Re: radian, UNITS, & explanations

["John D. \"Chip\" Sample" <sample@IDCOMM.COM>]

At 10:18 AM on 12/10/98, Mike Wilson wrote:

> <From Mike mwilson@colosys.net>
>
> Hi List,
>
> Thanks for the many replies.  I am rather new to this list and may
> have hit a nerve with no such intentions.  Thanks to all for the
> calm and thoughtful replies.
>
> After reading several replies I've reached the following conclusions:
>
>      1. I would prefer to say radians have no units.
>
>      2. It does seem radian  refers to the size of an angle, so it
>          does have some dimension.
>
>  If the list disagrees with these statements please continue to provide
> information.
>
> Most useful have been those providing exlanations which seem to
> help to their students understand why we can drop the word radian in
> calculating angular quantities.
>
>
> Best Wishes,
>
> Mike Wilson
> mwilson@colosys.net
> Math/Science
> West Grand HS
> Kremmling, Colorado
I would disagree with both, and since I'm closest you should agree with
me.;)  Radian is the *natural* unit of a dimensionless quantity.  The ratio
of arc length to radius, if these lengths are expressed in the same units
is a constant and equal to the angle expressed in radians.  There is no
measure of mass, length or time (or other) involved.  But you are free to
express this angle in inches of arc length per light-year of radius.

A conversion factor is similar.  The number of inches in a meter has units
(in/m) but no dimension.

I like to think of the radian measure as a pure ratio for our flat geometry
and when one chooses to use degrees or gradians or other *units* for
angular measure that you are applying a dimensionless conversion factor to
either arc length or radius (or maybe both).  One degree = 1 darc/meter
where 1 darc = pi/180 meters, a new length measure.  The ratio stays
dimensionless, but the units change.  (Kind of like switching from dollars
per pound to pounds per kilo.  Dimension doesn't change, economics doesn't
change, but the units and therefore the number changes.)

And then there are *natural* units of "dimensionful" quantities.  That's
when we start setting c=1, h=1 etc.

Chip