This is a known wrinkle in dimensional analysis and people have considered making angles a fundamental quantity such as mass, length and time but have not done so because of the disruption it would cause.
More details here
> The current state of affairs leads inevitably to ghostly appearances and disappearances of the radian in the dimensional analysis of physical equations.
In "A spectral unit", Nature Physics (2020) - https://www.nature.com/articles/s41567-020-0997-3.pdf Giacomo Prando summarizes the troubled history of the radian, a unit with the odd property of appearing and disappearing seemingly at will in dimensional formulas
It led me to reading about "dimensionless quantity".
> There have been periodic proposals to "patch" the SI system to reduce confusion regarding physical dimensions. For example, a 2017 op-ed in Nature argued for formalizing the radian as a physical unit.
SI units need reform to avoid confusion (2017) - https://doi.org/10.1038%2F548135b
> The idea was rebutted on the grounds that such a change would raise inconsistencies for both established dimensionless groups, like the Strouhal number, and for mathematically distinct entities that happen to have the same units, like torque (a vector product) versus energy (a scalar product).
Don't tamper with SI-unit consistency (2017) - https://doi.org/10.1038%2F549160d
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What's surprising is that these discussions are so recent, considering the wide implications on so much practical work in mechanics and engineering. Maybe people got so used to working around the question, that any proposed solution would be too disruptive - so it's better to keep things backward compatible rather than conceptually simple/clear and explicit.
In another comment someone said, "types" (in programming and type theory, I'd guess) are a poor model of physics units. But I wonder about that, it seems "units" are something like types with associated quantities, like degrees with 360, or meters with the speed of light.
This line of inquiry also led me to "dimensionless physical constants". https://en.wikipedia.org/wiki/Dimensionless_physical_constan...
Certainly pi (and e) must be one of those fundamental constants regardless of any unit of measurement. A "turn", on the other hand - well, if we consider 1 as a dimensionless constant that means a "whole"..
How Many Fundamental Constants Are There? (2011) John Baez https://math.ucr.edu/home/baez/constants.html
I think the fact that action and angular momentum have the same units played an illuminating role in making sense in Bohr's model regarding why only certain orbits are allowed.
Not sure how that would play out once angle is considered a fundamental entity.
This sure is a rabbit hole.
Thanks for your submission
https://news.ycombinator.com/item?id=49372847
hope it gets picked up.
But that's more for analysis of your code / formulas than when you actually go and compute things.
Dimensionless, sure, but what do you mean here? Radians and degrees are units, are they not?