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Not if you wish to maintain the convenient rule that units combine
One can use this, { (1) and (2) above }, as an opportunity to distinguish
between *units* and *physical dimensions*. E.g. *length* is a physical
dimension, the SI unit for length is *meters*. If you make this distinction
then the *radian* is not a psuedo-unit, but is an actual honest-to-gosh
unit; it does happen to be a dimensionless unit!