Talk: $e$ (mathematical constant)

E (mathematical constant) has been listed as a level-3 vital article in Mathematics. If you can improve it, please do. This article has been rated as GA-Class.

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July 2020 changes to lead[edit]

I have destroyed Rjdeadly's version of the lead as the consequence of an edit conflict. Their version is not convenient, at least because of the too vague first sentence, and their lead is written for a specific public, which should not be the case of a Wikipedia article.

The are two problems is the previous lead. Firstly, the phrase "the unique number whose natural logarithm is equal to one" can be understood as a part of the definition of $e$ , while is simply a parenthetical explanation of "base" therefore I have replaced it by a link to base of a logarithm.

The second issue is more subtle, and is related to the use of the word "definition": a constant has not to be defined, it is characterized by some property. It is the name of the number with this property that is subject of a definition. With my formulation, it becomes clear that readers can choose the characterization/definition which fits best with their background. Even if the first characterization may be viewed as circular (this depend on the chosen definition of the natural logarithm), those that follow are undoubtly not circular. I have kept as the first one the relationship with the natural logarithm, because it seems that for most students this is their first meet with $e$ . D.Lazard (talk) 11:25, 25 July 2020 (UTC)

To start with:

"...at least because of the too vague first sentence"??? That has not changed.
"....their lead is written for a specific public" ??? not at all. see MOS:INTRO. The noteworthiness of e needs to be pointed out first of all in an accessible way.
"It is the base of the natural logarithm" does not belong as a key sentence in the introduction to explain e as that explains nothing. That is simply the definition of the natural logarithm.
a whole series of "characterisations" does not belong here when it is covered later; the summary is supposed to be general and brief.
"I have kept as the first one the relationship with the natural logarithm, because it seems that for most students this is their first meet with $e$ ". Doubtful that's true and not a good reason to keep it there.
"...related to the use of the word "definition": a constant has not to be defined, it is characterized by some property" So you say that Mathematical constant is completely wrong where it states "A mathematical constant is a number whose value is fixed by an unambiguous definition..." Are you going to change this wiki page?

Rjdeadly (talk) 13:55, 25 July 2020 (UTC)

That is simply the definition of the natural logarithm This is at best naive and at worst definitely wrong, as is explained in Deacon Vorbis's comment and in some of the discussions I linked above. --JBL (talk) 14:15, 25 July 2020 (UTC)

A few comments: the "unique number whose natural logarithm is equal to 1" bit is probably good to keep because it simply reinforces what the base of a logarithm is; a little redundancy is helpful here. Stylistically, the changes are rough – they leave the intro with several short, choppy (often one-sentence) paragraphs. Neither this nor the "as described on this page" self-reference are appropriate. Mathematically, simply stating that

e

is defined to be the base of the natural logarithm is misleading at best; it can be defined in multiple ways, as the current lead attempts to convey, and I think it's important for it to do so. I'm not overly enthusiastic about some of the structure of this article, and there's probably some room for tweaking the lead, but what was there couldn't stand. –Deacon Vorbis (carbon • videos) 15:47, 25 July 2020 (UTC)

The previous lead completely misses the point. e is not just another number like any other, but has a significance not just in maths but physically in the real world, just as Pi is a fundamental ratio inherent in all circles and all matters related. e is not an obscure, seemingly random number that can be "applied" to or derived from mathematical problems but represents something from the real world e.g. population, radioactive decay, interest calculations and the idea that all continually growing systems are scaled versions of a common rate. This is badly lacking in the previous introduction.Rjdeadly (talk) 15:55, 25 July 2020 (UTC)

Digits redux[edit]

(Disclaimer: I hold five world records of mathematical constants now.) I would like to respark this debate on whether to add digit records later than Steve Wosniak. I admit that since the Taylor Series has been the most efficient algorithm for hundreds of years, the pure mathematical relevance has diminished. However, world records of e currently being broken are still relevant in experimental mathematics scopes and computational improvements (such as binary splitting merged with Advanced Vector Extension multiprocessing) that facilitate astronomical digit computations are still of value to Wikipedia. Since the plight to the known digits of Pi is not necessarily purely because to expand mathematical significance, other mathematical constants that have the known digits after 1978 have relevance in public perceptions and understanding of mathematics. Not to forget that other major mathematics constants such as Apery, Catalan, Ln(2), and Euler-Mascheroni keep track of all recent computations. I hope that the mathematician moderators here understand that this isn't just a pure mathematics thing, and it is a debate on whether adding a few lines on this table helps public awareness on mathematics vs making the document longer unnecessarily. I don't think this is a thing that makes this document longer for no reason. 39.119.42.102 (talk) 08:30, 10 August 2020 (UTC)

Can you clarify two things? (1) Are you the same person as Owlice1 or Rick314? (2) Are you proposing to do something different from what they were proposing, and if so, then what? --JBL (talk) 11:09, 10 August 2020 (UTC)

Answer: No, I am my own person and this is my first time writing. I would like to add in the Known digits section more recent known digit values and also add in some information about the development of multiprocessing capabilities assisted by the binary splitting of Taylor series, since the Euler-Mascheroni Constant, Catalan's Constant, Apery's Constant, etc all have recent known digit records on its page.39.119.42.102 (talk) 16:05, 10 August 2020 (UTC)

I've moved this to a new section even though it's related to the old discussion to help keep things organized. I've also indented your replies; see WP:THREAD for more info. I'm still pretty tepid on the digit records. On the other hand, if there are sources that discuss new algorithmic improvements to digit computation, that could be worth a short mention at least. Can you be any more specific? –Deacon Vorbis (carbon • videos) 16:25, 10 August 2020 (UTC)

1. [[1]] 2. [[2]] 3. [[3]]. Important points are that the first-ever multithreaded program to compute both the world record of Pi and e was developed in the late 2000s by [Alexander J. Yee] and binary splitting coupled with FFT has been a very crucial component. e is the simplest form of the Taylor Expansion, so binary splitting for e was one of the first computationally implemented as a proof of concept for the usage of binary splitting for multiprocessing. The rapid progression of digits would have been impossible with multiprocessing and this has influenced binary splitting algorithms of more complex mathematical constants. This is why I think recent digits are still meaningful. I also repeat that Apery, Catalan, Ln(2), and Euler-Mascheroni documents keep the recent history starting from the 1700s to just a month ago and multiprocessing of such constants would have never been possible without the implementation of binary splitting for e. Last but not least, it's still worthy to keep track of digit history for the general audience not only as a mathematics encyclopedia, since this is Wikipedia. 39.119.42.102 (talk) 15:11, 11 August 2020 (UTC)

First image in the article[edit]

Shaded area cannot be 'equal to 1'. --5.43.102.127 (talk) 18:16, 1 September 2020 (UTC)

Why not? It is true that it equals 1. D.Lazard (talk) 19:00, 1 September 2020 (UTC)

(and hence one may define e as f(1)[edit]

maybe this has been flogged to death already, but that statement confuses me beyond belief. Does it really have any meaning? As far as I can see f(1) means e to the power 1. and defining any number as itself to the power one seems bizarre. I would just delete it, but I don't want to interfere in contentious waters. Maybe you could explain it to me here. JohnjPerth (talk) 15:44, 9 October 2020 (UTC)JohnjPerth

It seems fine: define

f (x)

to be the function which is equal to its own derivative such that

f (0) = 1

, and then define

e

to be

f (1)

. Proving that such a function exists and is unique can be done, as can showing that it's equivalent to other definitions of the exponential function. –Deacon Vorbis (carbon • videos) 16:02, 9 October 2020 (UTC)

In fact, pretty much any definition of

e

can be stated as something like: Define the exponential function to be ____, and then define

e

to be the value of that function evaluated at

1

. –Deacon Vorbis (carbon • videos) 16:18, 9 October 2020 (UTC)

Thank you Deacon. To say that 2 is defined as f(1) where f(x)= 2 exp x would be silly, and those were the terms that I was thinking in. I was not realizing that 'we know what 2 is', but we don't know what 'e' is in the same way. 'e' is only defined by things like the properties of f(x)=e exp x and therefore the function defines 'e' rather than 'e' defines the function. Your answer did not seem to explicitly say that, but it brought me to that understanding. Thank you. JohnjPerth (talk) 08:35, 31 October 2020 (UTC)JohnjPerth

Complex Number - Possible Edit War[edit]

Both Sine(x) and Cosine(x) can be expressed in terms of $e^{ix}$ and $e^{-ix}$ The expression for Sine is divided by 2i whereas the expression for cosine is divided by 2. (Not 2i)

There is a disagreement between editors. Twice an attempt has been made to write $\cos x={\frac {e^{ix}+e^{-ix}}{2i}}$ (sic) and twice this has been reverted to the correct expression $\cos x={\frac {e^{ix}+e^{-ix}}{2}}$ .

Please if you disagree then discuss it here. Failing that I shall revert any attempt to change these expression (unless someone else gets there first). OrewaTel (talk) 02:21, 12 October 2020 (UTC)

anhducwiki

Saturday, November 28, 2020