Talk:Radian

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)

@Mathnerd314159: @Bunkerpr: 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]⁻²), whence the chosen base units (e.g. meter, second, etc.) produce corresponding derived units (e.g. newton, equal to kg m s⁻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⁻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 1 Hz is well and truly equal to 1 Bq.

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.

References

was deleted by AE16 in [1] 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, N • m = J and [torque] = [force] • [length] = [energy] just as much as 2×3 = 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#History as an SI unit 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 [2] / [3] 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)

@A1E6 and Mathnerd314159: 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 Wikipedia: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)

 Partly done 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 θ_C, 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 θ_C 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 θ_C 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 Wikipedia: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)

References

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)

Relationship of angle to arc length and radius.

The verbal definition says (correctly): More generally, the magnitude in radians of a subtended angle is equal to the ratio of the arc length to the radius of the circle. The following mathematical formula has that ratio on the right-hand side. So the left-hand side (denoted by Greek letter theta) also must be the magnitude in radians of the subtended angle. This is often colloquially called "the angle in radians". And is the source of most of the confusion and "controversy" about plane angle, in general, and the radian, in particular. In the SI, what is called "angle" is not the physical angle but rather the magnitude in radians of the physical angle, which is the number of radians in the physical angle: (physical angle)/rad. Or, in other words: the numerical value of the physical angle when that angle is expressed in radians. So the SI "radian" is the number of radians in one radian: SI "rad" = (1 rad)/rad, which, of course, is identically equal to 1. This habit seems to have arisen in the terminology of complex numbers, where (physical) lengths of lines and curves in the complex plane represent (real) numbers; and (physical) angles in the complex plane also represent (real) numbers. The unit for the physical lengths is the (physical) length of the unit circle. But, because plane angle is independent of scale, the (physical) unit for angles in the complex plane is ALWAYS one radian.

Please read my Metrologia Review article: "Proposal for the dimensionally consistent treatment of angle and solid angle by the International System of Units (SI)", Metrologia 58 052001, Published 16 July 2021.

Dr B P Leonard, Emeritus Professor of Mechanical Engineering, The University of Akron.

Boppennoppy (talk) 22:24, 2 November 2024 (UTC)

@Boppennoppy Regarding your review article, I do not have access to that. It would help if you could upload it to ResearchGate, or if you can't share it publicly, I can send you an email via the Wikipedia interface so you can email it to me. Having not read the article, perhaps I am inaccurate on some of the details, but from the abstract it seems your proposal is to distinguish the "physical angle", the geometric construction of two rays from a point, from the "magnitude of the angle", which is simply a real number. There is then a further distinction between the "measure of the angle", which is a dimensional quantity with units of radian and a dimension of angle, and the "numerical magnitude of the angle", which is a nondimensional (pure) quantity, equal to the measure divided by 1 radian.

I am struggling through because, in your comment, you admit that SI equates "angle" with "the magnitude in radians of the physical angle". In fact, you say that, colloquially, the magnitude in radians of the physical angle is simply called "the angle in radians", and in SI, which assumes radians, simply "the angle". So from an SI perspective there is no issue with abbreviating the description of θ the way the article did before your edits. Per previous agreement, the way the article is written is that it assumes use of SI throughout the article, except in the dimensional analysis section where it discusses alternative systems. That's also why I find it strange to write "the magnitude in radians of 1 SI radian = 1", when the SI brochure simply says "1 rad = 1" - it's adding words that are not there in SI.

Certainly it would be nice if the article was able to phrase things in a manner that didn't require favoring SI, and somehow accommodated more dimensionally-correct unit systems, but to me it doesn't seem justifiable to add this magnitude verbiage - it doesn't particularly clarify the situation for those trying to understand dimensionalized angles, and for anyone used to SI it looks incredibly verbose and out of place. Mathnerd314159 (talk) 02:42, 3 November 2024 (UTC)

Because of copyright restrictions, I am unable to share my Review article publicly. But I can send single copies to individuals. If you send a private message to me, I'd be happy to transmit a copy to you that way. In the mean-time, please review Proposition 33 in Book VI of Euclid's Elements. This is the (correct) basis for the relationship of angle to arc length and radius—and leads directly to the dimensionally correct definition of the radian. Everything in rotational kinematics and dynamics (if treated correctly) should stem from that. Needless to say, that is not the way the SI (or any physics or mathematics textbook or dictionary) handles plane angle and rotational mechanics. Using the same name and symbol for both a physical angle and the numerical value of the angle when expressed in radians (i.e. the SI "angle") is the primary cause of the well-known widespread confusion that has permeated this subject for centuries. Ken Brownstein's paper is also of fundamental importance, and is worth reviewing. Boppennoppy (talk) 04:14, 3 November 2024 (UTC)

The stated formula l_arc = 2π r (θ/360º) is correct, provided θ represents a real physical angle (expressed in any angle unit). This can be written l_arc = r θ/(180º/π) or:

l_arc = r θ/rad

This is the fundamental (dimensionally correct) formula relating arc-length, radius and central angle in a circle (stemming directly from Proposition 33 in Book VI of Euclid's Elements), where rad is a dimensional constant of Nature. [It makes sense to choose this constant of Nature as a unit for plane angle.] In the SI, the ratio θ/rad is called the "angle in radians"—which, at best, is a confusing shorthand for "the numerical value of the angle when expressed in radians". But, even more confusingly, this ratio is usually just called the "angle" and given the same symbol as the physical angle, namely: θ. So we see:

"l_arc = r θ, where θ is in radians"

In the dimensionally correct formula (with the explicit rad), when l_arc = r, θ = 1 rad. However, in the SI formula, when l_arc = r, "θ"= 1. Since the SI (correctly) defines the radian as the central angle in a circle for which the arc length is equal to the radius, this defines the SI "radian" as being a special name for the number 1. Since the SI "angle" is actually the number of (real) radians in the (real) angle, the SI "radian" is the number of (real) radians in 1 (real) radian: SI "rad" = (1 rad)/rad = 1, exactly.

There is no wonder that there is widespread confusion permeating this subject. This confusion would immediately evaporate if the dimensional constant rad were to be kept explicitly in the fundamental formula and all relationships deriving from it. There is no need to introduce a different symbol for the constant of Nature (or its reciprocal). The same symbol, rad, can be used for both this dimensional constant and the (chosen) unit. This is particularly useful in computational software, where all units work out correctly—without the need for any "human insight" and post-calculation "adjustment". Boppennoppy (talk) 14:22, 3 November 2024 (UTC)

OK, I finally had a chance to look at this, thanks to the recent snow days. I will first preface this by explaining that it is quite tricky to write this article while adhering to Wikipedia's pillar of neutral point of view as it is unclear how much weight to give SI vs. you and Quincey and the others writing in Metrologia. There was the discussion here where A1E6 argued that peer-reviewed journal papers do not trump the multitude of textbooks, although I argued most textbooks only give the discussion of the radian's dimensional problems a passing mention at best. I don't know if A1E6's stance has changed since then.

But let's see if everyone agrees on this summary of the different positions:

• Undisputed: The formula for arc length, l_arc = 2π r (θ/360º), and the definition of one radian as "the central angle in a circle for which the arc length is equal to the radius", are undisputed. It also is not disputed that SI's definition of the radian as dimensionless is confusing in some cases, (It is disputed as to why this confusion arises and how serious this confusion is) or that dimensional definitions of the radian are more suitable for use in symbolic manipulation in computer programs.
• SI: SI uses the equation s = rθ or θ = s/r, which implicitly makes the radian a special name for the number 1. Because 1 is the multiplicative identity, there is no distinction between "the numerical value of the angle when expressed in radians" or the "angle in radians". SI also assumes radians, so all of these quantities are referred to simply as "angle". Some explanatory changes in wording have been made for the forthcoming edition of the brochure to make these points clearer. However, the CCU has no plans to change the equation s = rθ. The majority position within SI is that the current definition of the radian is not confusing enough to warrant a change.
• Leonard (Boppennoppy): You argue that Proposition 33 in Book VI of Euclid's Elements leads directly to a dimensionally correct equation, s = (rθ)/[rev/(2π)]. You propose using the unit rad (radian) as the dimensional symbol, leading to the equation $\theta =s/r\cdot {\text{rad}}$, and argue this is superior to introducing a separate dimensional symbol, based on considerations of the close relationship between nondimensionalized quantities and physical angles in radians. You hold that confusion is "widespread", is inherent to the SI's conflation of angle with its numerical value, and that all physics and mathematics textbooks / online resources should be updated to use equations with dimensional information.
• Quincey: Quincey has a separate line of argument based on analysis of force and torque. He ends up with a similar proposal, although some equations differ. He says he finds it better to not use unit symbols within equations and instead use a separate dimensional constant $\theta _{c}$ to emphasize that the quantity is independent of the choice of unit angle. Quincey does not advocate phasing out traditional equations, instead viewing them as simplified or nondimensionalized equations with $\theta _{c}$ set to 1.
• Authors generally: Authors in Metrologia and other places, such as Ken Brownstein, generally have proposed to make the radian dimensional, with only a few isolated cases arguing for the status quo. The proposals can be separated into the (inconsistent) proposals to make the radius of a circle have units of m/rad, and the dimensional constant proposals, which generally agree in adding a constant but have disagreements on the notation for this constant and in, some cases, how it should be inserted into equations.

Mathnerd314159 (talk) 18:10, 8 November 2024 (UTC)

I don't watch this article anymore and I won't oppose any of your edits here. But researching various resources is a very good thing... keep it up! Regards. A1E6 (talk) 19:04, 8 November 2024 (UTC)

What should be undisputed is that: (arc length)/(circumference) = (central angle)/(one revolution), where one revolution is a fixed reference angle (dimension A), not a unit (unless or until someone chooses it to be). Writing 360º for this reference angle does introduce the common unit of degree—which goes against the philosophy of all those authors who have studiously avoided introducing the (unit) symbol "rad" into the fundamental relationship between translational and rotational motion, derived indeed, directly from Proposition 33 in Book VI of Euclid's Elements. In a plane, c = 2π r, so (using s for arc length) the fundamental relationship becomes:

s /(2π r) = θ/rev or s = (2π r)θ/rev

where rev (with an italic font) is the reference angle of one full revolution (not a unit). I'm guessing that if it is "undisputed" that s = (2π r)θ/(360º), where the degree unit has been introduced, then it should be even "more undisputed" that s = (2π r)θ/rev, which does not involve any units! This, of course, can be rewritten as:

s = r θ/(rev/2π) = r θ/rad

where the reference angle denominator (in an italic font) rad = rev/2π is just the symbol for a different reference angle called a radian, not (yet) a unit.

The main source of the (widespread) confusion is that what the SI calls "angle" is not the physical (dimensional) angle θ but rather the quotient θ/rad, sometimes calling this quotient of two angles "the angle in radians". And it compounds the confusion by writing this as θ, as if this were a (dimensional) angle.

If we were doing this properly, we would CHOOSE the base unit for (dimensional) angle as one radian, defined as (written in roman font, as all units are):

rad = rad = rev/2π

This, of course, means that the SI unit "rad" is actually rad/rad = 1, exactly.

If this is recognised, then, wherever "θ" appears in SI-type (i.e. textbook) formulas, it should be replaced by the physical angle (dimension A) divided by (the reference angle) rad. Similarly, wherever "ω" (angular velocity) appears, it should be replaced by the physical angular velocity (dimension A/T) divided by rad. And similarly, for angular acceleration, and higher time derivatives. And this should be done throughout all the rotational kinematics AND dynamics equations. If computational software is written this way, then, where the end results involve rad, this can be interpreted as the unit rad. [Or, more practically, there is no real need to distinguish between rad and rad, so that rad can be used throughout—giving dimensionally correct units in the results automatically.]

In illumination technology, the unit steradian is (now) used correctly: with the symbol sr appearing explicitly in the basic relationship between solid angle, Ω, spherical area, A, and radius, R, given by:

Ω = A/R² sr

This is necessary in order to distinguish between the candela and the lumen. [With the SI "steradian" = 1, the current (2019) supposed definition of the candela is actually, in terms of defining constants, a definition of the lumen!]

There is absolutely no "reason" (except "inertia") why the (even more basic) translational-to-rotational formula should not be written:

θ = s/r rad

where the (symbol for the) unit radian appears explicitly here—and throughout all the other equations of rotational mechanics. Boppennoppy (talk) 21:51, 9 November 2024 (UTC)

References

Definition of radian and fundamental formula.

As originally stated in words, in a circle, the magnitude in radians of an angle subtended at the centre by an arc on the circumference is equal to the quotient of the arc length and the radius. In symbols:

θ = s/r

where θ is the magnitude in radians of the subtended angle, s is arc length and r is radius. Thus implying that θ is the quotient of the physical angle and an angle equal to one radian. Boppennoppy (talk) 15:15, 10 November 2024 (UTC)