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Somebody should tell us what the difference is between ${\displaystyle \rho }$, ${\displaystyle \rho _{n}}$, and ${\displaystyle \phi _{n}}$. Otherwise, we mix them all up. – Nomen4Omen (talk) 14:00, 16 August 2020 (UTC)[reply]

The first example of adding the angles is clear and easily understood. It could talk about modulo arithmetic when "adjusting" the sum to the range 0-360°.

The second example is confusing. Are we adding real or complex numbers? I would assume the numbers are in the set T but don't see the imaginary part. How do we get to the final result 0.155 from the previous 2.155? Throw away 2 does not explain anything. Jonne6v (talk) 07:15, 13 November 2022 (UTC)[reply]

I tried to make it clearer. –Nomen4Omen (talk) 10:18, 13 November 2022 (UTC)[reply]

One question I have is: When people say “circle group” do they necessarily mean the unit complex numbers per se? Or do they mean the group of orientations or rotations in the plane, for which the unit complex numbers are one possible representation (among many isomorphic alternatives)? This depends on context, and I am sure there are many sources which define it that way (in general, mathematicians are usually pretty cavalier about glossing the difference between isomorphic structures), but I guess my question would be: what would the most pedantic experts say is the proper definition / conception of the “circle group”? –jacobolus (t) 16:21, 13 November 2022 (UTC)[reply]

The redirect SO(2) has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at Wikipedia:Redirects for discussion/Log/2026 February 19 § SO(2) until a consensus is reached. 1234qwer1234qwer4 10:25, 19 February 2026 (UTC)[reply]

One paragraph reads as follows:

"In the same way that the real numbers are a completion of the b-adic rationals ${\displaystyle \mathbb {Z} {\bigl [}{\tfrac {1}{b}}{\bigr ]}}$ for every natural number ⁠${\displaystyle b>1}$⁠, the circle group is the completion of the Prüfer group ${\displaystyle \mathbb {Z} {\bigl [}{\tfrac {1}{b}}{\bigr ]}~\!/~\!\mathbb {Z} }$ for ⁠${\displaystyle b}$⁠, given by the direct limit ⁠${\displaystyle \varinjlim \mathbb {Z} ~\!/~\!b^{n}\mathbb {Z} }$⁠."

This is very intriguing, but it omits all explanation of how these limits work.

I hope someone knowledgeable about these limits will explain (maybe later in the article) how these limits work. ~2026-28336-51 (talk) 14:43, 23 May 2026 (UTC)[reply]

@Sławomir Biały – Your general approach here is to somewhat elide the distinctions between various isomorphic groups. This is common in advanced mathematics, though potentially confusing for less prepared readers. But in any event, if we're going to say that the circle group "is the group of symmetries of the circle which preserve its orientation", should we just lead with that as our first sentence, instead of starting with unit complex numbers? –jacobolus (t) 09:12, 30 May 2026 (UTC)[reply]

I changed the lead to remove "isomorphism" early on because it seemed to me that belaboring this rather fussy point is likely to make things less clear to likely readers, not because I am trying to take a fundamentalist stance about them all being "the same group" or whatever. It might make sense to define it as the orientation-preserving symmetries of the circle, and then discuss the unit complex numbers. However, I think the unit circle is actually better, because it is something that most students already learn about before college, and provides a clearer model up front. The one-level-down reader in my mind is more comfortable with the idea of unit complex numbers (e.g., a first-year physics student) than with the idea of a group of symmetries. So the progression of "concrete model -> symmetry and group structure -> other models" seems natural to me. I don't think your basic student is going to fret much over whether complex numbers represent rotations in the plane (after some explanation of the identification), and the point about isomorphism seems more likely to cause confusion than clarify the matter.

All that having been said, there may be a better way to order the concepts in the second paragraph, which might currently be a little bit abrupt. I think hanging the term "circle group" on "group of symmetries of the circle" feels a little bit wrong, and is probably a bad transition for that discussion. Sławomir Biały (talk) 09:46, 30 May 2026 (UTC)[reply]

I'm not convinced about the latest version of the lead. This seems logically reversed: "Because multiplication of complex numbers commutes, several rotations of the plane can be composed in any order with the same result," Rotations of the plane commuting seems like an intuitively clear geometric fact, which doesn't really seem to be "caused" by the algebraic details of complex numbers.

This version also makes it seem like just the particular "complex unit circle" is the "circle" whose symmetries we are discussing, but really the symmetries of any circle in the Euclidean plane are the same. I think this gives a misleading impression to less-technical readers who may not immediately realize that the complex unit circle can be made to stand for any arbitrary other circle.

If anything, I'd say the geometric facts about rotation and scaling of the Euclidean plane are logically primary and complex numbers are a way of encoding them algebraically by letting the identity transformation be the unit 1 and giving the symbolic name i to a quarter-turn rotation. I wouldn't say that here explicitly though, as it would need excessive explanation to unpack.

(If we want to model geometry, we often should formally distinguish points, vectors, and "complex numbers" [rotation and scaling transformations], rather than conflating them. Multiplication of such a "complex number" by a vector serves to rotate it, but such vectors are not complex numbers per se and do not have the same multiplicative structure. Taking complex numbers to be points and vice versa confuses these differences and can lead to misconceptions.)

(Aside: Complex multiplication links to something different than what is intended here.) –jacobolus (t) 15:33, 31 May 2026 (UTC)[reply]

I don't exactly disagree on any of the minor points here, and didn't mean for my edits to be taken as some final version of what the lead should do; just possibly a step in the right direction to remove some of the abruptness I noted before. That being said, I am not really convinced that group of rotations of a circle is actually more elementary, intuitive, or informative than the unit complex numbers. Even leading with this, there is still the question of which circle, which center, is the plane rotated as well, orientation-preservation, etc. It doesn't necessarily preempt the problem that you identified: that the second paragraph describes rotations of the unit circle, which is a particular model of "the circle". Now, part (or maybe most) of my objection to refocusing the lead exclusively on rotations is largely a matter of taste: as an analyst by training, I tend to prefer concrete models of things to more abstract avatars of the same object. But to my training the unit complex numbers is concrete and immediately gives the composition law, the connection to rotations, and the topology at once. As for commutativity, maybe it is obvious that rotations commute, I don't really know since I can't easily step back and look at the question from behind a veil of ignorance. But numbers! Aha, multiplication of numbers surely commutes, and I don't have to spend extra effort thinking about it. Sławomir Biały (talk) 16:13, 31 May 2026 (UTC)[reply]

I think of turning circles around as much more "concrete" than twiddling symbols on paper which represent numbers. :-) As for commutativity: it's true for rotations of the 2-dimensional plane about a fixed point and their numerical representation, the unit complex numbers, but is false for rotations of 3-dimensional space and their numerical representation, the unit quaternions. If a "complex number" is a "number" then a quaternion certainly also is, so this "surely" turns out to be false in general. I think it's fairly intuitively clear that 2-dimensional rotations commute, but many people are initially surprised when they learn that 3-dimensional rotations are not. –jacobolus (t) 18:34, 31 May 2026 (UTC)[reply]

You are arguing against a joking aside. ;-) My point was only that in the unit-complex model, the group law is ordinary complex multiplication, so commutativity is immediate. I was not intending to make a claim about every algebraic object that has ever been called a number. More generally, I think we have different ideas about what counts as concrete. To me, ${\displaystyle \mathbb {T} =\{z\in \mathbb {C} :|z|=1\}}$ is very concrete: it gives a specific set, a specific operation, its algebraic properties, the topology (it's a circle!), and the exponential parametrization all at once. The rotation interpretation is then immediate. By contrast, "the group of rotations of a circle" still has to specify what transformations are being considered: rotations of an ambient Euclidean plane, orientation-preserving isometries of the circle, choice of center/radius, and so on. And it just seems to me that the abstract group of symmetries of an abstract circle that represents all circles simultaneously is the opposite of a concrete model: it's necessarily an abstraction unless you have a particular circle in hand. And that's before you get to topology, associativity, commutativity. (E.g., why is the group pf rotations of a circle a circle? If I rotate a circle trough 360 degrees, why is that motion "the same" as doing nothing at all?) None of that is fatal, but I do not see it as simpler or more elementary than the unit-complex-number model. So I still prefer to begin with the unit complex numbers and then explain the rotation interpretation immediately afterward. Sławomir Biały (talk) 20:52, 31 May 2026 (UTC)[reply]

${\displaystyle \mathbb {T} =\{z\in \mathbb {C} :|z|=1\}}$ uses notation and concepts that people don't encounter until an undergraduate math or science program, built on a tall tower of other abstract concepts. Rotations of a circle are a topic from preschool. You can physically hold some object with circular symmetry in your hands and you can rotate it. That's as concrete as it gets. That a wheel (or any other object) returns to its original displacement when you turn it all the way around is an immediately accessible physical fact. The best way to explain "why" this happens depends on what kind of conceptual context and level of preparation the reader has, but is ultimately kind of a philosophical question. "why is the group of rotations of a circle a circle?" – personally I would say that it isn't a circle per se, but shares structural properties with a circle. But if we start with a physical wheel, we can make a mark on it, and then as we spin the wheel the mark traces out a circle in space. We can associate each possible rotation of the wheel 1:1 with the corresponding position of the mark. –jacobolus (t) 21:40, 31 May 2026 (UTC)[reply]

I mean, a pre-schooler surely "knows" when they rotate a wheel through a full turn that they have actually done something that is not the same as having done nothing. Or that rotating a ship's wheel will not rotate a car's steering wheel, or perhaps even that rotating a wheel from the front is not the same as rotating it from the back. I think the whole idea that "rotating the circle is concrete" collapses on inspection. It is literally an abstraction: the circle group acts on every circle, the ideal circle, etc. That, by definition, is abstraction. Moreover, the circle group is a circle, metrically and topologically, and that is an extremely important aspect of the topic, just as important as its grouphood. Anyway, I think preschoolers are definitely the wrong target for this article. The most important aspects of this topic are not things for which even many college students have exposure to, being that the circle group is at the foundation of Fourier analysis, electromagnetism, ergodic theory, etc. One level down for this article is a student who already knows what the complex numbers are. I am not saying that it is impossible to explain certain things to someone else, but I strongly believe that it would sacrifice precision, clarity, and the concision required for a lead. The unit complex numbers is the ideal model because it contains all of the key properties of the group, is a concrete, explicit set, is not overly technical. Perhaps most importantly the unit complex numbers is the definition that actually appears in the majority of reliable sources. I have not been able to find any define that define it as the group of rotations of a circle. Sławomir Biały (talk) 05:35, 1 June 2026 (UTC)[reply]

Explaining to someone that every circle can rotate in the same way is "abstract", but much less so than anything to do with numbers, in my opinion. In any event, I agree with you that most reliable sources seem to explicitly say that the "circle group" means the set of unit complex numbers. I would recommend sticking to that throughout this article, instead of sometimes using the symbol ⁠${\displaystyle \mathbb {T} }$⁠ to use various other isomorphic groups. –jacobolus (t) 07:17, 1 June 2026 (UTC)[reply]

Well, many sources also call ${\displaystyle \mathbb {T} =\mathbb {R} /\mathbb {Z} }$ the circle group. I have given several that make this clear, some web-accessible ones are [1], [2], [3], [4], [5]. That is an important viewpoint in the traditional coverage of Fourier series, for example. So I strongly disagree with that proposal. It is inconsistent with the neutral point of view policy, which states that all significant viewpoints of a topic must be represented. Fortuitously, we also have high quality sources that explain the identification. (I would perhaps add the book of Hofmann and Morris The structure of compact groups, who use the symbol ${\displaystyle \mathbb {T} }$ for the group ${\displaystyle \mathbb {R} /\mathbb {Z} }$, and identify it with the circle which they denote ${\displaystyle \mathbb {S} ^{1}}$, referring to both as "the circle group". And in the sequel The Lie theory of connected pro-Lie groups, the circle group explicitly refers to ${\displaystyle \mathbb {T} =\mathbb {R} /\mathbb {Z} }$.) Sławomir Biały (talk) 08:04, 1 June 2026 (UTC)[reply]

Sure, and I can also find sources which say that SO(2) is "the circle group". But do any sources use the same symbol interchangeably for different isomorphic groups within the same source? –jacobolus (t) 14:16, 1 June 2026 (UTC)[reply]

Quite a few do use the symbol interchangeably, yes. Folland, and Bekka et al, are explicit about the identification. Hoffman and Morris use ${\displaystyle \mathbb {S} ^{1}}$ for the unit complex numbers and ${\displaystyle \mathbb {T} }$ for ${\displaystyle \mathbb {R} /\mathbb {Z} }$, but refer to both as the circle group in different places. I haven't seen any source that defines the circle group as a group of linear transformations of a two-dimensional real vector space preserving a positive-definite quadratic form on the space. The closest I found was Stillwell, who defines it as the unit complex numbers and later (without elaboration) identifies it with ${\displaystyle SO(2)}$. Sławomir Biały (talk) 15:19, 1 June 2026 (UTC)[reply]