, may say so myself) flow beautifully from the arithmetic to the geometry to the probability, and to $e$.  The current version misses the crucial idea behind isolating $e$ between 3 and 2, shown in the proposed sentence "after...but not before".

<div class="boilerplate" style="background-color: #efe; margin: 2em 0 0 0; padding: 0 10px 0 10px; border: 1px dotted #aaa;"> In addition to analytical techniques and expressions involving $e$, there is a unique stochastic process that acertains $e$.

Consider the terms of a sequence of partial sums generated by n random trials from the interval $[0,1]$ until an $e$ term exeeds 1. A population of 100 of such samples will often have an average number of N = 2.7 terms. This computation holds true for zero-based, continuous or discrete, intervals. A large population from a large interval will have exactly $e$ terms on average. In other words the mean number of trials needed for the sum of the trial values to exceed a uniform interval, is $e$.

A more visual approach considers all the possible sequences with two terms, with three terms and so on. Here, the unit square contains the vector sum of any two-term, and the unit cube any three-term sequences. We use $[0,1]$ on the axes of these unit spaces so they equal the probability space and so that the sampled space for a sequence of n trials that equals 1 or less is now in general $${1/(n!)}$$: 1/2 for the square, 1/6 for the cube, and so on. The probability that a total of 1 is exceeded after $e$ terms is the complementary event $$1-{1/(n!)}$$. The probability that a total of 1 is exceeded after $e$ terms but not before simplifies to $${(n-1)/(n!)}$$. The expected number of terms until a total of 1 is exceeded is therefore an exactly as expected, probabilistic expression of $e$
 * $$e = \sum_{n=2}^\infty { n (n-1)/(n!)}$$

derived from the unique stochastic process that would generate such trials.

More formally if $e$ trials of continuous random variables $e$, $e$, ..., $e$ from the standard uniform distribution form a sample of size $N^{th}$, limited such that
 * $$N = {\min \left \{ n \mid X_1+X_2+\cdots+X_n > 1 \right \} },$$

then the expected value of a discrete random variable $e$ is $e$, or $n$. I think I have adopted all your reasonable remarks (thanks). Do you have any other (positive) criticism, or rebuttle concerning "illuminates"? (Since when was Wikipedia "irrelevant"?!) &mdash;  Cp i r al <u style="color:#2820F0;">Cpiral  00:56, 17 November 2011 (UTC)


 * Your text is long. The current text is short. I prefer short. (Yours does have the advantage of explaining why the statement is true, but I don't think that outweighs its verbosity.) Ozob (talk) 00:34, 18 November 2011 (UTC)
 * "Too long" for what? "Why" which "statement" is true, the formal statement? "Why" indeed could be how, could it not? and if so offers the "meaning of the theorem" defense (per MOS:MATH, the theorem being "e has a stochastic property"). It is up to an informed concensus.  Discuss? &mdash;  Cp i r al <u style="color:#2820F0;">Cpiral  02:53, 18 November 2011 (UTC)
 * But to intervene before much debate about the two sentences that look like "correctness" proofs:
 * "The probability that a total of 1 is exceeded after n terms is the complementary event 1 − 1 / (n!)" ties the geometrical (triangular area or pyramidal volume) with the probabilistic.  Because this was forshadowed in the previous sentence where it says "unit space = probability space", it obviously seems to me (the author) to be a point that bares repetition, but that's because at my level, I notice with glee the subtle differences of 1 throughout it.  So it can be sacrificed.
 * "The probability that a total of 1 is exceeded after n terms but not before simplifies to (n − 1) / (n!)" holds the crucial idea that e is 'after' two but 'before' three. In an earlier version this was explained in an entire paragraph ala "stochastic e develops from 0 (never), 1 (never), 2 (not expected), 3 (usually), and 4 (over the population [0,1,2,3])".
 * As such they are not mathematical handholding or correctness proofs. They're compactions. &mdash;  Cp i r al <u style="color:#2820F0;">Cpiral  05:17, 18 November 2011 (UTC)


 * The stochastic business has little weight in the context of e. I would have though none of it should have been in the article in the first place. The current bit is short enough to creep in as an extra but it really has not established its weight for inclusion. Dmcq (talk) 01:22, 18 November 2011 (UTC)
 * Please consider the Encyclopedia argument above (near Heuristic). &mdash; Cp i r al <u style="color:#2820F0;">Cpiral  04:09, 18 November 2011 (UTC)


 * (ec)The section is interesting as an example of how to compute the expected value of a stochastic process, but the relation to the subject is rather weak. It's an application of the series for e, but it doesn't really help explain the place of e in mathematics. As a method of computing e it's problematic for a number of reasons, not the least because it will converge very slowly. If the section is given at all then some explanation why it's true might help, though I agree that the version proposed is too verbose.--RDBury (talk) 01:33, 18 November 2011 (UTC)
 * I don't believe that there is the (banned) how-to spirit there except for its exemption: to outline the 'biology' of 'the stochastic nature of e'.  I hope another, careful read of the just edited version will brighten your view.  (I promise not to edit the content further.) On the second point, that the proposal does not "explain the place of e in mathematics" (may I say "is not mathematical enough for you?") e is really more physical than mathematical, and "the series for e" I would counter, is just (another) probabilistic coincidence, that supercedes "discoveries" in mathematics.  Thus because it is said 'e is everywhere', the article should have some non-mathematical accompaniment; however, you may argue that it depends on the definition of encyclopedia where it says "from either all branches of knowledge or a particular branch of knowledge." On the third: it's not meant to compute, but to inform by elementary arithmetic, underlings not unlike myself 19 days ago.  On your "verbosity" complaint: maybe I could scrape a few phrases off the skinny. &mdash;  Cp i r al <u style="color:#2820F0;">Cpiral  02:53, 18 November 2011 (UTC)

Concern about compound interest formula
I have a problem with the last statement in the compound interest section. It appears to be true for a single year, but not for multiple years. Should it be corrected?

Suppose I start with $1 at 5% interest, for 10 years. Simple interest would lead to $1.62889 Using the equation, R is 0.62889, and the continuously compound interest result should be e^0.62889 = $1.8755. That's not correct. The compound interest is actually e^0.5 = $1.6487. You can verify that by trying it with small increments. I also found an online calculator.

The problem is much greater with larger time periods. Consider 20 years. The simple interest leads to a final value of $2.6533, for an R of 1.6533. This suggests a continuously compounded interest of e^1.6533 = $5.224, which is much too high. Continuous compounding doesn't produce any such result. The correct value is $2.71828 (since .05*20 = 1 we get e itself as the new value).

Can somebody say what that formula should be? Or should it be replaced by a more standard one? Ed Gracely (talk) 16:51, 24 November 2011 (UTC)


 * The section is only discussing the effects over a single year.  Sławomir Biały  (talk) 12:25, 25 November 2011 (UTC)


 * The sentence in question starts "More generally..." and is confusing even for one year. I've replaced it with a general formula.--agr (talk) 12:51, 25 November 2011 (UTC)


 * Good point.  Sławomir Biały  (talk) 13:01, 25 November 2011 (UTC)


 * Thanks, Arnold (agr). I thought that your formula was the correct one, but didn't feel quite expert enough as to replace it myself, and I wasn't sure if the original author was making some specific point that might be lost. This is better. Ed Gracely (talk) 14:45, 25 November 2011 (UTC)

1731 > 1728
As of this revision, the History section appears to be self-contradictory. Is anyone able to straighten this out? Sławomir Biały (talk) 22:00, 25 November 2011 (UTC)

Recent Stochastic Representations
You may ignore this, but I think the recent changes to Stochastic representations are a funny misinterpretation of the mathematics in general and the pheneomena in particular, and that it was correct before those recent changes, but only almost. My opinions are laughable. I mean that literally, for I am not a mathematician.

The nature of e in chance theory:
 * It's not a representation, its a property. (Resist the temptation to use it as a competitive algorithm.)
 * It's not "representations", its a unique stochastic property that might be called "exceed me".
 * U(0,1) is "a closed standard realm" for random variables, the equivalent of some function random(X), as simple as the simplest operation we might call "pick a card". (Please don't say "U=min ", say V or something.)
 * One sample is usually three random variables, after that it's a simple average of a secondary population of n. There is no more sampling done after the simple ones that makes the n's happen.

What it should return to, almost: In addition to analytical techniques and expressions involving $n$, there is a unique stochastic process that acertains $e$. If random variables $n$, $X_{1}$, ..., $X_{2}$ of sample size $X_{n}$ from the standard uniform distribution are limited such that
 * $$V= \min { \left \{ n \mid X_1+X_2+\cdots+X_n > 1 \right \} },$$

then the expected value of $n$ is $N$, or $e$. I say all this because I'm working hard on the discussion section above this one, and I really want to get lambasted by an understanding mathematicians so my proposal for this section will become correct. But in all respect, I am not wholy unqualified to speak, or even revert the edit, just (perhaps overly) curious about the resolution of these recent discussions. &mdash; Cp i r al <u style="color:#2820F0;">Cpiral  07:52, 18 November 2011 (UTC)


 * Agree calling the section something like 'Stochastic estimation' would be better and U was a bad choice for the sum. Otherwise the English bit there at the moment is better, we need to remember the target audience. The method is simple not unique. We should be pretty explicit rather than just referring to 'standard'. I would go for something like

There is simple stochastic process that estimates $e$. The number of random variables with a uniform distribution on $[0, 1]$ that need to be added up to first exceed $E(N) = e$ is on average $e$. In other words if $e$ and
 * $$V= \min { \left \{ n \mid X_1+X_2+\cdots+X_n > 1 \right \} },$$

then the expected value of $V$ is $e$, or $e$.
 * and if it can be simplified a biut more then well and good. Dmcq (talk) 10:17, 18 November 2011 (UTC)


 * Looks good, though there's no need to write 'simple' as readers can decide that for themselves, per WP:EDITORIAL.-- JohnBlackburne words<sub style="margin-left:-2.0ex;">deeds 10:43, 18 November 2011 (UTC)
 * No. It is simple to anyone, for who can count using natural numbers? even most crows. The arithmetic part is taking the mean.  What you are refering to is more complex, is generally misunderstood, is the nomenclature and naming conventions the equation packs in.  It's too formal.  Please review the proposal without bias. Four days ago in our discussion, starting at "Encyclopedia" near "Heuristic" I was doubly sure of it. As always I have you to thank. Thank you. &mdash;  Cp i r al <u style="color:#2820F0;">Cpiral  16:39, 18 November 2011 (UTC)


 * No. The stochastic process does not "estimate e", the mean does.
 * The uniform distribution is complex conceptually, but I have not seen its equivelent, some unit line segment, to refer to as I have seen unit square and unit cube. We could omit mention of the thing except for the formal need of this section to use "random variable X", and say a much wordier "a random selection n over the closed, real interval [0,1]", but... the section is only a compact, formal explication.  The uniform distribution is not a number line, its a equiprobability space that may contain much more mathematical tooling than a simple output of one number.  The standard uniform distribution is "like" the missing "unit line segment" except that a random variable may "return" an interval, not just a number.  Where is the unit line segment when you need it to build a unit square or cube?  Nowhere that I know of.
 * Unit interval &mdash; Cp i r al <u style="color:#2820F0;">Cpiral  11:06, 22 November 2011 (UTC) Embarrassed red.  (Was green)
 * Furthermore about the U: the original version was mis-copied from the less-than stellar reference I just now studied, where they decided to use U. (They were high school teachers?)  The copy left out the difference between n and N, which I pointed out 12 days ago.  The other reference bar graphic from some fancy math-suite app is inferior to to the numeric output the C program (above section) performed for me: ten steps, three numbers (illuminated me 20 days ago.  BTW, 4 days ago, I finally got the terminology and nomenclature.  Ya know... I am the wider audience I write for in the proposal above.  What's 20-4 days to me?  A lot of effort for widening the audience per WP:CLASSES. )
 * About the section title. It should be Stochastic under Properties. &mdash;  Cp i r al <u style="color:#2820F0;">Cpiral  16:39, 18 November 2011 (UTC)

No no no. These remarks all miss the point of this section. The article has now been saying for 12 hours that there are an infinite number of X's. You should watch more carefully what the words are saying everywhere: in your watchlist, in my proposal, and in these talk page sections here and the one above, and in the references. Now I'm becoming bolder to add my proposal outright and get some more eyes. Let us conclude. Let us resolve, not the courts please. That's inefficient. &mdash; Cp i r al <u style="color:#2820F0;">Cpiral  16:39, 18 November 2011 (UTC)
 * And I've reverted it as it made things much worse. In particular: it's not unique; 'acertains' apart from being misspelt does not make sense; there are not only n variables, as n is undetermined; the standard uniform distribution is not well known and should be specified; 'limited' like 'acertains' is poor English and also does not make sense.-- JohnBlackburne words<sub style="margin-left:-2.0ex;">deeds 17:00, 18 November 2011 (UTC)
 * Remarkably, you restored the version with infinite X's, reverting my reversion. (I made an honest mistake between "::" and ":" and interpreted your "Looks good" as a go ahead, when in actuallity it was your comment to Dmcq. My apologies. Additionally, thanks for correcting the discussion's order.) More importantly do you mean that n is infinite?.  I thought it was usually 3. &mdash;  Cp i r al <u style="color:#2820F0;">Cpiral  21:10, 18 November 2011 (UTC)
 * Yes, "ascertain". But reconsider "limited".  In fact the infinite X's error is an easy misunderstanding without it.  The standard uniform distribution is well known.  That's why they call it a standard. I am still emboldened, even without "proper English", and maintain that the section needs to be renamed and moved (see above), and re-written (see previous).  &mdash;  Cp i r al <u style="color:#2820F0;">Cpiral  17:13, 18 November 2011 (UTC)


 * Question by an outsider: is it really unique? I would think that there are many stochastic techniques that approximate e. CRGreathouse (t | c) 20:10, 18 November 2011 (UTC)
 * I'd guess there's a way of transforming the secretary problem for instance into one so I doubt one would have to look far. I think they must have had a different idea of unique like saying 'very unique' to mean something specially interesting. Saying very unique grates on me so please don't! :) Dmcq (talk) 20:53, 18 November 2011 (UTC)
 * Looks better as it stands now. And, as CRGreathouse and Dmcq suggest, there are other stochastic techniques that approximate e. -- 203.171.197.251 (talk) 23:35, 18 November 2011 (UTC)
 * Pray tell, most honorable and welcome non-parameterized statistic, why does not the most recent version
 * $$U= \min { \left \{ n \mid X_1+X_2+\cdots+X_n > 1 \right \} }$$
 * not say
 * $$U= { \left \{ \min { n \mid X_1+X_2+\cdots+X_n > 1 } \right \} } $$
 * More to the point I ask you hopefully and expectantly, what, recently, is "infinite" about X, a sequence or a series? (Does "+" mean "," i.e. the Complex plane?) According to the recent wording its an "infinite sequence".  Assuming standard notation is used, its a finite series. All this assumes the awesome E(U)=e, and therefore that my personal role as (Cpiral) has an an unstated function domain U = {2,3,4,3,2,3,2,5,3,3...} &mdash;  Cp i r al <u style="color:#2820F0;">Cpiral  01:55, 19 November 2011 (UTC)
 * Because this
 * $$U= \min { \left \{ n \mid X_1+X_2+\cdots+X_n > 1 \right \} }$$
 * is correct while the other is meaningless. As for infinite that's needed for the problem. U could be any integer: with decreasing probability it could be 2, 3, 4, ... but every integer has a finite probability. Therefore the range of U is infinite and you have to have an infinite number of Xi to test it. If there are only finite Xi, 1000 say, then E(U) will not be e but slightly less.-- JohnBlackburne words<sub style="margin-left:-2.0ex;">deeds 02:20, 19 November 2011 (UTC)
 * Thank you so very very much. &mdash; Cp i r al <u style="color:#2820F0;">Cpiral  21:07, 19 November 2011 (UTC)


 * I think it's worth pointing out that U (which I've just changed to V in the article) is a random variable. It's a random variable defined in terms of the infinite (yes, infinite) sequence of random variables X1, X2, X3, ....  It's because V is a random variable that it makes sense to take its expectation.  Ozob (talk) 03:21, 19 November 2011 (UTC)
 * Thanks! But about the infinite X's part: the equation has metastasized. The heralded section has gone beyond "too formal" being one step worse for me. The English use of the symbols are wrong because they are not  about the equation, but rather they are, for me, now biased and intuitively comfortable, in the advanced algebraic equation, which homes sui generis, multi-purposed symbols.  I am also for homophonics in English, but not when it is, as it should be here, about a formal math equation. I've acquired a bias for this equation, except the new "infinite X's" part, the one putting what I will now call Me, the prototypical "next step audience", Ne, back into confusion from a curiosly driven talkpage breakthrough.  It is for the turn of Me now as JohnBlackburne says "worse".  (Given my authorship of the previous proposal for the section as the only step forward, I am actually not confused myself, but Me is now two steps backward.)  Let me explain My position, analyzing the one, too obvious confusion from the several angles offered therein, and I quote.
 * Xn: The Xn is either a series of trials "X1 + X2 + X3, ...." in a sample limited by "> 1", or a sequence of samples "X1, X2, X3, ...." not limited at all. Xn is portrayed as both a trial—what constitutes the continuous random variable X, and is what the binary operator + on Xn require—and a sample of trials—what the discrete random variable V accumulates for E—each denoted Xn. The same symbol, Xn is now heralded in the English Wikipedia to mean, both at once, the continuous random variable trials and the discrete random variable, whose samples comprises the number V. Which is it?
 * 'n' or 'N':The succesor to the last n < 1, i.e. the first n > 1, i.e. the min n, let us denote as N, as the referenced source to the equation does. The article says "Let V be the least [discrete random variable] n", so "n" is now associate with both X and V. Is n multipurposed here?. Isn't that situation unacceptable by Wikipedia standards?  Granted, both n and N are on [2,∞), but n is stochastic, and N is a blip at the ending in an infinite-loving mean operation on E(V(N1, N2, ...N∞)) To Me, n seems finite, limited by "< 1" with an X that seems finite by association. Is there an unstated-in-English, but implied step with N = the successor to the nth term, and a similarly implied step that V is multivalued? I think these are so, for I use implied steps plentifully in my proposal (previous section). Let it stand, then, but without "ininite X's". Let it be stated though, in a way that differentiates n from N, not so error-frought in it's implications, as does my proposal (in the previous section).
 * sum(X) and avg(V): Each Xn is a trial, and each VN is a sample. The article says "sum of the first n samples", similar to where historical versions have said "sample averages of U". The new version confuses "sample" with "trial". The older version was more accurately stated.
 * Random variable: The article says "independent random variable" X, while the wikilink says only that there is either a "discrete random variable" (multivalued, like V) or a "continuous random variables". Why say an "independent random variable"?
 * Please help Me if you can. &mdash; Cp i r al <u style="color:#2820F0;">Cpiral  21:07, 19 November 2011 (UTC)
 * No, the current version is clear and correct. You are simply seeing problems that aren't there that seem to arise from your own misunderstanding of the techniques being used. As such, because this article is not about stochastic methods, you probably need to look elsewhere for help with this, to other articles or more probably sources other than Wikipedia, as Wikipedia serves poorly as a textbook on topics you don't understand.-- JohnBlackburne words<sub style="margin-left:-2.0ex;">deeds 22:12, 19 November 2011 (UTC)
 * I do not understand your first paragraph at all.
 * In the present context, a trial is the same thing as a sample.
 * You should not denote the minimum n by N. It already has a name, V.
 * You seem to be misunderstanding some of the terminology used in the article. I believe that it is within your reach, but you will have to study more probability first. Ozob (talk) 22:44, 19 November 2011 (UTC)
 * My pained style is discomforting, and yours is an admirable example.
 * We should differentiate "trial" from "sample" precisely as is done in Sample (statistics) because, as it says in the sourced reference [11], "Students will generally be comfortable with the notion of the sum of a fixed number of random variables, but the sum of a random number of random variables may cause some difficulty." N is a sample, and n is a trial. Now make both variables random variables. Yes?
 * Xn: Still, the X's are each presented, indistinguishable symbolically, as two different entities differing in cardinality. Are you saying that's not confusing? Yes infinitely? &mdash; Cp i r al <u style="color:#2820F0;">Cpiral  00:11, 20 November 2011 (UTC)
 * It's perfectly clear. Again, if you don't understand what is a straightforward example you need to look elsewhere for explanation. -- JohnBlackburne words<sub style="margin-left:-2.0ex;">deeds 00:28, 20 November 2011 (UTC)
 * I'm sure "it" is, but this does not seem about you or me in the previous section. What is "it"? "It" remains unspecified.&mdash;  Cp i r al <u style="color:#2820F0;">Cpiral  01:11, 20 November 2011 (UTC)
 * By 'it' I mean the current version, the subject of my previous comment. It's clear and correct and if you don't understand it you need to look elsewhere for such understanding.-- JohnBlackburne words<sub style="margin-left:-2.0ex;">deeds 01:29, 20 November 2011 (UT
 * Specifically which part of X and "sample" is clear, and how am I to be satisfied by more than mere strokes of keyboards? You did ask the WikiProject:Mathematics to help us. Now I'm going out, to help someone I love.  Goodbye for now. &mdash;  Cp i r al <u style="color:#2820F0;">Cpiral  02:27, 20 November 2011 (UTC)

The whole of the subsection is clear: if you do not understand it then you won't find the information here: the articles linked in that section would be a good place to start. I posted to the WikiProject Mathematics talk page as you and I were at a standstill. It is normal in such circumstances to ask for a third opinion. There is a semi-formal mechanism for this at WP:3 but as it was a mathematical disagreement it made sense to ask mathematicians for help resolving the dispute.-- JohnBlackburne words<sub style="margin-left:-2.0ex;">deeds 02:40, 20 November 2011 (UTC)
 * OK, thank you for your perseverence. Then the three links and two refs in the section should work, and


 * 1) the n of Xn really is both limited (the section rightly says "the least number n") and infinite (the section strongly implies "an infinite sequence" with n terms), making that
 * 2) trial and sample really are the same word because V is a function that generates one sample with infinite "trials" in a single probability space (not a multiset): all possible values of V and similarly all possible values of each Xn at once, making
 * 3) a "+" operator with really unusual meaning that is
 * 4) not unlike the $$sum$$ of probability spaces I document in my proposal, and also
 * 5) not unlike the "+" in a complex number
 * 6) and an unusual set notation really has extensional definitions that go outside the curly bracket.
 * I just might enjoy trying on the previous section (style and content) much later. (You are all right.  I did not know that, but I have a curious intuition Zeno, Reimann, and Deutche are blah blah blah .) &mdash;  Cp i r al <u style="color:#2820F0;">Cpiral  08:36, 20 November 2011 (UTC)
 * The "+" and the set notation are used in the same manner that they are used in thousands of first textbooks on probability theory, as is the word "random variable" (as opposed to "sample", "trial", and other terms that have been tossed around). This isn't the place to argue against completely standard conventions in probability theory.  Also, there really are infinitely many Xi.   Sławomir Biały  (talk) 14:25, 20 November 2011 (UTC)
 * The siren call of any noumenon or textbook arouses my sentiment, but I rather feel a multiply beneficial duty to the wiki, and to the participants here, whose every attention fills me with honor. Most importantly I will take my three weeks of study on it, turn it into three minutes for an audience at my level.
 * Prime Obsession (2004) p 15, says "Every mathematical statement that contains the word 'infinity' can be reformulated without that word" and "Modern analysis does not admit [the concept of the infinite and the infinitesimal]" because they "created serious problems in math during the early nineteenth century...and eventually they were swept away altogether in a great reform." Ugh. Ugh.
 * Perhaps something like


 * Consider the terms of a sequence of partial sums generated by $X_{1}$ random selections from some finite, zero-based, continuous or discrete, interval until an $X_{2}$ term exeeds the size of the interval. The expected value of N is $X_{n}$.


 * A more visual approach considers the random field of each N-term event, each in its own probability space, and then adds them. Here
 * $$e = \sum_{n=2}^\infty { n (n-1)/(n!)}$$.


 * The temporal version of this is e = E(N) where N is a random variable composed of random variables $n$, $V$, ..., drawn from the uniform distribution on [0, 1] such that
 * $$N = \min { \left \{ n \mid X_1+X_2+\cdots+X_n > 1 \right \} }.$$.


 * Besides, "infinite X" is excess verbiage, like the finest print in the rules of a ball game, since it is implied in the theories employed, and N is usually around 3. &mdash;  Cp i r al <u style="color:#2820F0;">Cpiral  21:52, 21 November 2011 (UTC):
 * That makes no sense: how is a formula a 'visual approach'? How is derived? How is it related to what's there? This could maybe fixed with the lengthy geometric digression in your original proposal, but we've already determined that has no place here.-- JohnBlackburne words<sub style="margin-left:-2.0ex;">deeds 22:00, 21 November 2011 (UTC)
 * Thanks. I'll take "maybe" and the questions you ask, and I'll remember that you only make it seem very much like, really like, you are the consensus.  (Others like(d?) it short.) That's a fearsome burden that works to motivate only some few offerings.  I'll get it right.  Happy editing, Gonzales! &mdash;  Cp i r al <u style="color:#2820F0;">Cpiral  10:09, 22 November 2011 (UTC)

Well, it is possible to say it without infinite numbers of random variables, but it would be much more complicated. If we're going to include the iid in the more complicated version, we should do it in the simpler version, even though it may not be necessary.

Let $e$ be constructed as follows: Let $E(V) = e$, $1$, ..., $X_{i} ~ U(0, 1)$ be independent random variables with uniform distribution $E(V) = e$. Let $n$ be the random variable taking the least number $N^{th}$ such that the sum of the first n of the $e$ is greater than 1, defining $X_{1}$ if no such $X_{2}$ exists. In symbols,
 * $$V_N =

\begin{cases} \min \left \{ n \mid X_1+X_2+\cdots+X_n > 1 \right \} &   X_1+X_2+\cdots+X_N > 1 \\ N+1& \text{otherwise} \end{cases} $$

Then $$\lim_{N \rightarrow \infty} E \left( V_N \right) = e$$

I fail to see how this is simpler than:

Let $V_{N}$, $X_{1}$, ..., be independent random variables with uniform distribution $X_{2}$. Let $X_{N}$ be the least number $~U(0,1)$ such that the sum of the first $V_{N}$ samples exceeds 1:
 * $$V = \min { \left \{ n \mid X_1+X_2+\cdots+X_n > 1 \right \} }.$$

Then the expected value of $n$ is $X_{i}$: $V_{N} = N + 1$.

I might include an additional note that V almost surely exists. — Arthur Rubin (talk) 16:16, 22 November 2011 (UTC)


 * Based on Cpiral's quotation, he seems to believe that modern mathematics is not supposed to use "infinity" (like "infinite sequence", "infinite set"). This, no doubt, is based on his misunderstanding of a pop-math book that he read.  Read a book on modern analysis (like Rudin or Apostol).  Flip to the chapter of "Infinite sequences and series".   Sławomir Biały  (talk) 12:53, 28 November 2011 (UTC)

Confusing Antiderivative Proof/Section
The following section shows a calculation regarding the antiderivative of $n$. It states that the antiderivative of $X_{1}$ is $X_{2}$, but I do not feel that it shows that clearly. The last statement shown does not imply this directly to the layperson (in my opinion.) Can someone review it? Thanks! Sdegan (talk) 17:57, 25 November 2011 (UTC)

Calculus
As in the motivation, the exponential function $~U(0,1)$ is important in part because it is the unique nontrivial function (up to multiplication by a constant) which is its own derivative


 * $$\frac{d}{dx}e^x=e^x$$

and therefore its own antiderivative as well:



\begin{align} e^x & = \int_{-\infty}^x e^t\,dt \\[8pt] & = \int_{-\infty}^0 e^t\,dt + \int_0^x e^t\,dt \\[8pt] & = 1 + \int_{0}^x e^t\,dt. \end{align} $$

Sdegan (talk) 17:57, 25 November 2011 (UTC)


 * It doesn't prove it but that's a bit of work to do, and not especially enlightening. It does give two forms for ex, as a single integral and as 1 + an easier/finite integral, and relates them, which is useful though I'd do it all in one line, perhaps clarifying that it's just a statement:

e^x = \int_{-\infty}^x e^t\,dt = 1 + \int_{0}^x e^t\,dt. $$
 * -- JohnBlackburne words<sub style="margin-left:-2.0ex;">deeds 18:35, 25 November 2011 (UTC)


 * Thanks for the clarification JohnBlackburne. That makes more sense. It's positioning had me feeling that it was intended to be more of a proof. Sdegan (talk) 06:24, 16 December 2011 (UTC)

A consistent style for series notation
I find no recommendation in the MOS, so I'm not surprised we have an inconsistent style for the series notations. Compare:
 * $$p_n = \sum_{k = 0}^n \frac{(-1)^k}{k!} = \frac{1}{0!}-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+-\cdots+\frac{(-1)^n}{n!}.$$
 * $$e = \sum_{n = 0}^\infty \frac{1}{n!} = \frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \cdots$$
 * $$ e^{x} = 1 + {x \over 1!} + {x^{2} \over 2!} + {x^{3} \over 3!} + \cdots = \sum_{n=0}^{\infty} \frac{x^n}{n!}$$

Recommend?:
 * $$p_n = \sum_{k = 0}^n \frac{(-1)^k}{k!} = \frac{1}{0!}-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}\pm\cdots$$
 * $$e = \sum_{n = 0}^\infty \frac{1}{n!} = \frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \cdots$$
 * $$ e^{x} = \sum_{n=0}^{\infty} \frac{x^n}{n!}= 1 + {x \over 1!} + {x^{2} \over 2!} + {x^{3} \over 3!} + \cdots$$

&mdash; Cp i r al <u style="color:#2820F0;">Cpiral  22:14, 28 November 2011 (UTC)


 * I would get rid of the 0! in all cases, replacing it with one. I also don't like the recently-added $$+-\,$$, and would suggest that we get rid of that as well.  I prefer the (original) $$(-1)^n$$ to the (proposed) $$\pm$$, because it more clearly implies that the signs alternate.  (Also, your recommendation for $$p_n$$ is missing its last term.)  Sławomir Biały  (talk) 22:22, 28 November 2011 (UTC)
 * plus-minus sign (±) is commonly used "to indicate a value that can be of either sign". We don't need to repeat the same mechanism on both sides of the equation for pn.
 * Zero in the denominator is not pretty? 0! is not intutive? Agreed on both. And if aesthetics is not to be over-ruled by mere mechanics, then, we might smooth the compaction's echo:


 * $$p_n = 1+\frac{1}{2!}-\frac{1}{3!}\pm\cdots+\frac{(-1)^n}{n!}$$
 * $$e = \sum_{n = 0}^\infty \frac{1}{n!} = 1 + 1 + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \cdots$$
 * $$ e^{x} = \sum_{n=0}^{\infty} \frac{x^n}{n!}= 1 + x + {x^{2} \over 2!} + {x^{3} \over 3!} + \cdots$$
 * &mdash; Cp i r al <u style="color:#2820F0;">Cpiral  23:06, 28 November 2011 (UTC)
 * This looks just wrong for $$p_n$$. I've never seen an alternating series written this way.   Sławomir Biały  (talk) 23:38, 28 November 2011 (UTC)


 * Right... Then I conclude here. (Edited 18:10, 30 November 2011 (UTC))
 * $$p_n =1-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+\cdots+\frac{(-1)^n}{n!}= \sum_{k = 0}^n \frac{(-1)^k}{k!}$$
 * $$e =\sum_{n = 0}^\infty \frac{1}{n!} = 1 + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \cdots$$
 * $$ e^{x} = 1 + \frac{x}{1!} + {x^{2} \over 2!} + {x^{3} \over 3!} + \cdots= \sum_{n=0}^{\infty} \frac{x^n}{n!}$$
 * Article consistency "rules", but sectional aesthetics, and practice in the field, matter more.
 * &mdash; Cp i r al <u style="color:#2820F0;">Cpiral  00:06, 29 November 2011 (UTC)

Summary: There are two, stylistic edits to make. $$ e^{x}$$ is the "model equation".

Exigesis: Having 0! there is not aesthetic, Having "1! + 2! + 3!" is now "the model". 0! is tempting to see when $$\textstyle\sum$$, is in on the left.

&mdash; Cp i r al <u style="color:#2820F0;">Cpiral  20:07, 29 November 2011 (UTC)

Continued fraction notation
This notation from the article
 * $$e = 2; 1, \textbf{2}, 1, 1, \textbf{4}, 1, 1, \textbf{6}, 1, 1, \textbf{8}, 1, 1, \ldots, \textbf{2n}, 1, 1, \ldots, \,$$

is not supported at Continued_fraction. May I change it to single square brackets here, or should someone update Continued fraction? &mdash; Cp i r al <u style="color:#2820F0;">Cpiral  02:38, 30 November 2011 (UTC)

Next I would like to change the < ; > to a <, >. Shouldn't we show only the changes provided by the borrowing of the one, and write:

or in its more concise form
 * $$e = \,$$   $$[2, 1, \textbf{2}, 1, 1, \textbf{4}, 1, 1, \textbf{6}, 1, 1, \textbf{8}, 1, 1, \ldots, \textbf{2n}, 1, 1, \ldots], \,$$

which can be entirely harmoniously written by inserting the zero:


 * $$ e = [1, \textbf{0} , 1 , 1, \textbf{2}, 1, 1, \textbf{4}, 1 , 1 , \textbf{6}, 1, 1, \textbf{8}, 1, 1, \ldots]. \,$$

&mdash; Cp i r al <u style="color:#2820F0;">Cpiral  09:42, 4 December 2011 (UTC) which can be written


 * If you're going to follow a standard then you should follow it so why remove the semicolon? Also there is no particular point to putting in the zero, one might as well then make an argument to go back to -2 or something like that. Dmcq (talk) 13:36, 4 December 2011 (UTC)


 * I thought the point was that there was a clearer pattern if you insert the zero, whereas without it there is an exceptional term.  Sławomir Biały  (talk) 14:03, 4 December 2011 (UTC)


 * OK, then let's keep the semicolon in both versions. But the semicolon is only an option:
 * "it is customary to replace only the first comma by a semicolon."
 * "The semicolon ... is sometimes replaced by a comma." &mdash; Cp i r al <u style="color:#2820F0;">Cpiral  21:39, 4 December 2011 (UTC)

Showing the "chalkboard" continued fraction when we're mainly interested in representative forms is unecessary, but I encourage it for it's "inviting read" effect. To be consistent with this style, we need to make them both semicolons, for that removes distraction, and keeps the focus on this page. As far as standards go, I have just learned that zero violates the nature of the intention of the concise form, which is have no redundancy; but neither the fact that it is non-standard to insert a zero in the continued fraction's standard nomenclature, nor that one form is more "harmonious" need be said. We want the reader to take in many things on faith while staying on track to finish the article in one sitting. Don't say "permitted" or "harmonious" or throw the nomenclature for loops, just say

or in its more concise form
 * $$e = [2; 1, \textbf{2}, 1, 1, \textbf{4}, 1, 1, \textbf{6}, 1, 1, \textbf{8}, 1, 1, \ldots, \textbf{2n}, 1, 1, \ldots], \,$$

which can be written:


 * $$ e = [1; \textbf{0}, 1 , 1, \textbf{2}, 1, 1, \textbf{4}, 1 , 1 , \textbf{6}, 1, 1, \textbf{8}, 1, 1, \ldots]. \,$$

&mdash; Cp i r al <u style="color:#2820F0;">Cpiral  21:39, 4 December 2011 (UTC)


 * I don't think it really matters. Although from the point of view of thinking of the semicolon as the analog of a decimal separator, it doesn't really make sense for the second expansion.  So I would just leave it out of both.   Sławomir Biały  (talk) 13:59, 5 December 2011 (UTC)


 * But the second expansion has got an integral part 1, just like the first has an integral part 2. Isn't this right:
 * $$1 + \cfrac{1}{0 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{2 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{4 + \ddots}}}}}}} = 2+\cfrac{1} {1+\cfrac{1} {{\mathbf 2}+\cfrac{1} {1+\cfrac{1} {1+\cfrac{1} {{\mathbf 4}

+\cfrac{1} {1+\cfrac{1} {1+\cfrac{1} {{\mathbf 6}+\cfrac{1} {1+\cfrac{1} {1+\ddots}}}}}}}}}}$$? &mdash; Cp i r al <u style="color:#2820F0;">Cpiral  07:57, 6 December 2011 (UTC)


 * Both expressions, being equal to e, have integer part 2. This is why the semicolon is misleading.   Sławomir Biały  (talk) 14:01, 6 December 2011 (UTC)
 * Could it be, pray tell, that when zero is allowed, the semicolon is not allowed? In other words, does a generalized fraction have any such a notation?   Please see the discussion at talk:Continued fraction.&mdash;  Cp i r al <u style="color:#2820F0;">Cpiral  10:27, 7 December 2011 (UTC)
 * I do not agree that a semicolon implies a specific integer part; it is as Sławomir Biały says: like a decimal separator. A parallel is that a decimal number may be denoted $\overline{1}$.234=−0.766 (typically when using log10 with tables); this uses a relaxation like the one being spoken of. So the semicolon makes perfect sense in both cases. I suggest using the semicolon or comma consistently to avoid the distraction of the difference. Both the regular/harmonious/neat and generalized/nonstandard aspects should be mentioned; the first is the point of the example, the second might confuse if not clarified. Quondum talk<sub style="margin-left:-2.0ex;">contr 11:41, 7 December 2011 (UTC)
 * A more general formula is e1/n=[1;n-1,1,1,3n-1,1,1,5n-1,1,1,7n-1,...], which is a proper continued fraction when n is an integer greater than 1. Plugging in n=1 to get [1;0,1,1,2,1,1,4,1,1,6,...] seems natural even though it's now a generalized continued fraction. The bracket notation still defines a valid expression which can be evaluated even if the entries are not positive integers, the only problem is you lose some of it's nice properties such as uniqueness and guaranteed convergence.--RDBury (talk) 16:28, 7 December 2011 (UTC)
 * I'm guessing that this general formula is valid for all real n. It would be a nice addition to Exponential function#Continued fractions for ex if you have a reference for it.  It might also be cooler in this article to mention it as a special case (for n=1) of the general formula, rather than pre-substituting n. I think that the loss of the canonical form is not an issue. Quondum talk<sub style="margin-left:-2.0ex;">contr 16:53, 7 December 2011 (UTC)
 * Apparently it can be found as an exercise in Knuth's Art of Computer Programming, I'm only going by the snippet view in Google Books though. In any case it's fairly straightforward to derive it from Gauss's continued fraction.--RDBury (talk) 20:39, 7 December 2011 (UTC)

I consider myself the target audience for this article. I need a consistent representation between the two bracket notations so I don't have to concern myself with the confusion between a formula and its notation. A student need not know the esoterica of the numeral to do the numbers, nor a reader of e that of the bracket notation. I was confused by the unnecessary inconsistency. Now I'm not, but I want to make the edit reflect the me without my cognitive bias. &mdash; Cp i r al <u style="color:#2820F0;">Cpiral  19:37, 7 December 2011 (UTC)
 * I think this is an excellent way to approach it. My immediate reaction is to remove the unfamiliar "concise" notation and to put in the explicit continued fractions you have higher in this talk section. The pattern is still pretty clear; some mention of "regular" would focus the reader's attention on the pattern. Would that achieve the objective? Quondum talk<sub style="margin-left:-2.0ex;">contr 19:55, 7 December 2011 (UTC)
 * The objective has seemed to me to be concise, more precisely in Ozob's terms "I prefer short". Thus I would take your suggestion and propose that we replace the current generalized fraction of e and replace it with the generalized fraction that has the zero in it.  Furthermore I would add a citation needed tag on the concise notation that has the zero in it.
 * $$e = 2+\cfrac{1} {1+\cfrac{1} {{\mathbf 2}+\cfrac{1} {1+\cfrac{1} {1+\cfrac{1} {{\mathbf 4}

+\cfrac{1} {1+\cfrac{1} {1+\cfrac{1} {{\mathbf 6}+\cfrac{1} {1+\cfrac{1} {1+\ddots}}}}}}}}}} = 1 + \cfrac{1}{{\mathbf 0} + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{{\mathbf 2} + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{{\mathbf 4} + \ddots}}}}}}}$$
 * or in continued fraction notation
 * $$[2; 1, \textbf{2}, 1, 1, \textbf{4}, 1, 1, \textbf{6}, 1, 1, \textbf{8}, 1, 1, \ldots, \textbf{2n}, 1, 1, \ldots]$$
 * which can be written
 * $$[1; \textbf{0}, 1 , 1, \textbf{2}, 1, 1, \textbf{4}, 1 , 1 , \textbf{6}, 1, 1, \ldots].$$
 * See also Continued fraction#Regular patterns in $V$.&mdash; Cp i r al <u style="color:#2820F0;">Cpiral  01:24, 8 December 2011 (UTC)

Q&A out loud:
 * The compact notation is only used for generalized continued fractions if there are, as required per the convention of the compact notation, "all ones" in the numerators of the continued fraction represented. Is there any other way to get to a generalized continued fraction with all ones than to add a zero by borrowing from the whole part?
 * The semicolon is optional because like a "mixed number" expression $2&amp; 71⁄100$, the plus sign is optional. The first number in compact notation is usually the "whole number part" or "integral part" of the number represented, which is "an integer". The semicolon is an operator, while the commas are just separators.
 * The semicolon is not like a radix point because the optionally signified "two compartments" have different bases (to say the least).
 * The objection to using the semicolon to represent e when the first number is one is understandably like the objection to saying "I have one dozen and fourteen eggs". It involves the "measuring vs counting" aspect of that first number in that representation.  Zero was born when people started counting in a positional notation that, previous to zero, measured "nothing yet" in that column or heap.  Yes, we can now count on zero to count to zero.
 * Yet "If zero is allowed" confuses math (zero) with audience (allowing). It really asks if the writer may take the unusual step of presenting something that does not match at the level of the "whole number part".  The math allows it.  &mdash;  Cp i r al <u style="color:#2820F0;">Cpiral  23:30, 8 December 2011 (UTC)


 * You are putting a lot of energy into this. I must point out that this probably belongs in List of representations of e, and that this section is at presently inadvisedly attempting to duplicate a lot of the material there.  It is already linked to as the "main article".  In view of this, I'd suggest brutally trimming this section and putting the material that is here but not yet there on that page. I'd agree with your "Q&A" points, with the exception of the third bullet. Quondum talk<sub style="margin-left:-2.0ex;">contr 09:00, 9 December 2011 (UTC)


 * Such discussions are brewing. The article should document all of e's characterizations and properties and  select a few of the many remarkable representations and applications.  The sectioning and titling and inaccessibility and redundancy issues the article has are related to Representations, but my energies to address those issues are moving more towards checking the philosophy of the layout, and it's implications. &mdash;  Cp i r al <u style="color:#2820F0;">Cpiral  04:36, 12 December 2011 (UTC)

Now I'm leaning towards some representations in continued fractions read something like:

$$ e =\lim_{n \to \infty}[2;1,\mathbf 2,1,1,\mathbf 4,1,1,\mathbf 6,1,1,\mathbf 8,1,1,...,\mathbf {2n},1,1] = [1;\mathbf 0,1,1,\mathbf 2,1,1,\mathbf 4,1,1,...] $$

which written out looks like

$$2+ \cfrac{1} {1+\cfrac{1} {\mathbf 2 +\cfrac{1} {1+\cfrac{1} {1+\cfrac{1} {\mathbf 4 +\ddots} }        }      }   } = 1+ \cfrac{1} {\mathbf 0 + \cfrac{1} {1 + \cfrac{1} {1 + \cfrac{1} {\mathbf 2 + \cfrac{1} {1 + \cfrac{1} {1 + \cfrac{1} {\mathbf 4 + \ddots} }         }        }      }    }  } $$ &mdash;  Cp i r al <u style="color:#2820F0;">Cpiral  04:50, 12 December 2011 (UTC)


 * I think this is an improvement on all counts, so I'd suggest replacing what's there for now. Then the question of how much detail to recap from the main article on representations can be considered separately. Quondum talk<sub style="margin-left:-2.0ex;">contr 07:25, 12 December 2011 (UTC)

"Is there any other way to get to a generalized continued fraction with all ones than to add a zero by borrowing from the whole part?" Yes, if fractional denominators are allowed:

e =\lim_{n \to \infty}[1; 0.5, 12, 5, 28, 9, 44, 13, ..., 4(4n-1), 4n+1, ...], $$

which is a special case of

e^{2/y} =\lim_{n \to \infty}[1; (y-1)/2, 3y\times 2, 5y/2, 7y\times 2, 9y/2, 11y\times 2, 13y/2, ..., (4n-1)y\times 2, (4n+1)y/2, ...]. $$

The above can be derived from the generalized continued fraction for $$e^{2x/y}$$, where x=1. — Glenn L (talk) 07:05, 4 March 2012 (UTC); revised 11:23, 03 November 2012 (UTC)