Same thing with radians (a "dimensionless" unit) -- angle is the ratio of two lengths, arc length/radius, so technically dimensionless, but those two lengths ar

Same thing with radians (a "dimensionless" unit) -- angle is the ratio of two lengths, arc length/radius, so technically dimensionless, but those two lengths are not of the same thing. I always felt more comfortable saying "pi radians" rather than just "pi" because radians feel like a real unit in the same way that units-east-per-unit-north do. I wonder if there's been any study into when it's ok to cancel two of the same unit and when it's better to leave it as a ratio of two quantities with the same unit but describing different dimensions. Is there a way to make this notion precise?