"Quantity dimension" has a very precise meaning in SI, as does "quantity",
"derived quantity" and other terms. There are many examples of dimensionless
derived quantities (with a quantity dimension of one), and of products of those
dimensionless derived quantities with other non-dimensionless derived
quantities. You may find the SI nomenclature arcane, but it is very precise. (01)
The SI indicates a property of derived quantities, "kind", which is to be used
for distinguishing between derived quantities having the same quantity
dimension. Its meaning is not fully developed in the SI, rather it is left to
others, such as yourself, to do so. To be consistent with SI nomenclature, I
urge you to use the concept of "kind". (02)
If you are interested in units and their meanings, I strongly recommend
purchasing and reading INTERNATIONAL STANDARD ISO/DIS 80000-1, "Quantities and
units" if you haven't already. It is the first, and most complete source I have
read for international metrology standards. I suggest this because it seems it
would clear up much of the lack of knowledge and misconceptions often exchanged
about quantities, units, physical dimensions.
It does cost money, but I urge you to overcome, as I did, the cheapness
instilled by "free" information elsewhere. (03)
Regards,
Joe C. (04)
Patrick Cassidy wrote:
> I would like to pursue the issue of the unit of measure of an angle, since
> it also relates to other "ratio'-like quantities that may be used as
> measures. The gist of this note is that I would not like the 'dimension' of
> an angle to be considered as null, or 1, or dimensionless, but as something
> that means 'angular measure'. For an ontology that is intended to represent
> meanings, I am very leery of oversimplifications that work fine in
> restricted contexts but may prove confusing in missed contexts.
>
> The last note from Ingvar Johansson had this portion of a discussion:
>
> [John Sowa] > > I also like that analysis. But it has to be extended to
> angles,
>>> since we have to support multiple functions that map angles to
>>> numbers: degree and radian.
> [IJ] > I agree, and in a sense so do also the metrologists that (as I said
> in
>> an earlier mail) I criticize. In my opinion, one should say that radian is
>> a unit of the derived dimension length/length, but the SI system and VIM
>> says that it is a dimensionless unit or a unit of dimension-one.
>> However, everyone agrees that angles can be measured by (or mapped on)
> scales
>> whose magnitudes are 'x degree' or 'x radian'.
>>
>> I think, by the way, that it is misleading to say that "angles are
>> mapped to numbers"; angles are mapped to magnitudes of a scale.
>>
> Although an angle in radians can be expressed as a ratio of linear
> measures, the linear measures themselves do not measure arbitrary straight
> lines, but are quite specific regions of some imaginary circle. I think it
> is a misleading oversimplification, when taking ratios of things that are
> not themselves pure numbers, to ignore the meanings of the measures that are
> being divided. A similar issue has arisen in the past about how to express
> things like "weight percent" which, if one ignores the objects that are
> represented by the numerator and denominator, can appear to be a
> dimensionless number (grams/grams). Such ratios have an actual conceptual
> "dimension" though the SI and VIM committees may have found it possible to
> ignore the meanings in the case of radians, knowing that the dimensions will
> likely be interpreted properly in applications. One way to recognize the
> problem is to note that if one wants to represent a weight ratio, it is
> possible to use micrograms per gram or grams per gram, and the "dimensions"
> will appear to cancel out in either case, leaving a "dimensionless" number,
> though the resulting numbers differ greatly depending on what units are
> chosen for the numerator.
> I would suggest that we promiscuously include all quantifiable "units"
> that carry meaning in any application, and not take as "dimensionless" any
> measures that are in fact distinguishable in their intended meaning. A
> weight ratio does *not* have the same dimension as an angle, though one can
> oversimplify either to some dimensionless number.
>
> In this view, a 'radian' is a unit of measure, as is a 'degree-of-angle',
> and if the dimension is represented separately from the unit of measure, the
> dimension in either case would be 'angular measure'. The dimension of a
> weight ratio is the ordered pair of objects or types of objects whose
> weights are being divided (weight ratios might better be treated in a
> different way, but if they were treated as measures with a unit, that would
> be my preference for the unit).
>
> It may be possible to consider certain ratios as the 'base unit' as in
> the case of a radian, where the subtended arc length and radius are the
> defining measures being divided. In the case of weight ratio,
> grams-of-X/grams-of-Y might be the base unit for each X/Y pair. Measures
> that are related to other ratio measures by some constant number, such as
> angle degrees or micrograms/gram, would then be related to the base unit as
> "prefix"-unit is to other base units, where "prefix" may be micro, kilo,
> etc. or a special non-SI prefix.
>
> Pat
>
> Patrick Cassidy
> MICRA, Inc.
> 908-561-3416
> cell: 908-565-4053
> cassidy@xxxxxxxxx
>
>
>
>
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>
> (05)
--
_______________________________
Joseph B. Collins, Ph.D.
Code 5583, Adv. Info. Tech.
Naval Research Laboratory
Washington, DC 20375
(202) 404-7041
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B34, R221C
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