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Re: dimensionless units



At 01:20 PM 12/1/00 -0500, David Bowman wrote:
Rather than append the inverse angle dimension directly
to the [epsilon]_i_j_k symbol, suppose we include it as a separate
dimensioned scalar factor that multiplies it.

Fine. At that point David's proposal becomes isomorphic AFAICT to what
other people (including me) have been proposing.

> S = f t
>should be replaced by
> S = f t cycle^-1

This case doesn't seem to be very related to the previous one's related to
angular measure. In this case the linkage results in a 1 - 1 relationship
between cycles of the crank and bongs of the bell, so S (bongs of bell) =
S (cycles of crank) = f*t. The answer in this case *ought* to have the
cycles remain uncancelled so they can be reinterpreted as bell bongs.

If you want something to be interpreted as bongs, that's fine. But leaving
the cycles uncancelled is a step in the wrong direction, because it leaves
S = f t
ambiguous (worse than ambiguous, actively misleading) as to whether the
frequency f is in Hertz or radians per second. So why don't we fix it up
this way instead:
S = f t (bongs / cycle)

That is, if we are going to make radians (and therefore cycles)
dimensionful, we might as well make bongs dimensionful. (In for a dime, in
for a dollar!)

David speaks of a "nonserious" proposal, but I'm semi-serious. Of course
this type of dimensional accounting will never become mandatory, but it is
a useful option.

================================

The fundamental physical idea here is that some formulas are (and some are
not) homogeneous, i.e. invariant when we change our choice of units.

Example: There is a centrifuge being rotated by a physical clockworks
mechanism, rotating at a definite unchanging rate.

Question 1a) What is the rotation rate in radians per second?
Question 1b) What is the rotation rate in degrees per second?

Question 2a) What is the centrifugal field (in Gees), assuming the rotation
is measured in radians per second?
Question 2b) What is the centrifugal field (in Gees), assuming the rotation
is measured in degrees per second?

The point is that question 1 is not homogenous (it manifestly depends on
the choice of angular units) while question 2 is homogeneous.

This way of looking at it (homogeneity / invariance) includes all of
dimensional analysis, plus a little bit extra.