[UNITS] Re: Dimensionless units

[Ed Shaya]

David,

I would say that your list is indeed dimensionless, but that the class 
dimensionless has various subclasses such as "count of something", 
"ratio", "pure number", and "index".  Angle and logarithms can be  
subclasses of ratio.   Angle is basically asin(length/distance) which is 
dimensionally a ratio (note that for large distance it can be 
approximated as length/distance).  Logarithm is integral (1/x)dx
which is dimensionally a ratio.
These subclasses are going into the ontology that I am working on.
Ed


Brian Thomas wrote:

> David,
>
>On Thursday 17 February 2005 06:15 am, David Berry wrote:
>  
>
>>On Wed, 16 Feb 2005, Brian Thomas wrote:
>>
>>    
>>
>>>Good point and true. There are such things as "dimensionless" units
>>>(e.g. "angle" and "number of <thing>") which *do count* as "dimensions".
>>>
>>>      
>>>
>>Tables 5 and 6 in FITS WCS Paper I includes the following dimensionless
>>units:
>>
>>deg  (plus rad, arcmin, arcsec, mas)
>>sr
>>mag
>>count (plus photon, ct, ph)
>>pixel (plus pix)
>>Sun
>>chan
>>bin
>>voxel
>>bit
>>byte
>>adu
>>beam
>>    
>>
>
> Right, most of these are in the 'dimension' "number of <thing>".
> "pixel" is in the dimension "number of pixels" and so on. "sr" and 
> "degree" (and etc) belong to the dimension of "angle". Magnitude
> is an interesting case as it is, of course, 'dimensionless' being the
> log of a brightness and scaled relative to a reference source.
>
>  
>
>>Obviously, not all of these are equally important (the top 3 in the list
>>are probably the most important), but it makes the point that almost any
>>noun could be used as a unit. A string parsing scheme has no problems
>>with this, but I'm not sure how a dimensional analysis scheme would handle
>>it.
>>    
>>
>
> For the purposes of dimensional analysis, all of these represent separate
> dimensions I would think, (with the exception of "bit" and "byte" which 
> surely belong to the same 'number of' dimension and differ by a scale factor 
> of '8'). There may be other connections that one may make between the 
> 'noun' dimensions (as you refer to them) above, depending on the experiment
> (for example bin equivalent pixel in cases).
>
> So, at any rate, yes, the dimension "number of <thing>" is not one singular 
> dimension but a 'super class of a type of dimension' wherein each sub-class
> is orthogonal to one another (if properly designed...again I don't thing 'bit' 
> and 'byte' are in different 'dimensions') and <thing> is implicitly defined by
> the conditions under which the measurement was taken ('bin' in one experiment
> is not necessarily equivalent to 'bin' in another). 
>
> But its not so hopeless that we have to go out and select candidates from a 
> dictionary to build up a set of these 'noun' dimensions.  For the purposes 
> of the VO, I think we would be able to get by quite well with the list you just 
> included above.
>
> Regards,
>
> =b.t.
>
>
>  
>
>>David
>>
>>    
>>
>
>  
>