# [UNITS] Re: Dimensionless units

Brian Thomas brian.thomas at gsfc.nasa.gov
Thu Feb 17 07:29:10 PST 2005

``` 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
> 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
>

--
--------------------------------------
|
| Dr. Brian Thomas
|
| Dept of Astronomy
| University of Maryland-College Park
|
| Phone: (301) 405-2312
| Fax: (301) 314-9067
|
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