Re: when to write radians

[Joe Heafner <heafnerj@VNET.NET>]

On Tuesday, Jul 1, 2003, at 10:55 US/Eastern, Robert Cohen wrote:

> On Tuesday, July 01, 2003 10:38 AM, Joe Heafner wrote:
>
> > I used to think that all dimensionless quantities have
> > radians for units, but I don't think that's quite correct.
> > Radians are only needed if the dimensionless quantity is the
> > measure of an angle.
>
> Stating that one uses radians for angles is overlooking the meaning of
> radian.  It isn't simply a dimensionless unit for angle.  Rather, it is
> a particular one, i.e., length of arc per length of radius when both
> are
> measured with the same units.

I'll have to think about this.

> Wouldn't "degree" also be a dimensionless quantity?  Just because it is

Yep.

> dimensionless and refers to an angle does not mean it is equivalent to
> a
> radian.  Suppose we define a smidgen as 1/360 of a meter.  Then a
> degree
> can be defined as the ratio (smidgen of arc)/(m of radius).  Wouldn't
> this be a dimensionless quantity?

Yep. Now you're getting into why radians, rather than degrees, are
"Nature's preferred unit of angular measure". It sounds as thought I've
contradicted myself doesn't it, with all my talk of how units are
arbitrary? ("Do I contradict myself? Very well then I contradict
myself, ..." -- Walt Whitman) This arbitrariness allows us to use units
of convenience when we want to. With radians, there are no extra
numerical factors (e.g. pi/180) to carry around as there are with
degrees or smidgens or revs.

> P.S. Wouldn't the unit conversion "180 degree per radian" also be a
> dimensionless quantity?

Yep.

Cheers,
Joe Heafner

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