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From: Noah Stein (noah_at_[hidden])
Date: 2005-10-13 23:43:51

> Matt Calabrese wrote:

> 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
> 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.

I personally have never seen radian used as merely a synonym for ratio.
Your reference to the wikipedia seems less than authoritarian on the matter.
Your quote is mentioned at the start of the article. In the remainder of
the article, substantially larger than the single sentence you quoted,
radians are only described in their traditional usage regarding angles.

Assuming your reference does stand on its own in the article, I still have
issue with the reference. Given the large amount of literature in regard to
the accuracy of the wikipedia, I looked up other references on the net.
Mathworld only makes mention of angles
( ). A search on ( ) returned definitions from
4 different dictionaries that mention only angular measure. One of them
makes specific reference to its definition in SI. On the US National
Institute of Standards and Technology SI units web page, radian is listed as
a derived unit (m * m^-1) of plane angle
( ). If you read a footnote, it
would appear your quote from the wikipedia is actually, since it is
unattributed, a plagiarism from the NIST web page. In a further derived
quantity table, radians are only used in the context of angles.

I can only conclude that radians are specifically related to angles. Thus
the example of a ratio of one shrub to another, although also a derived unit
of (m * m^-1), is not, by definition, a radian.

> 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.

Although I obviously disagree with the first part, I agree with the second,
with some reservations. Radian is the perfect example of a situation where
there is a clear semantic "dimension" even though there is no unit
dimension. In this case, tracking that semantic dimension would be just as
helpful as tracking regular unit dimensions. And if I'm doing something in
radians, I'd like it to be restricted to those computations that are
semantically correct. In addition, I can also have the compiler enforce
radian-degree interaction in my handling of angles.

On the other hand, sometimes a dimensionless quantity is just a ratio, no
more and no less. Sometimes I want to compare two quantities and use that
ratio for scaling another quantity that is linked to it. Going back to the
shrub, if I run a nursery and I price shrubs based on height, I might figure
out the price of any given shrub based on scaling the value of a reference
shrub. It's a little contrived, but I think you get the point.

-- Noah

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