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I had a neat realization during a conversation with my colleague Cameron today, so I thought I'd share. Feel free to tune this out if you're not a math/physics type.
Cameron was pondering the various "units" for angles: the common ones are degrees, radians, and rotations (a.k.a. revolutions). Physics students learn that many common equations (such as v = r ω) only work when angles are in radians: they given wrong answers if you use degrees or rotations. It's a standard story: radians are defined in terms of a ratio of lengths (arc length/radius) so they are automatically dimensionless (since a ratio like "meters/meters" cancels out).
But in what sense is "rotations" not dimensionless? After all, you're just counting things, so what you end up with ought to be a pure number (which has no units by, definition). For that matter, "degrees" is just counting things too (albeit more finely spaced things). Given that multiplying by a pure number can't possibly change the units of your answer, it's hard to understand on a conceptual level why angles measured in degrees or rotations shouldn't work in the equations. (The equations themselves are entirely clear about what works and what doesn't, mind you! It's just understanding them that's subtle.)
The resolution that I've come to is that "degrees" and "rotations" are not units in the sense that "seconds" or "meters" are units. They can't be, as noted. Instead, "degrees" and "rotations" have the same status as SI prefixes like "kilo" and "micro". After all, "kilo" effectively translates as "times 1000", a pure number, but you wouldn't expect your equation to give the right result if you removed a "kilo" that was supposed to be there! So I would assert that one can think of "degrees" as an SI "prefix" meaning "times 2π/360" and "rotations" as an SI prefix meaning "times 2&pi". Explicitly plugging in these definitions then just gives your result in terms of the base unit, the most natural one: radians.
It may not actually be all that helpful in practice (my advice is still "just use radians"), but I think this gives a cute new perspective on what these angular "units" actually mean.
Update: Another useful note: The choice of "base unit" occurs when we define the angle in radians: θ = s/r. That choice then carries through the rest of the equations that we use for angular motion. If we had instead defined the angle as θ = s/(2πr), the "base unit" would have been rotations, and that factor of 2π would show up in every other rotational equation. Radians are clearly the simplest choice, but there isn't anything privileged about it. (Just imagine the ugliness of equations throughout rotational motion if we decided to choose our fundamental definition of angle to be θ = 360s/(2πr)!)
Cameron was pondering the various "units" for angles: the common ones are degrees, radians, and rotations (a.k.a. revolutions). Physics students learn that many common equations (such as v = r ω) only work when angles are in radians: they given wrong answers if you use degrees or rotations. It's a standard story: radians are defined in terms of a ratio of lengths (arc length/radius) so they are automatically dimensionless (since a ratio like "meters/meters" cancels out).
But in what sense is "rotations" not dimensionless? After all, you're just counting things, so what you end up with ought to be a pure number (which has no units by, definition). For that matter, "degrees" is just counting things too (albeit more finely spaced things). Given that multiplying by a pure number can't possibly change the units of your answer, it's hard to understand on a conceptual level why angles measured in degrees or rotations shouldn't work in the equations. (The equations themselves are entirely clear about what works and what doesn't, mind you! It's just understanding them that's subtle.)
The resolution that I've come to is that "degrees" and "rotations" are not units in the sense that "seconds" or "meters" are units. They can't be, as noted. Instead, "degrees" and "rotations" have the same status as SI prefixes like "kilo" and "micro". After all, "kilo" effectively translates as "times 1000", a pure number, but you wouldn't expect your equation to give the right result if you removed a "kilo" that was supposed to be there! So I would assert that one can think of "degrees" as an SI "prefix" meaning "times 2π/360" and "rotations" as an SI prefix meaning "times 2&pi". Explicitly plugging in these definitions then just gives your result in terms of the base unit, the most natural one: radians.
It may not actually be all that helpful in practice (my advice is still "just use radians"), but I think this gives a cute new perspective on what these angular "units" actually mean.
Update: Another useful note: The choice of "base unit" occurs when we define the angle in radians: θ = s/r. That choice then carries through the rest of the equations that we use for angular motion. If we had instead defined the angle as θ = s/(2πr), the "base unit" would have been rotations, and that factor of 2π would show up in every other rotational equation. Radians are clearly the simplest choice, but there isn't anything privileged about it. (Just imagine the ugliness of equations throughout rotational motion if we decided to choose our fundamental definition of angle to be θ = 360s/(2πr)!)