If we want to be picky, all numerical ratios, while dimensionless, still
have units.
For example, suppose we want to measure the ratio of a person's arm span
to his/her height. The answer would have units of length of arm span to
length of height. For example, if both had values of 2 m, the ratio
would be 1 m of arm span per m of height.
Similarly, the mixing ratio of water vapor (mass of water vapor per mass
of dry air) can be written as kg of water vapor per kg of dry air.
Actually, it is typically written in units of grams of water vapor per
kg of dry air (in order to have numerical values around 1).
Similarly, as already noted earlier, the radian can be rewritten as m of
arc per m of radius. The radian deserves no special treatment in this
regard.
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Robert Cohen; 570-422-3428; www.esu.edu/~bbq
East Stroudsburg University; E. Stroudsburg, PA 18301