  The International System of Units (SI) is supposed to be coherent. That is,
when a combination of units is replaced by an equivalent unit, there is no
additional numerical factor. Here we consider dimensionless units as defined in
the SI, {\it e.g.} angular units like radians or steradians and counting units
like radioactive decays or molecules. We show that an incoherence may arise
when different units of this type are replaced by a single dimensionless unit,
the unit "one", and suggest how to properly include such units into the SI in
order to remove the incoherence. In particular, we argue that the radian is the
appropriate coherent unit for angles and that hertz is not a coherent unit in
the SI. We also discuss how including angular and counting units affects the
fundamental constants.
