Can a quantity have dimensions but still no units?
A quantity cannot have dimensions without units. That is illogical.
But it is logical to have units without dimensions.
For example :
"Mach" is a unit which determines the ratio of the speed of a body with respect to the speed of sound.
Since it's a pure ratio, it does not have dimensions But it is a unit.
no ph. quantity can have dimension but no unit because we derive thedimensions by directly or indirectly using the units.
there are quantity having units but no dimensions.Plane angle,angular displacement, solid angle.These physical quantities have units but they dont have dimensional formulas.