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From: Matt Calabrese (rivorus_at_[hidden])
Date: 2005-10-13 21:13:38

On 10/13/05, Scott Schurr <scott_schurr_at_[hidden]> wrote:
> I'm no where close to the expert that you guys are, so I'll quickly
> toss in my two cents and get out of your way.
> On 10/13/05 Matt Calabrese <rivorus_at_[hidden]> wrote:
> >
> > On 10/13/05, Deane Yang <deane_yang_at_[hidden]> wrote:
> > >
> > > Mathematically, I think Andy is entirely correct in what he says
> above.
> > > Once you take the ratio of two measurements of the same dimension (or
> > > quantity), you end up with a pure number that really can't be
> > > distinguished from other ratios or pure numbers.
> >
> >
> > I tend to disagree here. A ratio of two lengths, for example, is in
> > fact a value in radians.
> Um, careful there. Suppose I'm measuring the height of a shrub at
> various points in time. If I take one of those measurements and
> divide it by a measurement taken at a different time, then I get a
> ratio that expresses the shrub's change in height in a unitless
> way - I get the same value whether I made my measurements in meters,
> inches, or furlongs. This is an example of a common case where
> dividing two lengths doesn't give radians - it's just a ratio.

This is where I disagree. Radians ARE just ratios. More specifically,
radians are ratios, degrees are ratios after a transformation, as are
decibels and other similar units. Any quantity of one classification divided
by a quantity of the same classification results in an untransformed ratio,
which is what radians are. I gave the example with arc length and radius
length since that's the easiest case for people to see the radian result,
however the same holds true for energy divided by energy, or time divided by
time. Results are radians. The arc example wasn't a specific case, just one
that people would recognize.

A lot of you here are under the assumption that radians are just "angles,"
which is not the case. They are just ways of representing a relationship
between two quantities of the same classification directly, just like
decibels and several other quantities of empty classification. It just so
happens that we usually only explicitly attach the name "radians" to angles,
since in other areas where ratios are used, it's understood that we aren't
working under a transformation. Any length divided by any other length
results in a value in radians regardless of whether you decide to interpret
that result as an angle. A persons height divided by another persons height
gives you a value in radians. A velocity divided by another velocity yields
a value in radians. A radian is merely an untransformed ratio.

I hate to use wikipedia as a source, but after searching, their description
agrees with mine . Note their definition
"The radian is useful to distinguish between quantities of different nature
but the same dimension." Note that radians aren't always angles. They are
simply just untransformed ratios relating two quantities of the same
dimension, just as I also described. They serve the same purpose as decibels
do -- they describe a relationship between two quantities of the same
dimension. Most people only think of radians as angle measures and not
simply ratios because the only time they would usually need to explicitly
state it is with angles, since we have other common ways of representing
angles through transformations of simple ratios.

Any quantity divided by a quantity of the same dimension is an untransformed
ratio, which is precisely what radians are. Storing it as a raw value
without units associated with it is a mistake and ambiguous.

-Matt Calabrese

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