Rijgersberg, Hajo wrote:
> Please allow me to make a comment on dimensions and measures in OUM (from
> the forthcoming concept paper):
>
>
> Units of measure and quantities have a dimension. Dimensions are abstract
> properties of units and quantities neglecting their vectorial or
> tensorial character and all numerical factors including their sign.
> Dimensions can be expressed as the products of powers of base quantities
> of a system of units. (01)
Being careful, this definition of "dimension" specifies what is ignored
in the representation of a "quantity (magnitude)". It doesn't say what
the abstraction _is_. (02)
One possible interpretation of "dimension" is 'quantity kind' -- the
nature of the quantity, the nature of the property being measured, such
as linear spatial displacement (length), or force distributed over a
spatial area (pressure). This is the way Pat Hayes used the word. (03)
The other interpretation is a relationship to 'base quantities'
(quantity kinds) from a 'system of units', which relates the nature of
the quantity to the natures other quantities. (The 'quantity kind' is
_not_ a relationship to base quantities; it acquires that relationship
when a system of units is chosen.) (04)
As Hajo says, this latter interpretation can be "represented by" a
formula that is a product of powers of base units, where each unit
represents a base quantity. But, because that formula is in "lowest
terms", a given units formula does not uniquely identify a dimension.
For example, torque and energy are different dimensions (in either
sense), but they have the same 'dimensional formula' in base units. And
obviously, each choice of a system of units will produce a different
formula. (05)
(The problem is that the nature of 'angle', for example, can be conveyed
by arc length / radius, but the two elements of the relationship are
both lengths, and in the units formula, the result is length to the
power 0. length/length and length^0 are different relationships.
Powers of units alone do not convey the relationships to the base
quantities.) (06)
> The above is mainly based on Cohen, E.R. and Giacomo, P. (1987):
> Symbols, units, nomenclature and fundamental constants in physics.
> 1987 revision. Document I.U.P.A.P.-25 (SUNAMCO 87-1). International
> Union of Pure and Applied Physics, SUNAMCO Commission. (07)
-Ed (08)
--
Edward J. Barkmeyer Email: edbark@xxxxxxxx
National Institute of Standards & Technology
Manufacturing Systems Integration Division
100 Bureau Drive, Stop 8263 Tel: +1 301-975-3528
Gaithersburg, MD 20899-8263 FAX: +1 301-975-4694 (09)
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