[VOUnit] How to express the unit of a dimensionless quantity?

[Norman Gray]

Alberto and all, hello.

On 6 Mar 2024, at 13:33, alberto micol via semantics wrote:

> The latest version of the VOUnits/20231215/REC-VOUnits-1.1 document
> states that:
> 1.- In §2.2: empty strings are invalid
> 2.- In §2.2: the string ‘1’ indicated a dimensionless quantity
> 3.- In §2.8: data providers should use an empty string for unitless quantities
> (excerpts from the standard are shown below)
>
> 3. clearly contradicts 1. and 2. May I kindly ask for an errata?

I think you're right, and that there is an omission here, but it's not quite a contradiction.

There are three cases here:

  1. Things which are properly dimensionless, such as dlambda/lambda -- the standard says that this can and should be represented by the unit-string "1"

  2. Things where the units are unknown, as a matter of ignorance rather than because they are absent -- the standard reserves "unknown" for this, but doesn't mandate using that, and notes that these omissions can be represented any way.

  3. Things which do not have units, such as a comment-string.  Sect.2.8 is addressing this third case, and could be better phrased.

Sect.2.2 notes that the empty string is not a valid 'units string', but if you want to put that into the 'units' column of your database, this standard isn't going to forbid you -- it's your database, after all -- but a units parser is obliged to report that as an invalid string.

Sect.2.8 is therefore proposing a good practice, namely to put an empty string in this column of the database, rather than some other flag, which might cause confusion or uncertainty in future.  That is, it's suggesting which particular invalid units string might be the best choice in this location.  An empty string shouldn't cause confusion, since the database owners presumably know that the column is a non-unit object, and the fact that there's a 'units' field associated with it, which requires a value, is merely an artefact of schema design.

As an edit, I'd propose adding a suitably compact version of the above trichotomy into Sect.2.2, and changing the end of Sect.2.8 to refer back to that.  The back-reference would be useful, since this 'other remarks' section is where I might look for a discussion of this.

I can write some words if folk agree this is a suitable interpretation, and which of the above points is and is not illuminating to a reader.

> Given the above, what shall I use for the units of the following unitless/dimensionless quantities:
> - spectral resolving power (lambda/delta_lambda)
> - SNR
> - NCOMBINE (number of exposures co-added in a stacked image)
> ?

I think that each of these would be straightforwardly "1", on the grounds that the dimensional analysis of each of these -- in terms of M, L, T, and so on -- would have no dimensions with a non-zero power.

I have the nagging feeling that we're not sufficiently clear about the scope of the VOUnits standard, in terms of both exclusion and inclusion.  We try to say something to that end in Sect.1.2, but I'm sure we could be accused of both saying too little and of going on at too much length.

I've just looked at the SI Brochure for inspiration, and it says:

> The value of a quantity is generally expressed as the product of a number and a unit. The unit is simply a particular example of the quantity concerned which is used as a reference, and the number is the ratio of the value of the quantity to the unit.

...which still has an element 'you know it when you see it' when it comes to the question 'what is a unit?'  I don't want to get too philosophical here.

Best wishes,

Norman


-- 
Norman Gray  :  https://nxg.me.uk
SUPA School of Physics and Astronomy, University of Glasgow, UK