Dimensionless units

[Ed Shaya]

Pedro,

    If you are given units of w/cm^2/um then there is no way for a 
computer to know that the um refers to wavelength.  A human can decide 
based on complex reasonable judgement and experiences but it is always 
just a guess.  If I have light emanating from  boxes that are 1 cm in 
height and width and 1 um in depth, then I have the right to express 
this as w/cm^2/um.  Note that this is dimensionally watts/volume.  In 
this case, we would not want a conversion to Janskys with the assumption 
that the um is a wavelength.  You need a system in which the physical 
objects within the units are revealed, such as Light Energy 
Flux/wavelength or Light Energy Flux/Frequency where Flux is already 
defined to be Quantity Flow/Area/Time.  This is to separate it from 
Quantity Change/Area/Time because you do not want to mix up quantity 
sources with quantity motion.   There is also possible confusion between 
Flux at the observer and Flux at the surface of the source, so area 
needs to be more specific.
   Likewise with the problem of Jy/arcsec^2, this needs to be described 
as Light Energy Flux/Frequency/angle^2. 
    Now, maybe the idea is that the UCD or a more precise class of the 
quantity has this detailed description of what it is and then one 
proceeds to do conversions between two quantities of the same class.   
Any  automated tool will need to understand the class as described here 
to know how to go about a conversion.  After breaking things down to 
just dimensionality, some of the information needed to do the 
comparisons and numerical conversions is lost.  Dimensional analysis is 
what you do when you are ignoring amplitude and want a sanity check that 
unit agree.
   Does anyone know the origin of the expression "Nothing is simple"?

Ed