Re: Representation of Dimensionless Units
aneilbaboo_at_gmail.com
Date: 02/04/05
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Date: 4 Feb 2005 07:53:17 -0800
Shmuel,
The more I think about it, the more I like your approach to
interpreting the Taylor expansion. In fact, if we replace the NIST
approach (radians as a derived dimensionless unit) with "angles as
first-class dimensions" (AFCD), I think we have a workable (i.e.,
computable) system for correct dimensional arithmetic for scalars based
in part on angular measurements (e.g., lumens, angular velocity,
radians, etc.).
The standard (NIST) definition rejects this, stating instead that a
measure in radians <i>represents</i> the ratio of the length of an arc
to its radius. This definition is not only impractical (for reasons
I've outlined in my previous post), but also <i>incorrect</i>. I think
the correct definition is that radians represent <i>the angle
subtended</i> by an arc <i>of a circle</i>. The unit 1 radian is
defined to be that angle which is found when the length of the arc is
the same as the length of the radius. Under this interpretation the
arc and radius appear in the definition of the conditions of
measurement (1 radian is defined when arc=radius) of a new dimension,
not in the measurement itself. This is similar to the definition of
other base units in terms of conditions of measurement.
For derived dimensions, it is unnecessary to define such conditions of
measurement, since no new definitions are required (other than the
meanings of multiplication and division). When new semantic content
appears (conditions of measurement), I think we must define a new
dimension, so that this content is carried through our calculations.
Here, the new content is the notion of an angle subtended, which is
distinct from a simple ratio which does not carry the new content
implicit in the special conditions of measurement required to determine
an angle.
There is a way that radians are unlike the other standard base units
(SBUs), however. The conditions of
measurement of the SBUs are defined with respect to some
<i>physical</i> property of the world, for example some number
determined from an experiment using a "real" object. But for radians,
the conditions of measurement are defined with respect to a
mathematical object (the circle). Is it a stretch to call this a
mathematical property of the world? If so, perhaps the choice about
whether to include radians in the SBUs may hinge on one's philosophy
with respect to these issues.
This strikes me as a weak reason to reject AFCD. On grounds of
practicality, I think the NIST definition loses out to the AFCD
approach.
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