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From: Matt Calabrese (rivorus_at_[hidden])
Date: 2005-10-13 23:23:38

Also, the fact that the result is in radians makes sense with my description
of radians, since sine describes the ratio of the the opposite side of a
right triangle to the hypoteneuse. This is an untransformed ratio of two
quantities having the same dimension, which is, I believe uncoincidentally,
exactly how I described radians. They are merely untransformed ratios which
we only most commonly explicitly use when refering to angles. I believe the
abstract concept of "radians" is always there for ratios of quantities
having the same classification, it's just that we only write them down on
paper when working with angles since it's used for disambiguation with
common transformations of ratios such as degrees.

I know it's strange, but comparing radians to decibels I think is another
good way to attempt to explain the standpoint. Decibels and bels and nepers
are all dimensionless types which represent the ratio of two values in
logarithmic scales. My system can represent them perfectly fine and in a way
that I believe makes logical sense. They are related to radians and degrees
since all of them are just ways of representing ratios, either transformed
or untransformed, just like other quantities having an empty classification,
and a conversion exists between them. Since they are just transformations of
ratios, expressions such as my_decibel_quantity = length / length should
work fine, since the result of the right-hand operation is just a ratio, and
decibels are merely transformations of ratios -- exactly like the
relationship of two units of another classification such as time, energy,

Raw ratios are convertible to decibels since they both have units of empty
classification and a defined relationship exists between them. I believe
that stating that ratios exist in the same classification as quantities with
an empty classification, only with no units attached is a mistake. My stance
is that the unit type is not "unitless", but that the ratio has
untransformed units in that derived classification. This unit type happens
to be analogous to radians only perhaps with a broader definition than most
people use. I wonder if the name is all that we are arguing about, so call
it what you want -- "untransformed ratio" or something else, as the name is
unimportant, but I believe the abstraction definately exists which is why we
have different unit types which describe ratios, such as decibels. Using raw
values without units associated loses that abstraction. That natural form
isn't unitless, it's just that we don't normally attach a name to it.
Following with geometry, radians are untransformed ratios (note here I'm
purposely stating that radians are untransformed ratios not that
untransformed ratios are radians, since I apparently have hit a nerve with
some of you). If you don't want to call all ratios radians, a conversion
still definately exists between them -- that conversion happens to be an
identity conversion, where you leave the value the same, which is why the
length of an arc divided by the radius can be converted to radians without
changing the value. If you want, then look at the relationship similar to
that of kelvin vectors and celsius vectors -- conversion between them is to
simply leave the values the same and is implicit. Both concepts have units
and can be looked at logically differently, but they are represented exactly
the same, and they can be converted between.

-Matt Calabrese

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