I actually (embarrassingly) got tripped up by that a bit.
So, you're saying that radians/degrees _are_ a unit because we can use them to describe a count of something, but they're still in fact dimensionless?
Is that because 2*pi or degrees never refer to an actual physical length? I think that's what trips me up because I generally have thought of degrees and radians as essentially units of length around a circle. But I suppose that since we're working with circles of arbitrary rather than specified diameter, they're still unitless.
I think I got to it on my own at the end there ha but I'm happy to be corrected!
One way to think about radians is "feet (of perimeter) per feet (of radius)" -- the dimensional units cancel and you're left with a dimensionless unit.
Taxes are a familiar dimensionless unit -- dollar owed to the government, per dollar spent on snacks. The dollars cancel and you're left with a raw percentage. (actually, to convert to a percentage you multiply by 100 percent per whole; another dimensionless conversion)
If they are truly dimensionless, when is it meaningful to multiply a non-currency value by them? There’s still a semantic that attaches the rate to dollars obtained in a transaction.
So we can discuss precisely: how do you define dimensionless? There are well-accepted definitions, I’m just asking you to lay it out so we can refer to it specifically.
Ratios in which the numerator and the denominator have the same unit are only dimensionless if you decide to apply the division and cancel the common unit. If you do this cancellation, and you forget/hide the semantic role of numerator and denominator, you will indeed get mistakes.
Part of the issue is being able to capture enough semantic role to be useful. That’s usually ignored in units discussions.
Radians are a unit, but are dimensionless. You'll end up in a world of trouble if you got rad/s and rpm mixed up.