Talk:Euler's identity

Counterpoint

I believe there is more of a variety of opinions on the beauty of Euler's identity than is currently reflected in the article. Accordingly, I made this edit, which was WP:BRDed by @Purgy Purgatorio:. I'm interested to hear what people think; thanks. Danstronger (talk) 00:53, 8 January 2019 (UTC)

I reverted the addition of the 𝜏-aspects of beauty not because I had the slightest doubt that beauty is in the eye of the beholder, but because I am myself in doubt whether any 𝜏-PR beyond Turn (geometry) is considered as (non-)neutral POV in WP. Since I have absolutely no fundamentalist personal preferences about the content of sections in WP dealing with beauty in mathematical formulae, I also look forward to reading about other people's opinions.

Neglecting any attitude of WP wrt 𝜏, and having enjoyed reading Hartl's manifesto, I consider ongoing propagation of "𝜏 is the better π" as fringe, especially after reading the "final blow" against π, using 𝜏/4(!) in the section on n-dim volumes.

As a rather funny side effect of comparing $e^{i\tau }=1$ to $e^{2i\pi }=1,$ I am flattened by the beauty of using the only even prime in all numbers in this Beauty Queen of Formulæ. :D Purgy (talk) 08:17, 8 January 2019 (UTC)

I agree with the removal. The removed text is really about tau, it is not about Euler's identity in any substantive way. This article shouldn't be a coatrack for fringe ideas, even harmless ones like tau. --JBL (talk) 19:00, 8 January 2019 (UTC)

My thinking was that the relevant guideline is WP:UNDUE, which draws a distinction between opinions of a minority and opinions of a "tiny minority". Sure the opinion that humanity should switch from pi to tau is very fringe, but the notion that 2 pi is a more fundamental constant that makes math more elegant is probably not a tiny minority. For some evidence of that, see this old blog post (note the comment by Terry Tao that perhaps 2 pi i is even more fundamental than 2 pi or pi). In Tau Manifesto (which has received third party coverage), Hartl spends a whole section making the argument, essentially, that Euler's identity, as it is normally written, contains a significant, fundamental, and instructive aesthetic flaw. I don't see why this should be treated differently from the opinions of Devlin and Nahin that the equation is beautiful. Danstronger (talk) 01:08, 10 January 2019 (UTC)

WP:UNDUE states in the four occurrences of "tiny":

- ... the views of tiny minorities should not be included at all ...

- Views that are held by a tiny minority should not be represented except in articles devoted to those views.

- ... to include that(=a view) of a tiny minority, might be misleading as to the shape of the dispute.

and the fourth occurrence deprecates confronting a by far less than representative number of sources for one side (Devlin, Nahin) with an almost exhaustive list consisting of Hartl, a "perhaps"-side note by Terry Tao in a (imho ridiculous) blog, not to beef about the refuting "third party coverage" of Hartl.

This is already more than enough to quarrel about, I think I stop commenting this. Purgy (talk) 11:09, 10 January 2019 (UTC)

I agree that tau definitely should be mentioned in the article, but not as a "contrast" or as an opinion of "proponents", but as a viewpoint that further clarifies the true meaning of the identity; something along these sentences from the referenced website: "The complex exponential of the circle constant is unity." or "A rotation by one turn is 1." Lemondevon (talk) 00:09, 9 April 2021 (UTC)

Good suggestion. I did my best to add a sentence along the lines of what you suggest, at the end of the geometric interpretation section. Danstronger (talk) 00:44, 7 October 2021 (UTC)

about priority scale

Why is this article the top priority (importance) ? This article is closely related to Euler's formula, but it is a separate article.--SilverMatsu (talk) 15:07, 6 September 2021 (UTC)

Short description

I have twice tried to fix the short description per WP:SDFORMAT and WP:SDEXAMPLES, which says that the format should follow the statement "[Article subject] is/was a/an/the ... ". I have been reverted twice. The SD at this time is "e^(iπ) + 1 equals 0", which is a definition or restatement, not a short description. I suggested "Mathematical equality". – Jonesey95 (talk) 14:29, 6 February 2022 (UTC)

"Mathematical identity" is a definition of the second word of the title. So, it breaks twice WP:SDFORMAT ("A short description is not a definition" and "avoid duplicating information that is already in the title"), and is therefore totally useless. On the other hand, "e^(iπ) + 1 equals 0", follows the spirit of WP:SDEXAMPLES as it is naturally read as

Euler's identity
[says that] e^(iπ) + 1 equals 0

Moreover, this clearly says that this is mathematics, which is the most important information for most readers, who are generally not interested in mathematics. Also, it disambiguates from Euler's formula, which is another identity established by Euler. So, there is absolutely no reason to prefer a totally uninformative short description. D.Lazard (talk) 14:57, 6 February 2022 (UTC)

"Euler's identity is the e^(iπ) + 1 equals 0" does not follow WP:SDEXAMPLES. I encourage followers of this page to come up with a short description that follows the pattern of the vast majority of other short descriptions and also successfully disambiguates this article from others with similar titles. A bare formula does not do the job. The short description at Euler's formula, "Expression of the complex exponential in terms of sine and cosine", follows the correct format and gets the job done, even if it is a bit long. – Jonesey95 (talk) 19:44, 6 February 2022 (UTC)

How about "exp (iπ) +1 equals 0"? But I would also like to hear(see) a short description using the base of natural logarithm, the imaginary unit, the additive identity, the multiplicative identity, and the fundamental circle constant.--SilverMatsu (talk) 03:05, 11 February 2022 (UTC)

Practical Application(s)

Euler's Identity is an elegant special case of Euler's Formula [e^ix = cos x + isin x] where x = pi. It is of little value other than beauty. The generalized Euler's Formula does have practical uses in any discipline that requires wave modeling, Electrical Engineering is a good example. Mxyptlck (talk) 17:03, 22 July 2022 (UTC)

The above expression is still WRONG in Generalizations section, if you accept (i,j,k) a Quaternion and state it in a math exam, teacher will fail you; because there is NO Quaternion as Quaternion[i, j, k] in mathematica or in any other math program at all. In Mathematica they are shown as, Quaternion[a, b*i , c*j , d*k] . where (a) is a pure constant and the others are vectors (i,j,k) . For more information ask it to gemini, a famous (AI) program as---- (how quaternions are expressed in math?) OR Just open https://en.wikipedia.org/wiki/Quaternion — Preceding unsigned comment added by Germanvas (talk • contribs)

Euler's identity expressed with tau

In my experience, $e^{i\pi }+1=0$ has little to no applications, but $e^{i\tau }=1$ is very important in solving complex exponentiation related problems. I wonder if this could be included in the article. NutronStar45 -- T / C 09:05, 12 November 2022 (UTC)

No. JBL (talk) 18:30, 12 November 2022 (UTC)

Please elaborate. NutronStar45 -- T / C 14:08, 16 November 2022 (UTC)

See WP:UNDUE (linked in JBL's reply). D.Lazard (talk) 14:30, 16 November 2022 (UTC)

Proving Euler's identity by Taylor series expansion

We can prove e^iπ = -1 by Taylor series expansion.It will expand our perspective. Yuthfghds (talk) 13:05, 25 July 2023 (UTC)

This is discussed in the relevant article Euler's formula. --JBL (talk) 20:52, 25 July 2023 (UTC)