First of all, thanks very much for the code - this essentially solves my problem (I don't like that it doesn't yell at me if I try to "drop" units that shouldn't be dropped, but the original function doesn't do that either so I'll just have to add that functionality). It's both obvious in retrospect (which is good, since it is easy to explain to students!) but also very clever, so thank you for that!

In case there are any non-mathematica experts who are as confused by how it could work as I was, I think you must have changed your notation part way through (or I am still confused). To make it work I had to rewrite it as:

qmr[q_, unit_] := QuantityMagnitude[q/Quantity[1, unit]]

Is there a better/more standard way of doing this with your args__ argument?

As for the reference, thank you. I had read a longer (earlier) paper by the same authors (the one I referenced) where it seems that they were making a slightly different case - that radians are the natural (coherent) unit that should be treated as dimensionless (and that Hz should be treated as cycles/second and not 1/s). But I was wrong in my belief that this opinion was anywhere near unanimous - I'm sorry about that.

So thanks again, Eric

Besides the interesting discussion about radians considered or not as dimensionless units (just read about why they shouldn't be dimensionless [1]), you can have your own simple QuantityMagnitude that effectively normalizes radians:

quantityMagnitudeRadians[q_, unit_] := QuantityMagnitude[q/Quantity[1, unit]]

This would work for all these cases:

In[64]:= quantityMagnitudeRadians[5, "Radians"]
Out[64]= 5

In[59]:= quantityMagnitudeRadians[Quantity[4, "Radians"], "Radians"]
Out[59]= 4

In[60]:= quantityMagnitudeRadians[Quantity[4, 1/"Seconds"], "Radians"/"Seconds"]
Out[60]= 4

In[61]:= quantityMagnitudeRadians[Quantity[4, "AngularDegrees"], "Radians"]
Out[61]= \[Pi]/45

In[62]:= quantityMagnitudeRadians[Quantity[4, 1/"Seconds"], "Radians"/"Seconds"]
Out[62]= 4

In[63]:= quantityMagnitudeRadians[Quantity[4, "AngularDegrees"/"Seconds"], "Radians"/"Seconds"]
Out[63]= \[Pi]/45

[1] https://iopscience.iop.org/article/10.1088/1681-7575/ab27d7

Yes, radians are a unit of measuring angles, like degrees. But they are not at all arbitrary - they are defined in a very particular way (as I mentioned, as a ratio of arc length to radius) which is why they end up having 2Pi as one revolution.

Also, yes, Hz means cycles (revolutions, periods,...) per second, not radians per second. So perhaps counter-intuitively writing Hz is NOT the same as writing 1/s, because 1/s is the same as radians/second. You need to explicitly put in "cycles" or "revolutions" or something like that when you write it out.

You ask for informed opinions. I've been teaching this for the past 30 years but here are some other sources: https://en.wikipedia.org/wiki/Radian "Although the radian is a unit of measure, it is a dimensionless quantity... the units of angular velocity ... are s^{?1}" Or maybe better, a paper from NIST (National Institute of Standards and Technology) "Dimensionless units in the SI" (https://www.nist.gov/publications/dimensionless-units-si) which says "Here we consider dimensionless units as defined in the SI, e.g. angular units like radians...In particular, we argue that the radian is the appropriate coherent unit for angles and that hertz is not a coherent unit in the SI."

Thank you for the radrule suggestion and comments. You are absolutely right that as far as Mathematica is concerned Radians are not Dimensionless. That is the problem. Because Radians ARE dimensionless. An angle, in radians, is defined as the length of an arc on a circle subtended by that angle divided by the length of the radius. Length/Length = Dimensionless.

If that is unconvincing then think about a simple pendulum. It has an angular frequency of \Sqrt[g/L] (g is the gravitational constant with dimensions [Length]/[Time]^2; L is the length of the pendulum with dimensions [Length]). So the angular frequency has dimensions 1/[Time]. But angular frequency has to be in [AngularUnit]/[Time]. The "angular unit" here is Radians.

To be explicit: 1/"Seconds" is unambiguously converted to "Radians"/"Seconds". You are free to add in Radians units or remove them at any point you want, because they are dimensionless.

For a quick fix I've written a routine that tests whether they are compatible and if not throws in Radians to try to make them compatible. But I hope Wolfram can just fix this. Do they ever read these discussions?

Thanks for the quick response. I probably should have given a better example. This works:

QuantityMagnitude[Quantity[1, "AngularDegrees"/"Seconds"], 
 "Radians"/"Seconds"]

but this doesn't:

QuantityMagnitude[Quantity[1, 1/"Seconds"], "Radians"/"Seconds"]

Either way, the point is that you should be able to convert a dimensionless quantity to a quantity with units of Radians, because Radians are unitless (they are defined as the ratio of two lengths)