Re: Radians, dimensions, & explanations

[Ludwik Kowalski <kowalskiL@MAIL.MONTCLAIR.EDU>]

Doug Craigen wrote:

> ... A radian is usually defined as measuring the angle subtended by
> an arc as the ratio of the arc length to the radius of the arc.  The
> dimensions are distance divided by distance, so the dimensions cancel
> and you could say it is dimensionless.

As I recall, the radian was defined (in my high school) in terms of
degrees ( PI rad=180 deg) and degrees were treated as units (labeled
divisions along the protractor's scale, like mm but along the arc).
That was good enough to draw triangles, learn geometry, etc.. Only
later did I learn about the "dimensionlessness" of angles. Was I
deprived of something very important? I suspect that the emphasis
on dimentionlessness could, in some cases, create the unnecessary
learning barrier.

Every physical quantity can be expressed quantitatively in the
dimentionless form, if we want. Take temperature, for example.
Instead of saying 25 degrees C I can say 0.25 (one quarter of the
temperature difference between boiling and freezing). Or take the
difference of potentials. Instead of saying 48 volts I can say "four
times as much as in my fully loaded car battery". Or a charge of
seven electrons. Or an energy of 10, meaning ten times as much
as that of one galon of water rased to an elevation of one furlong.
Why not? Why yes?
Ludwik Kowalski