Talk:Radian

Dimensionless Analysis
Wandered by this article and was surprised to see protection status. I assume this is because the usual confusion/disagreement about angle dimensionality has led to problematic editing. As written, the dimensionless analysis section is a little confusing, since it mostly focuses on one of the many ad-hoc "patching" proposals (eta). The text refers to a few older ones, but there are still surprisingly many proposals even today. A 2017 editorial in Nature, SI units need reform to avoid confusion wanted to promote the radian to a physical unit, but this was refuted (M Wendl: Don't tamper with SI-unit consistency) because it would create contradictions for dimensionless groups like the Womersley number and for different physical entities having identical physical units, like torque (a vector product) versus energy (a scalar product). One sentence in this section is false, or at best very misleading: a majority of papers on the subject acknowledge that angles should be regarded as having an independent dimension and associated units, which cites a single preprint. Modern textbooks in physics and engineering refute this: they are crystal clear about the non-dimensional nature of angles, e.g. with respect to the entities of torque and energy just mentioned. To help readers with the non-intuitive fact that some physical things simply do not have dimensions, this section could perhaps refer to some other examples of inherently dimensionless entities, like mechanical strain or radiation emissivity. 2600:1700:8650:2C60:68D4:FED2:7981:B6EE (talk) 19:30, 13 April 2022 (UTC)
 * And all the citations involving Torrens' proposal are from Metrologia only (there are arXiv versions as well, but they're not peer-reviewed), I might as well label more stuff as WP:FRINGE and remove it. A1E6 (talk) 00:59, 14 April 2022 (UTC)
 * So Metrologia, "The leading international journal in pure and applied metrology", impact factor 3.157, is a fringe publication? As I understand WP:FRINGE that term is reserved for pseudoscience, not for peer-reviewed literature. For comparison the Nature letters in there now are not even peer reviewed, if anything they are what's fringe. Wendl's arguments are basically (1) I don't like it (2) the existing formulas are dimensionless (3) the units would change. None of these arguments are new and they are discussed in much more depth in the Metrologia articles. It is a poor refutation, and the IP's summary of it is wrong. There is no "contradiction"; Wendl's letter does not even use this term. As for torque vs energy, the discussion in shows that torque can either be defined to the same units as energy or to N m / rad. The difference is a matter of convention, as with the overall definition of radians as a unit.
 * As far as "a majority of papers on the subject acknowledge that angles should be regarded as having an independent dimension and associated units", the preprint makes this claim based on a survey of the literature (c.f. the 67 citations). Dimensional analysis is somewhat unusual to begin with, and only those concerned with correctness publish their results in peer-reviewed journals. Hence since angle as dimension is the "correct" choice the scientific literature is overwhelmingly in favor of it. I searched for peer-reviewed papers arguing against angles as dimensions, but concluded there are none. The only source that doesn't define angle to be a dimension is SI, and as said in, the decision was purely to avoid upheaval of current practice. The scientific consensus (as opposed to political consensus) is that angle should be a dimension, and this sentence is included to reflect that. The sentence is verifiable now and I assume it will become reliable once the preprint is published.
 * As far as textbooks, this and this say peer-reviewed journals trump textbooks, particularly when the textbooks are unsourced and don't really have any reasoning besides the circular "this formula assumes radians are unitless so radians must be unitless".
 * As far as presenting Torrens' proposal, I think the section is not complete without at least one proposal, as all the proposals are quite similar, differing mainly in notation. is a peer-reviewed literature review and says Torrens' is the best. An alternative could be the CCU 80-6 proposal, but that is not easily accessible.
 * Anyways, I will restore my version, so that at least the section does not contain misinformation. --Mathnerd314159 (talk) 03:00, 20 April 2022 (UTC)
 * If the Nature letters are problematic, I'll remove them.


 * The Metrologia journal is insignificant compared to the majority of modern physics textbooks and to the stance of the SI itself. From WP:SCHOLARSHIP, regarding the peer-review process in Metrologia:
 * Care should be taken with journals that exist mainly to promote a particular point of view. A claim of peer review is not an indication that the journal is respected, or that any meaningful peer review occurs. Journals that are not peer reviewed by the wider academic community should not be considered reliable, except to show the views of the groups represented by those journals.


 * WP:FRINGE is any view differing from the mainstream, which, for example, Mohr's and Phillips' view
 * "This gives the same result for the numerical value of the angle as the definition quoted in the SI Brochure, however by following similar reasoning, it suggests that angles have the dimension of length squared rather than being dimensionless. This illustrates that conclusions about the dimensions of quantities based on such reasoning are clearly nonsense."
 * ostensibly is. So I removed it.


 * I don't think you understand how WP:FRINGE theories work. If a group of people from one journal publishes several papers on a fringe theory, there is no need whatsoever for that fringe theory to be addressed by the same number of opposing papers.


 * Out of the "62 citations" in the preprint, 26 of them are from Metrologia, 2 are from P. R. Bunker (who regularly tries to edit the Radian article in a manner of egregious WP:COI under the name Bunkerpr, desperately wanting to promote the Metrologia articles – Metrologia's impact factor is totally irrelevant here). The rest of the citations supporting dimensional angles involve Brinsmade, Romain, Brownstein and Lévy-Leblond papers, but they're insignificant compared to the majority of all physics textbooks and to the SI, and mentioning them would be giving them WP:UNDUE weight, the same goes for Torrens' proposal from Metrologia. Again, if a fringe theory is proposed, there is no need whatsoever for it to be explicitly addressed and refuted by reliable sources.


 * "peer-reviewed journals trump textbook" – this is a false statement made by the user Reissgo, it is nowhere in Wikipedia policies regarding reliable sources, especially see WP:SCHOLARSHIP again. A1E6 (talk) 12:34, 20 April 2022 (UTC)

r.e. quote From WP:SCHOLARSHIP: Metrologia does not exist to promote a particular point of view. Per it has a single blind peer review system managed by a professional editor based at the BIPM. It is not a fringe journal. The reason all the articles cited are from Metrologia is because Metrologia is one of the most respected journals in this area.

r.e. "modern physics textbooks", they are tertiary sources. There are several policies recommending avoiding tertiary sources, such as WP:TSF and WP:DONTUSETERTIARY. Textbooks do not seem particularly reliable for this subject. There are textbooks that discuss giving radians a dimension and introducing a physical constant, e.g., actually very similar to Torrens' proposal. There are textbooks that discuss the CCU decision,, and that discuss the proposals to change it,. But for the most part textbooks simply paraphrase the SI in one or two sentences and are not worth citing. What does not exist are any textbooks providing a real argument that radians are dimensionless, because no such argument exists. But I'll concede that textbooks show that the dimensionless radian is popular.

Per WP:MAINSTREAM, "mainstream" relies on the highest-quality sources and may sometimes be a minority view in society. The highest-quality sources are the Metrologia articles, particularly Quincey's review article that compares the existing proposals (including the SI definition). The SI decision was based purely on practical considerations and does not have much scientific basis.

I don't think you understand how WP:FRINGE works either. Even supposing that trivial discussions such as the SI brochure and modern textbooks count as the mainstream "scholarship in the field", angle as dimension would be an alternative theoretical formulation. As such the article should explain the "context with respect to the mainstream perspective." WP:FRINGE doesn't explain how to do this, as it mainly discusses pseudoscience, so we fall back on WP:NPOV: the article should "fairly represent all significant viewpoints that have been published by reliable sources, in proportion to the prominence of each viewpoint in the published, reliable sources." It was a divided decision by the CCU to make radians dimensionless, so the article should also be divided. The lead and the definition section quote the SI brochure and describe the radian as dimensionless; I think this is sufficient weight to that viewpoint. But what about Quincey's comparison of options? Why do you keep removing it? The article needs to represent the viewpoint that angles can be dimensions, and with your deletions it does not. It does not put the dimension decision into any sort of context, or explain "how the minority view differs from [the majority view]". My version does both of these things, via the discussions from Quincey and the presentation of Torrens' approach to show the difference in formulas. It is due weight to describe the minority view. The current state of "section with two sentences" is simply laughable.

As far as Mohr and Phillips (2015), there is indeed a hole in their argument, as you said in. They forgot to divide by the unit area so in fact they should conclude that the angle is dimensionless rather than having units of length squared. But as far as removing the citation from the definition section, that I don't understand, because you simply changed attributed material into unsourced material, which does not improve the article in any way. --Mathnerd314159 (talk) 16:26, 20 April 2022 (UTC)
 * "What does not exist are any textbooks providing a real argument that radians are dimensionless, because no such argument exists." – I strongly disagree. Whatever...
 * Alright, I'll partially restore your version, but it must not contain statements like
 * 1) "However, a majority of papers on the subject acknowledge that angles should be regarded as having an independent dimension and associated units"
 * (I addressed this "majority" thing in my previous comment),
 * 2) "This definition of angle as dimension is mathematically consistent and can be extended to all mathematical and physical equations, allowing for defining formulas independent of the units used for angles."
 * 3) "but for correct measurements and numerical calculations the information about angular dimension must still be preserved"
 * 4) "The inability to use degrees in place of radians shows that angles are inherently dimensional"
 * 5) "Treating angles as dimensionless can be confusing and problems can arise"
 * 6) "Despite the benefits of applying dimensional analysis to angles"
 * I just think that something fishy is going on here, given Bunkerpr's connection with Metrologia and his history of illegitimate WP:COI and WP:REFSPAM edits.
 * "you simply changed attributed material into unsourced material" – the circular sector area formula is well-known and there doesn't need to be a reference promoting a fringe paper. A1E6 (talk) 17:00, 20 April 2022 (UTC)
 * (2) about consistency was mainly to ward off FUD like the Wendl paper that say you run into contradictions. But it doesn't seem to have helped in that respect so I guess leaving it out is reasonable. I'm fine with how the section is now. The only thing I think would improve it is making "Many scientists" more specific, something like the following:
 * At least a dozen scientists have made proposals to treat the radian as a base unit of measure defining its own dimension of "angle", as early as 1936 and as recently as 2022
 * The dozen scientists are Brinsmade (1936), Romain (1962), Eder (1982), Torrens (1986), Brownstein (1997), Lévy-Leblond (1998), Foster (2010), Mills (2016), Quincey (2021), Leonard (2021), and Mohr, Shirley, Phillips, Trott (2022) (with the last counting as 2). A dozen is close to the true amount and sounds a lot smaller than "many". --Mathnerd314159 (talk) 18:25, 20 April 2022 (UTC)
 * By the way, I can straightaway tell that Torrens' proposal is mathematically inconsistent: $$\sin$$ is an entire function and there exists one and only one sequence $$(a_n)_{n\ge 0}$$ such that $\sin z=\sum_{n\ge 0}a_n z^n$, by Cauchy's integral formula. So, any choice of $$\eta$$ other than $$\eta =1$$ is wrong. A1E6 (talk) 23:45, 20 April 2022 (UTC)
 * So actually there are two functions, the mathematical function $$\sin(x)$$ which is unchanged and a new unit-aware function $$\text{Sin}(\theta) = \sin(\eta \theta)$$. Then as an matter of notation $$\sin(\theta)$$ is written in place of $$\text{Sin}(\theta)$$, because the angular dimension of $$\theta$$ makes it clear that the unit-aware function is implied. With radians indeed the only reasonable choice is $$\eta =1$$, but using degrees one ends up substituting a value $$\eta =\pi / 180$$ into the equations, as illustrated in the Radian section. That's the main advantage I see for the dimensional approach: one can measure angles in a mixture of degrees and radians and use them in formulas with the dimensional analysis and $$\eta$$ producing the conversion factors, and it all works out nicely. --Mathnerd314159 (talk) 02:24, 21 April 2022 (UTC)
 * It should have been made clear in the article that it's not "the" sine function, but something else. So I propose using \overline, or something like that. A1E6 (talk) 03:09, 21 April 2022 (UTC)
 * And speaking of mathematical inconsistency, Torrens' "version" of $$\sin$$ (and $$\cos$$, for that matter) is mathematically unusable. For example, consider the fact that
 * $$\theta^2=\frac{\pi^2}{3}+4\sum_{n\ge 1}\frac{(-1)^n}{n^2}\cos n\theta,\qquad \theta\in (-\pi,\pi).$$
 * The dimensions don't "work out" nicely. A1E6 (talk) 13:51, 22 April 2022 (UTC)

The dimensions can always be made to match by inserting $$\eta$$ in appropriate places. Here I think it would simply be $$(\eta \theta)^2=\frac{\pi^2}{3}+4\sum_{n\ge 1}\frac{(-1)^n}{n^2}\cos(n\eta\theta),\qquad \theta\in (-\pi \text{ rad},\pi \text{ rad}).$$ I've updated the article to distinguish Sin from sin. --Mathnerd314159 (talk) 16:21, 22 April 2022 (UTC)


 * This arbitrary inserting of $$\eta$$ is completely ad hoc, done only for the purpose of "saving" Torrens' theory from being falsified, and is devoid of any rationale, other than getting rid of nonsensical angular dimensions. A1E6 (talk) 16:34, 22 April 2022 (UTC)
 * The factors arise naturally when you work out the equations using variables which are ratios of angles to a standard angle. They're no more ad-hoc than factors of c in the Lorentz transformation. And just like the factors of c disappear when you use Planck units, the factors of $$\eta$$ disappear when you use the radian convention. Mathnerd314159 (talk) 03:33, 25 April 2022 (UTC)

The above arguments (I don't like it, The scientific consensus...is that angle should be a dimension, peer-reviewed journals trump textbooks, etc.) are absurdly and patently false. It is remarkable that an important technical concept is allowed to be jerked around in the manner that this article (and its gatekeepers) continue to do. There is no dispute within scientific, engineering, and mathematical circles regarding the dimensionless nature of angle, but the "dimensionless analysis" section misrepresents technical consensus with "Torren's proposal". This is not mainstream, nor is this idea used by scientists, engineers, or mathematicians because it is superfluous over-complication. In this sense, it is indeed FRINGE. One can look at this issue from numerous different perspectives, for instance: Again, in technical environments, there is no confusion about this simple fact. I don't know what the motivation and/or agenda are here, but this section seems to be a vehicle by which to promote a number of articles in a particular journal. The vastly larger technical literature (papers, textbooks, etc.) are clear on this. If one wants an authoritative source, you might include Percy Bridgman's book Dimensional Analysis (1931), in which he treats this issue in the first 3 pages. I am not a regular Wikipedia editor, so I do not have a horse in this race. I am only offering the opinion as someone who claims expertise in this area that your "dimensional analysis" section, as it now stands, is misleading, at best, is FRINGE, and is the sort of thing that hurts Wikipedia credibility. 128.252.79.225 (talk) 18:40, 29 April 2022 (UTC)
 * The argument I already made above: Torque and energy have the same physical units: Newton * meters in SI. But torque, $$\vec r \times \vec F$$, is a vector (cross) product. Work has not been done (energy expended) until such torque is displaced through some unitless rotation, whereby energy $$= \vec r \times \vec F \cdot \vec \theta$$, which is a scalar (dot) product.
 * The simple tack-and-string experiment showing angle as the dimensionless ratio of arc length and radius
 * Frequency (in units of 1/s) mathematically integrated over a given time interval yields a dimensionless angular displacement (and vice-versa w.r.t. time derivative of angle)
 * The Buckingham-Pi Thm used in dimensionless analysis of physical problems leads to dimensionless groups in which various angular measures (angle, angular rate, rotational acceleration) have no physical dimension associated with angular displacement
 * differential equations for real-world phenomena, e.g. the simple spring-mass mechanical response equation $$x'' + C\cdot x = 0$$ have solutions indicating that angular displacement must be unitless


 * I don't think I have ever, in my life, seen angles treated as dimensionful. I mean, just to pluck an example off the top of my head, take Newton's second law for a pendulum not restricted to small angles: $$\ddot{\theta} = -(g/L)\sin\theta$$. The time derivatives on the left gives you units of T-2, which match the units of $$g/L$$, the length dimension canceling. So if $$\theta$$ had units, then they must be the same as the units of $$\sin\theta$$, which are the units of $$\theta$$ and $$\theta^3$$ and $$\theta^5$$ and... It's just nonsensical. Literally nothing is gained from trying to make that work out. Life is too short, and the number of actually interesting physics problems is too big. Trigonometric, exponential, and logarithmic functions take dimensionless arguments. Neither WP:TSF nor WP:DONTUSETERTIARY are policies. They are essays, which do not necessarily have any community consensus behind them. However, WP:UNDUE is policy, and it's very hard to see how the current "Dimensional analysis" section is compliant with it. XOR&#39;easter (talk) 23:08, 29 April 2022 (UTC)
 * On April 20, I stripped Mathnerd314159's original version off of statements which were blatantly going against Wikipedia policies. The current version is more neutral, but there are still some concerns regarding WP:UNDUE. Mathnerd314159's arguments "The reason all the articles cited are from Metrologia is because Metrologia is one of the most respected journals in this area." and "The highest-quality sources are the Metrologia articles" are dubious and need attention – in particular, to what extent should the Metrologia papers be covered in this article. A1E6 (talk) 00:14, 30 April 2022 (UTC)
 * This sort of argument from Mathnerd314159 should be recognized for what it is: a false dilemma. It frames our little debate here of whether angular measure has physical dimensions as an active research question. It is not. Angular measure was fully understood and resolved hundreds of years ago. Even Percy Bridgman's book Dimensional Analysis, which one could regard as the "Bible" on this topic, was already published by 1922. One perspective that this article could reflect is that this issue is confusing, especially for the lay person, which is true and which is why the issue only appears to be unsettled. It explains why there are still "research" papers that come out every few years that attempt to "patch" what is not even a problem in the first place. Before I close my parting comment here, I will again observe that Wikipedia's own rules seem to permit this sort of nonsense. The concept of technical correctness, the bedrock foundation of any encyclopedia, seems to be pushed ever further into the background by activists, editors who are topically ignorant, and agenda-based editing. I am again reminded why I am not a regular Wikipedia editor, which is that most of one's time is wasted trying to defend correctness against activism, ignorance, and agenda. Hope you all get this properly settled at some point. Over and out. 128.252.79.225 (talk) 12:58, 30 April 2022 (UTC)
 * Uh-oh. This is not the kind of argument that belongs on WP talk pages. It is unambiguously an "I'm right because I'm right" perspective that denies the possibility of a historical conceptual oversight. As a side note,  is sounding a little over the top too: just because Metrologia has the status of being the dominant and foremost forum for serious discussion of unit systems and thus that little is published elsewhere does not make it fringe.  This is a deeper discussion than most people (scientist included) seem to allow for.  WP should not go too deeply into presenting more than that this is controversial and that there are proposals to formalize angle as a dimensional quantity.  Even the "Sin" proposal is more than is needed in the article.  172.82.47.18 (talk) 13:37, 30 April 2022 (UTC)
 * Pardon, but I can't help one more comment to respond to such a bureaucratic viewpoint. Your policy-based relativism is another one of the root problems that will prevent Wikipedia from ever becoming a reliable, citable source. Instead of "I'm right because I'm right", what you should have said is "I'm right because it's right". I will reiterate, in the strongest terms, that there is no ambiguity about whether angular measure has physical units. You might as well debate whether gravity points down or whether the earth is flat. And I will also reiterate that this is the sort of thing that hurts Wikipedia's credibility, especially when someone with expertise reads this. (This is exactly how I came to comment here. I tried to offer some expert opinion, despite not being a regular editor.) The bureaucratic echo chamber here is remarkable. Good luck. 128.252.79.225 (talk) 14:05, 30 April 2022 (UTC)
 * This discussion is relevant. A1E6 (talk) 13:44, 1 May 2022 (UTC)

Can we not have a simple definition in the Lead for laymen?
Surely, we could just say "A radian is a unit of angular measurement of a circle that is equal to ~57.295779513082320876°". That way, I could read the header, and then know how to apply it in an example. Gehyra Australis (talk) 10:07, 20 January 2022 (UTC)


 * Simple English Wikipedia jumps to that punchline much more quickly, and is a great resource if you're looking for that kind of stuff, particularly when it comes to articles concerning subjects often steeped in rigour or technical detail, like mathematics and science. — JivanP (talk) 19:50, 11 May 2022 (UTC)

Dimensional analysis NPOV
The statement " the units of the left and right hand sides of an equation may not match" is a simple factual observation. For example with $y=rθ$, we might have $$ 10\ \text{cm} = (1\ \text{cm})(10\ \text{rad}) = 10\ \text{cm}\cdot\text{rad} $$ One side is cm, the other side is cm⋅rad. Saying that the units match because the radian is dimensionless is like saying that 90° = 450° because a circle has 360°; only true under specific assumptions, and quite confusing to the uninitiated. This is what the sentence "In general one must ignore radians during dimensional analysis and add or remove radians in units according to convention and contextual knowledge." explains.

As far as "minority viewpoint", it is the view of some physics teachers, and while they disagree on whether radians should dimensionless, they agree that the current SI definition leads to inconsistent units. This observation as far as I can tell is representative of the profession; it appears in the textbook I cited explicitly as an issue, while others are more focused on correct usage and simply imply that the radian may appear and disappear. I would be curious to know what the "majority viewpoint" on this issue would be; it certainly cannot be that the current definition of radian is flawless and perfect. --Mathnerd314159 (talk) 15:27, 3 May 2022 (UTC)


 * Since $$\mathrm{rad}=1$$,
 * $$10\,\mathrm{cm}=10\,\mathrm{cm}\cdot\mathrm{rad}.$$
 * "Some physics teachers" are still a minority and that observation is not representative of the profession at all. You're free to restore the content, but the sentences must include "according to some physics teachers" or something similar. A1E6 (talk) 15:48, 3 May 2022 (UTC)
 * Are you a physics teacher? Do you have a more reliable source than the American Association of Physics Teachers? I don't know where you get the gall to make these confident statements and categorize everything I cite as fringe. The AAPT source says the radian units appear and disappear, and provides a similar example. This is not a minority viewpoint at all. Mathnerd314159 (talk) 19:05, 3 May 2022 (UTC)
 * The idea that there is such a thing as a 'radian unit' that can match or not match is a minority view and should be clearly labelled as such. MrOllie (talk) 19:13, 3 May 2022 (UTC)
 * And your evidence is ... ? -- Mathnerd314159 (talk) 19:58, 3 May 2022 (UTC)
 * The International System of Units MrOllie (talk) 20:02, 3 May 2022 (UTC)
 * I see. So you're using a publication that identifies the radian as a unit to argue that there is no such thing as a 'radian unit'. Mathnerd314159 (talk) 20:14, 3 May 2022 (UTC)
 * A unit needs a dimension to 'match or not match' or to 'appear or disappear'. But that is the point of the whole deal and I suspect you know that most people (and sources) believe this. MrOllie (talk) 20:20, 3 May 2022 (UTC)
 * I'm using simple textual definitions here. The radian 'appears' in the unit rad/s because the unit symbol rad is in the unit. It does not appear in the similar unit s-1. So therefore rad/s does not 'match' s-1. If you're saying that people have used SI for so long that it's become innate knowledge and even a 5 year old will say that rad/s 'matches' s-1, I'd like to see a source. The dimensional analysis concept you seem to be referring to would be something like 'dimensionally equivalent'. --Mathnerd314159 (talk) 21:03, 3 May 2022 (UTC)
 * Mathematicians and physicists alike have already thought about these matters for a very long time. Ultimately, the dimensions of a measure are completely subjective. To produce a complete system of units, one first chooses a set of base dimensions (e.g. the 7 base dimensions of SI, being length, time, etc.), and then chooses a defining equation for each derived dimension (e.g. force, being derived via Newton's second law, $F = ma$, is equal to [mass][length][time]$&minus;2$), whence the chosen base units (e.g. meter, second, etc.) produce corresponding derived units (e.g. newton, equal to kgms$&minus;2$). To appreciate just how arbitrary these decisions/choices are, see my Quora answer on a related question.
 * The units of angle and solid angle are no different in this respect: one chooses a defining equation for a quantity (i.e. in SI, angle = arc length ÷ radius), and then applies it to decide the dimension of the quantity and the corresponding unit in the system in question (i.e. radian, equal to meter per meter, equal to 1).
 * There are many other instances in SI where equivalent units exist with different names, e.g. hertz and becquerel (both equal to s$&minus;1$). Under your logic, it would be incorrect to say that equations can relate quantities expressed in hertz and becquerel to each other without constants of proportionality that effectively convert between one unit and the other; but in fact, SI's chosen base dimensions and equations of derivation mean this is not the case, i.e. no such constant of proportionality is needed, because 1Hz is well and truly equal to 1Bq.
 * Fundamentally, systems such as SI and the one you (and Mohr, etc.) propose can coexist and be used independently. To boot, physicists already use natural units, whose sole purpose is to attempt to render as many quantities as possible as dimensionless. No system of units is absolute, it is all arbitrary. To appreciate this, consider: why is mole on the SI base units? What about ampere? Surely it should be coulomb instead, since electric current is the rate at which electric charge flows, a derivative (in the calculus sense)? The answers to these questions and more are all a matter of opinion, but a choice has to be made in order to get anything done, and the BIPM made those choices to yield the SI.
 * There are other bespoke arguments to be made in favour of radians in particular being dimensionless, such as the trigonometric functions being from the domain of real numbers but also being functions of absolute angle (as yielded by radians, being dimensionless), not of angles measured in an arbitrary unit of choice (which the radian would be if it were dimensionful), but I'll leave those aside as what I've said above is more than enough. — JivanP (talk) 19:20, 11 May 2022 (UTC)
 * To quote from the becquerel page: "Whereas 1 Hz is 1 cycle per second, 1 Bq is 1 aperiodic radioactivity event per second." If you wrote 1 Bq on an exam where the expected answer was 1 Hz, you would lose points for using the wrong unit. The units do not match - they are not equivalent in this context. Sure, they are dimensionally equivalent, but torque and force are also dimensionally equivalent and the SI is careful to distinguish the units N⋅m and J for those - N⋅m does not match J. Units are not just dimension salad where you throw in the powers of base units you want and out comes a unit. The conversion from a unit to its base dimensions is lossy - the dimension does not specify the unit, and several units may have the correct dimension.
 * I think you may have missed the context of this discussion, which is that the text
 * Because the radian is dimensionless, the units of the left and right hand sides of an equation may not match due to radian units. For example, a mass hanging by a string from a pulley will rise or drop by $y=rθ$ centimeters, where $r$ is the radius of the pulley in centimeters and $θ$ is the angle the pulley turns in radians. There is a unit of radians on the right but not the left. Similarly in the formula for the angular velocity of a rolling wheel, $ω=v/r$, the radians appear on the left in the units of $ω$ but not on the right hand side. This inconsistency has been "a perennial problem in the teaching of mechanics". In general one must ignore radians during dimensional analysis and add or remove radians in units according to convention and contextual knowledge.


 * was deleted by AE16 in Mathnerd314159 (talk) 21:02, 12 May 2022 (UTC)
 * I am aware of the context of the discussion; I disagree with the text that was removed, i.e. I agree that it should have been removed. Its claim that the units/dimensions do not match is false. You appear to treat e.g. [torque] and [energy] as different dimensions. In SI, they are honest to goodness equal. In SI, Nm = J and [torque] = [force][length] = [energy] just as much as 2&times;3 = 6. — JivanP (talk) 15:37, 11 July 2022 (UTC)

Radian "controversy"
@Bunkerpr added this sentence:
 * It is undeniable that there is controversy as to whether angles should be considered as having units and consequently whether they have dimension.

I removed it because it's unsourced and bunkerpr has a history of adding random junk to this article. But there are plenty of sources that refer to a controversy, e.g. "the controversy regarding the radian in SI", "controversy in the use of supplemental units", "controversies involving ... elimination of radians ... as being equivalent to the number 1", "The status of angles within SI has long been a source of controversy and confusion".

Note though that all of these refer to the SI though. So the question IMO is whether the section Radian gets across the point that there has been controversy around the SI decisions, or if additional sources could be added. The section is somewhat reliant on primary sources at the moment. There is /  which criticizes the 1980 decision as "unfounded" and says the 1995 decision used inconsistent arguments and introduced "numerous discrepancies, inconsistencies, and contradictions in the wordings of the SI". Usable? Mathnerd314159 (talk) 19:25, 3 July 2022 (UTC)


 * This was just one of Bunker's several attempts at self-promotion (see references #51 and #53 in https://arxiv.org/pdf/2203.12392.pdf), so I agree with the removal of his contribution. Regarding the second part: I think it's usable if you use quotes. A1E6 (talk) 19:50, 3 July 2022 (UTC)
 * Alright, I added some quotes and split up the first paragraph. It's not the best structuring as far as paragraphs but I think it gets the point across that there have been 3 major periods (supplemental unit, dimensionless derived unit, committee limbo). Mathnerd314159 (talk) 21:04, 4 July 2022 (UTC)
 * I'm not an expert on Wikipedia policy, nor have I been around on this article for long, but I had a quick look through User:Bunkerpr's edits on Wikipedia and it seems that they fall under CNH. Given this, do you reckon that it would be a good idea for one of you to submit a block request, since they seem to not be responding to COI things on their talk page?
 * Being a relatively inexperienced user, I probably misread something, so here's the link to their contributions page.
 * Cheers — MeasureWell (talk) 09:23, 10 July 2022 (UTC)
 * I'll take care of it. A1E6 (talk) 22:01, 11 July 2022 (UTC)

Euler's role in the history of the radian
I think it would be helpful to insert this text as the third paragraph in the History: 18th and 19th Century section. [The "h" that appears twice at the end should be the lowercase Greek eta.]

Leonhard Euler’s seminal work Theoria Motus Corporum Solidorum seu Rigidorum (Theory of motion of solid or rigid bodies), published in 1765, implicitly adopted the radian as the angle unit for all equations involving rotation. For example, his Definition 6, states (in Latin) that ”angular velocity in gyratory motion is the speed of some point, expressed in the unit of the distance from the axis of rotation”. He therefore defined angular velocity ω as ω = v/r, so that within his equations the angular velocity ω always represented radians per unit time, and the radian was treated as equivalent to the number 1. With an unspecified angle unit, the equation would need to contain a constant term such as ω = v/(hr), where h is described above. Pquincey (talk) 08:39, 5 September 2022 (UTC)


 * Be bold. Dondervogel 2 (talk) 08:47, 5 September 2022 (UTC)

Semi-protected edit request on 7/9/22
In the absence of any immediate objections, I would like to request that this text is inserted as the third paragraph in the History: 18th and 19th Century section:

Leonhard Euler’s seminal work Theoria Motus Corporum Solidorum seu Rigidorum (Theory of motion of solid or rigid bodies), published in 1765, implicitly adopted the radian as the angle unit for all equations involving rotation. For example, his Definition 6 (paragraph 316) states (in Latin) that ”angular velocity in gyratory motion is the speed of some point, expressed in the unit of the distance from the axis of rotation”. He therefore chose to define angular velocity ω as ω = v/r, rather than as the rate of change of an angle, which meant that within his equations the angular velocity ω always represented radians per unit time, and the radian was treated as equivalent to the number 1. If he had accommodated non-radian angle units such as degrees, the equation would have needed to contain a constant term, for example ω = v/(η r), where η is described above. Pquincey (talk) 07:32, 7 September 2022 (UTC)


 * The inference that this implicitly defined the radian as the unit of measure of rotation might be trying to read too much into it. If the interpretation as given here is taken literally, the statement "velocity is the speed of some point, expressed in the unit of metre per second" could be read as that m/s is equivalent to 1.  Another interpretation of the quote is that Euler measured rotational velocity in the unit "rotational radius per second", where "rotational radius" could be interpreted as a dimensional unit of angle.  That this should not be be interpreted with such nuance is further supported by the dimensional mismatch between "the speed of some point" and "expressed in the unit of the distance".  172.82.47.242 (talk) 15:40, 8 September 2022 (UTC)
 * The meaning of the quoted definition is clear in context. Paragraph 318 states “Therefore if the velocity of a point, that lies at a distance from the axis of gyration equal to x, is equal to v, then the angular velocity is v/x”. He was definitely setting out our familiar equation for angular velocity ω = v/r, which of course requires the angle unit to be the radian. Euler’s understanding of dimensions was quite shaky – in paragraph 322, he states that “angular velocity c is expressed by an absolute number” (i.e. it is dimensionless), so I wouldn’t read anything into “dimensional mismatches”.
 * Euler’s book is freely available on the internet, in an English translation, here: www.17centurymaths.com/contents/mechanica3.html (the relevant part is Chapter 2 in the Treatise), so you can read it for yourself.
 * I thought it was helpful to quote the Definition, to show that this equation had not been deduced from some deeper axioms, but if the language is a problem, how about simplifying the inserted text to:
 * Leonhard Euler’s seminal work Theoria Motus Corporum Solidorum seu Rigidorum (Theory of motion of solid or rigid bodies), published in 1765, implicitly adopted the radian as the angle unit for all equations involving rotation. For example, in his Definition 6 (paragraph 316) he chose to define angular velocity ω as ω = v/r, rather than as the rate of change of an angle, which meant that within his equations the angular velocity ω always represented radians per unit time, and the radian was treated as equivalent to the number 1. If he had accommodated non-radian angle units such as degrees, the equation would have needed to contain a constant term, for example ω = v/(η r), where η is described above. 86.14.37.0 (talk) 16:57, 8 September 2022 (UTC)
 * The the quantity defined by Euler just happens to correspond with the SI definition of angular velocity. This is incidental if it cannot be traced as directly influencing the definition of the radian.    Developing a thesis, as is done in this passage, is strictly disallowed in WP.  172.82.47.242 (talk) 02:03, 9 September 2022 (UTC)
 * Euler wrote the book for rotational mechanics in the same way that Newton did for linear mechanics. His definitions do not “just happen” to agree with the current familiar definitions – they are the original definitions. As Wikipedia rightly tells us: “The term moment of inertia was introduced by Leonhard Euler in his book Theoria motus corporum solidorum seu rigidorum in 1765.” Euler was not Nostradamus.
 * And there is nothing hypothetical about these definitions requiring that a radian is treated as equivalent to the number one. The passage simply reports relevant facts that are not widely known, which is exactly what Wikipedia was created for. Pquincey (talk) 08:06, 9 September 2022 (UTC)
 * If the history section wishes to trace the origins of quantities such as angular velocity, it may do so. This does not mean that it is dimensionally the same quantity (just as the Gaussian electromagnetic quantities introduced earlier are not dimensionally equivalent to their SI equivalents).  The best we can say is that Euler defined quantities that correspond to the modern quantities, but that does not imply that these should be mathematically equated.  "The passage simply reports relevant facts that are not widely known" seems blatantly false: it makes claims of "implicitly adopted" followed by a rationale.  Providing a rationale is not "simply reporting facts": it is trying to make an argument in support of an unreferenced claim.
 * Stripped of the disallowable synthesis, your suggested passage reads: "Leonhard Euler's seminal work Theoria Motus Corporum Solidorum seu Rigidorum (Theory of motion of solid or rigid bodies), published in 1765, includes Definition 6 (paragraph 316). This defines a quantity ω as ω = v/r, which is equivalent to the modern SI definition of angular velocity." 172.82.47.242 (talk) 23:43, 9 September 2022 (UTC)
 * Rereading my suggested text, I see no claims or rationale, only verifiable facts and clarification. I do not speculate why Euler chose the approach that he adopted, I just explain what it involved and its relevance to how the radian has been treated ever since. 172.82.47.242’s suggestion would leave the reader wondering why the text appears in the “Radian” article.
 * Perhaps someone else would like to comment. Pquincey (talk) 08:46, 10 September 2022 (UTC)
 * As Wikipedia editors, we don't provide our own explanations, unpack what's implicit in historic works, or construct hyperfactuals. Verifiability is at the heart of our work and it requires summarising and citing reliable secondary or sometimes tertiary sources rather than primary ones. Making deductions or bringing our own insights is original research for Wikipedia. This applies even if you feel your explanation is quite mundane and not particularly novel. You'll find more about this in the policies and guidelines I've linked. NebY (talk) 17:54, 10 September 2022 (UTC)
 * In that case, I would add the words "As pointed out by Roche," at the start of my suggested text. Reference: John Roche, The Mathematics of Measurement: A Critical History, Athlone Press, 1998, p.134. Pquincey (talk) 18:44, 10 September 2022 (UTC)

I wrote my own version based on Roche and a footnote of Quincey I'd been meaning to incorporate, please tell me if you have any issues with it. Mathnerd314159 (talk) 21:28, 10 September 2022 (UTC)


 * I should note that Roche doesn't mention Roger Cotes at all. He writes that it's Euler's definition of radian that Thomson adopted. It would be nice to clear this up - did Euler adopt Cotes's definition? did Thomson combine Cotes and Euler? Is there another significant publication that should be mentioned? Mathnerd314159 (talk) 21:41, 10 September 2022 (UTC)
 * Many thanks to Mathnerd314159 – you did a fine job.
 * I can fill in some of the gaps in the history. Cotes was investigating the mathematics of logarithms, and found a precursor to “Euler’s” equation: iθ = ln(cos θ + i sin θ). He noted (in Latin that is hard to follow) that if the trig functions are considered to be functions of angles, the equation only holds when the angle unit is 180/π degrees. He did this work before 1716, the year he died aged 33. Incidentally, he worked closely with Isaac Newton on the second edition of Principia, and wrote a long Preface for it. There is a secondary reference to Cotes’s use of the radian in “Roger Cotes, Natural Philosopher” by Ronald Gowing, Cambridge University Press, 1983, page 39.
 * It would have been obvious to any mathematician that the Cotes angle produces a circular arclength equal to the radius. It was this feature that Euler implicitly used in his 1765 book on rotational mechanics. I don’t think it ever occurred to anyone to think that the “radian” had two definitions. Euler may well have given credit to Cotes somewhere in his Complete Works – good luck with that one. Pquincey (talk) 13:04, 11 September 2022 (UTC)
 * Well, in a 1743 paper on an unrelated subject Euler says "these expressions were exhibited by the supreme mathematicians Cotes and de Moivre." And in this he mentions reading Cotes's Harmonia. So it seems that Euler was familiar with all of Cotes's work in this area, and most likely was significantly influenced by it. But I didn't see Euler giving explicit credit to Cotes on this subject. So I guess I'll leave it.
 * I added Gowing, that book seems like a more reliable source than a random bio website. I took out the 1714 date because it's not in Gowing and it seems like there's not actually enough information to give a precise date (Cotes wrote a now-lost note, but when?). Mathnerd314159 (talk) 16:37, 11 September 2022 (UTC)

The relation between the radian and the steradian; and other ways of writing the constant η
If angle is considered to be dimensional, and the radian to be a unit like any other (i.e. not somehow inherently equal to 1), it is inevitable that solid angle is also dimensional [Brinsmade, 1936]. It has the dimensions of (angle)2, and 1 sr = 1 rad2 (i.e. not 1 m2/m2 or 1) [Brownstein (1997), Leonard (2021)]. It would be helpful to say this in the Radian article, and also in the Steradian one. Among other things, this resolves the problem within the SI that the lumen and the candela, named units for distinct quantities, are formally the same thing (with 1 cd = 1 lm/sr where 1 sr = 1). The familiar solid angle equation Ω = A/r2 becomes Ω = A/η2r2, where the constant η is the one in the Radian article.

Incidentally, the constant η, which has a value of 1 rad−1, was I think introduced in Torrens (1986). Previously, people like Brinsmade had written “rad” as the constant in unit-invariant equations, equivalent to 1/η. It is easier to understand that a constant represents an angle, rather than the reciprocal of an angle, and, since Torrens, various symbols have been used for 1/η. Leonard (2021) continues to use “rad”; Mohr et al (2022) use Θ/2π, where Θ is the angle of a complete revolution; I now use θC, denoting the “Cotes angle” [Quincey, 2021], which I think is a good long-term symbol and name. This might be worth a footnote.

I will let others propose text, if the suggestions are thought to be appropriate. Pquincey (talk) 10:34, 12 September 2022 (UTC)


 * I think it is past time that the treatment of the radian (and consequently the steradian) as a dimensional unit is given proper treatment in the articles here. The primary discussion of angle as a dimensional quantity would occur in the article Angle as I see it, but this article should give fair treatment of the radian as a dimensional angle, rather than taking the view that it is "inherently dimensionless", as so many still seem to think it must be.  There is evidently a long history of considering angle as dimensional, and it is a serious topic.  My point is that this would mean a minor restructuring of the article to give the concept the space it deserves.  We are also free to report the sources fairly, unlike the SI, which has is constrained to one perspective.  Perhaps we could start by changing the subheading "Dimensional analysis" to something like "Dimensional angle".
 * On a suitable symbol (and this is obviously only my opinion), though θC might be a good symbol for the Cotes angle, adopting a symbol that appears in the numerator rather than the denominator of most common expressions is a strong consideration, and hence η (or any symbol for the same quantity) may be a good idea.
 * Digressing even further, angle and hyperbolic angle fit together rather well in pseudo-Euclidean spaces, including products of their powers. This suggests that hyperbolic angles should be considered on an equal footing from the start, and would suggest a matching dimensional unit of hyperbolic angle that would need definition.  172.82.47.242 (talk) 13:55, 12 September 2022 (UTC)
 * You can see the discussion I had with A1E6 above regarding how much how much weight to give the dimensional treatment. As I understand it, the agreement we worked out is that the dimensional radian can be discussed freely in the Dimensional Analysis section, but it cannot be presented as a mainstream view (of course for neutrality SI cannot be presented as "the" view either). I of course would like to give dimensional radians more weight, but the evidence just isn't there. Arguing against, there is SI and the general lack of secondary/tertiary sources. There just aren't many proponents of dimensional radians besides what's already cited in the article. For comparison the tau constant proposal has gotten a lot more popular attention (many news articles, software libraries, blog posts, etc.) yet it also is limited to a section and its mention on Pi is limited to a paragraph in the "In popular culture" section. Wikipedia is not a soapbox. I think bugging the people on the CCU task group to adopt a dimensional radian is a better use of effort than trying to push dimensional radians here. Regarding the Angle article, I added a link to the dimensional analysis section here, which seems sufficient.
 * I used Torrens's η because there is a secondary source (Quincey 2016) that said it's a reasonable proposal and used it in the paper. All the other constants like Θ or θC seem to be limited to their authors. I've listed the other notations in a footnote. Mathnerd314159 (talk) 00:06, 14 September 2022 (UTC)
 * On the subject of putting dimensional analysis on a less arbitrary footing, this paper: Paul Quincey and Kathryn Burrows, The role of unit systems in expressing and testing the laws of nature, Metrologia, 56 065001 (2019), which is freely available on ArXiv https://arxiv.org/abs/1910.11083, gives an argument for 5 necessary dimensions, including an angular one, based on the number of conservation laws. I don’t know a secondary source, so I am not proposing this for the Wikipedia article, but it may be of interest to anyone reading this. Pquincey (talk) 07:40, 14 September 2022 (UTC)

Stop removing references
I see in the edit history that citations and references for articles keep getting removed. Is this some new thing? Because it used to be a pattern that lead to "no citations" and later editors removing entire articles or the like. Removing this makes for less reliability and quality of content. Also, same user removed reference to a unit just because it's specific to firearms? 74.194.161.186 (talk) 11:14, 28 January 2023 (UTC)


 * What removals of citations in this article are you thinking of, and what reference to a unit specific to firearms has been removed from this article? NebY (talk) 16:09, 28 January 2023 (UTC)
 * I think the IP is talking about @Quondum's Dec 9 removal of the Wolfram Mathworld reference. I do question that removal. The only RS discussion I can find is Reliable sources/Noticeboard/Archive 264 which states that Mathworld is reliable for the topics it covers. I do remember reading something about a Wolfram refspam issue in the past, but the usage here seems reasonable. So I don't see why Mathworld isn't suitable.
 * As far as the firearms unit, the "angular mil" is the first unit that comes to mind. But although the paragraph for it got moved around a bit, it is still there. And it hasn't ever had any references AFAICT. And Quondum never edited it. So maybe the IP had something else in mind. Mathnerd314159 (talk) 20:05, 29 January 2023 (UTC)
 * Ah, I saw that as not so much removing the reference as removing the statement that a radian is $$\frac {180}{\pi}$$ degrees, which is arguably superfluous at that point. Does that paragraph need a source and if so, could any of the other sources support it too?
 * A lot of that discussion at RSN isn't about citing Weisstein's bit of Wolfram, which it might be worth discussing there some day; on the one hand, it seems he has such a free hand without editorial oversight that it verges on WP:SPS, on the other much of the content was published by CRC Press.
 * The OP's implication that Radian might be deleted if this goes on seems hyperbolic. NebY (talk) 20:58, 29 January 2023 (UTC)


 * The OP seems to be complaining in a non-specific way, leaving anyone to guess what they are referring to, as we can see from the discussion so far. Any interaction of this nature aught to be ignored as grousing.  If the OP wishes to be constructive, they can just point out where and why they think something was a change for the worse, and discuss it.
 * On the guessing, the removal of this reference is because that page is a logical mess, unsuitable for citing IMO, especially for this. It reads like it was written by someone who would say that F = ma in SI units, but not in imperial units, as though the units are not part of the quantities in the equations.  —Quondum 22:40, 29 January 2023 (UTC)

Radian is dimension unit because it must indicate the information about a perpendicularity of the radius to the angular values
Here is the key insight: in the context of circles, angles and rotation, units of length are either tangential or radial. Let there be two new units that replace good old $$\rm m$$ (meters): $$\rm m_{tan}$$ (tangential meters) and $$\rm m_{\rm rad}$$ (radial meters). The unit of a $$\rm rad$$ is the conversion or ratio between these two units.

$$\rm rad = \frac{\rm m_{tan}}{\rm m_{\rm rad}}$$

$$1 = \frac{\rm m_{tan}}{\rm m_{\rm rad}}\frac{1}{\rm rad}$$

$$1 = \frac{\rm m_{\rm rad}}{\rm m_{tan}}\rm rad$$

An angular frequency in this context should be $$\rm rad/s$$, not $$1/s$$. A formulation of the relationship of $$L$$ and $$\tau$$ is a little more clear I think if add $$\Delta$$s or $$d$$s like this: $$\tau \cdot dt = dL$$. Doesn't really matter for this question, and I'll use $$s$$ instead of $$ds$$ or $$\Delta s$$.

Take the left side, $$\tau \cdot dt$$. The units are $$(N\cdot \rm m_{\rm rad})\cdot s$$. Notice the use of radial meters.

Expand $$N$$ to give $$(\rm kg\cdot \frac{\rm m_{tan}}{s^2}) \cdot \rm m_{\rm rad}\cdot s$$ and notice the use of tangential meters. Simplify to give: $$\rm kg \cdot \rm m_{tan} \cdot \rm m_{\rm rad} /s$$.

Now multiply this by $$1 = \frac{\rm m_{\rm rad}}{\rm m_{tan}}\rm rad$$, which can also be thought of as "converting" the $$\rm m_{tan}$$ to $$\rm m_{\rm rad}$$:

$$\rm kg \cdot \rm m_{tan} \cdot \rm m_{\rm rad} \cdot \frac{\rm m_{\rm rad}}{\rm m_{tan}}\rm rad/s$$

Simplifying, this now gives the units you were looking for, with the $$m$$ now specified as radial meters:

$$\rm kg\cdot (m_{rad})^2 \cdot rad/s$$

Alternatively, you could multiply by $$1 = \frac{\rm m_{tan}}{\rm m_{\rm rad}}\frac{1}{\rm rad}$$ instead, and you'd end up with the expression that alfC gives, though those meters are now revealed to be tangential meters:

$$\rm kg\cdot (\rm m_{tan})^2 / \rm rad/s$$

This issue must will be addressed clearly and concisely yet. Firstly, thinking of $$\rm rad$$ as dimensionless is not useful, and thinking of $$\rm rad = 1$$ is not useful. In certain frameworks, *technically*, $$\rm rad = 1$$ and "$$\rm rad$$s are dimensionless" are workable, but these statements are somewhat counterproductive for grasping the key insight.

Additional related information is. Voproshatel (talk) 05:46, 5 May 2023 (UTC)


 * This is discussed in the dimensional analysis section, actually the Mohr/Phillips 2015 paper is already cited for its definition of angle in terms of sector area. Regarding your proposal for tangential/radial units (which is WP:OR? I haven't seen anything like it in the literature), I am not sure what it offers - just let $$\rm m = m_{rad}$$ and then naturally $$\rm m_{tan} = m \cdot rad$$, and it is simply the traditional unit system but with an angle dimension and radian base unit. Mathnerd314159 (talk) 15:11, 5 May 2023 (UTC)

SI multiples of radian
Are any of the multiples, from 101 (darad, decaradian) to 1030 (Qrad, quettaradian), ever used? NebY (talk) 20:57, 28 July 2023 (UTC)


 * Yes, I have seen kiloradian (Brown 1991, Koch et al 2012), megaradian (Ceja et al 2022) and gigaradian (Koch et al 2012) used.
 * * Brown 1991
 * * Koch et al 2012
 * * Ceja et al 2022
 * Dondervogel 2 (talk) 22:07, 28 July 2023 (UTC)
 * I see. Thanks. NebY (talk) 22:20, 28 July 2023 (UTC)