Talk:Euler's identity

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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]

number e

number e I suggest this could be described as the natural logarithmic base? Jokem (talk) 04:14, 10 February 2025 (UTC)[reply]

Remove wording in the lead and body

I have a big, big problem with "as it shows a profound connection between the most fundamental numbers in mathematics" in the lead and the subsections talking about mathematical constants. The truth is, this is purely specific to the complex analysis. This creates confusion because people think there's some beautiful connection between 0 and 1 and e and pi, but there's none. There's only connection between 0+0*i, 1+0*i, e^(whatever)+0*i and 0+pi*i. This is alluded to here and in other sources, so maybe it's time to be precise when we're talking about fundamental things in mathematics.

This needs to be reworded in the lead to highlight that this is strictly a complex analysis relationship and all those identity numbers are complex numbers, and this has zero beauty for zero and one in real numbers domain.

Maybe even it's criminal not to write this identity using numbers without the complex part. Maybe it's worth to write it out so people are very crystal clear this has nothing to do with the real numbers. It's obvious to us who know what's up, but this really is not helpful to anyone who is learning math. But also it's imprecise to write out one of the most fundamental identity of complex analysis without the complex part.

This is really important. This doesn't help people to learn to think. This wastes people's time. People should be confused about more important things that help them learn. Berkeleywho (talk) 05:58, 8 October 2025 (UTC)[reply]

"Complex" numbers (and hyperbolic numbers) are inherent in the continuum of "real" numbers and the geometry of space (even 1-dimensional space), and, another step removed, are inherent in the counting numbers.

What one person or another finds profound or beautiful is a matter of taste and is somewhat arbitrary; I personally think this equation's interest is exaggerated, but other people like it, so we cite them and summarize or quote their opinions.

If I had to describe in simple terms what this equation is "really" about, I would say that it shows that a half-turn rotation in the Euclidean plane is equivalent to a point reflection across the center of the rotation. This is a basic geometric fact that can be simply demonstrated and explained to young children without much trouble. But other people are more interested in what this equation says about the nature of the exponential function. YMMV. –jacobolus (t) 06:45, 8 October 2025 (UTC)[reply]

I mean, you want to be scalpel sharp precise about things and have that be a teaching moment to the public. It's just there's no identity to -1 and -1+0i. Euler's identity is purely a complex number theory artifact that has zero relevance to real numbers that are not a subset of complex number set. So it just needs to be driven home. Berkeleywho (talk) 09:18, 8 October 2025 (UTC)[reply]

I don't at all agree with this claim. I think it betrays a general lack of familiarity/experience with complex analysis. All sorts of behaviors of ostensibly real-valued functions turn out to crucially depend on behavior oat nearby complex values. –jacobolus (t) 09:29, 8 October 2025 (UTC)[reply]

Real numbers set does not know anything about complex numbers set. You are talking about real numbers that are a subset of complex numbers set. This is a crucial distinction. This distinction is also the one that can only be fully appreciated and understood after learning complex numbers relatively well (and discreet math and set theory probably). I'm not saying delete the page, I'm not saying don't study complex analysis. I'm saying it needs more precision because it's misleading the way it is right now. Berkeleywho (talk) 10:03, 8 October 2025 (UTC)[reply]

Complex numbers have been primarily introduced because they were needed for a better study of real numbers. An example is the fundamental theorem of algebra, which is primaritly about equations with real coefficients and cannot be stated without complex numbers. Similarly, for studying alternative currents modeled with real numbers, electrical engeners uses daily complex numbers. So, real numbers, like chairs, know nothing, but for well understanding real numbers one needs complex numbers. D.Lazard (talk) 11:21, 8 October 2025 (UTC)[reply]

Maybe just be rigorous and specify "is true in C". It's implied, but this is necessary for reasons I already mentioned. Is that not clear from what I've been saying?

Complex numbers may not even be the superset of real numbers in the cleanest sense. There is also a hyperreal numbers superset of real numbers that's completely disjointed from the complex number set, for example. It becomes a little muddy.

When you talk about real numbers having anything to do with complex numbers, you are talking about a projection of complex numbers with zero imaginary part onto the real axis. And notationally, you can't just say R={r∈C}, you have to state R={r∈C∣r=r+0i}. It's all nitty-gritty, but it's the important nitty-gritty when you describe fundamental things and want to be precise and want people to learn rather than accumulate confusion and be discouraged from intellectual pursuit. Berkeleywho (talk) 17:47, 8 October 2025 (UTC)[reply]

The claim that reading this article is going to discourage people from "intellectual pursuit" but that it wouldn't if we just add some parenthetical caveats somewhere does not really seem plausible to me. As for the formal definitional relation of real to complex numbers: this is really a matter of taste rather than anything inherent, and the most convenient choice of definitions depends on the context. –jacobolus (t) 17:55, 8 October 2025 (UTC)[reply]

What contexts does Wikipedia care about other than teaching as well as accurate and precise reference? Berkeleywho (talk) 08:58, 9 October 2025 (UTC)[reply]

Wikipedia is not for teaching, it's an encyclopedia based on what reliable sources say. The material you're complaining about is well supported in reliable sources. 2600:4041:5C39:3A00:A189:6D74:D612:F087 (talk) 11:17, 9 October 2025 (UTC)[reply]

This is just so wrong. There's zero context to understand i^2=-1 as anything other than some abstract frivolity up until the very end of linear algebra when eigenvalues are introduced for a 90 degree rotation matrix. Period. So to teach complex numbers for historical purposes in high school just to solve the quadratic equation is stupid. And to imply rotation of matrices has anything to do with something that looks like an equation with real numbers is so obnoxious. This is making math look messy. This is bad math. This can't be in wiki like this. Needs rigorous notation. Berkeleywho (talk) 16:35, 29 October 2025 (UTC)[reply]

I don't understand what point you are trying to make. –jacobolus (t) 17:12, 29 October 2025 (UTC)[reply]

Like jacobolus, I can't figure out what you're trying to accomplish. If you have concrete changes to propose, backed in some way by reliable sources, please propose them; otherwise it seems WP:NOTFORUM applies. --JBL (talk) 00:04, 30 October 2025 (UTC)[reply]

"The complication in understanding arises when we dare to put i in the exponent, as is the case in Euler’s Formula." Then the author makes sure that everything works in the complex numbers. It's not a trivial thing. This needed rigor to even stick a complex number into the exponent to say that it worked "over the whole complex plane". So it needs rigor in presentation too.

https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1357&context=rhumj Berkeleywho (talk) 05:59, 30 October 2025 (UTC)[reply]