Re: when to write radians

[Joe Heafner <heafnerj@VNET.NET>]

On Tuesday, Jul 1, 2003, at 10:12 US/Eastern, Robert Cohen wrote:

> Could someone define "dimensionless"?
>
> I had thought that "dimensionless" meant having no dimension and thus
> all ratios of similar dimensions were "dimensionless" (though not
> unitless).  But now I realize I have been using the term differently
> than others.  Is a mixing ratio of 2 g/kg dimensionless?  Is a
> conversion ratio like 3.2 ft/m dimensionless?

There's a difference between *dimension* and *unit*. We teach our
students that the basic *dimensions* are length, time, and mass.
Sometimes we can argue about whether charge belongs here too, but
that's more of an advanced topic. Therefore, I argue that 2 g/kg is
indeed dimensionless because both quantities have dimension of mass. A
ratio like 3.2 ft/m is also dimensionless because both quantities have
dimension of length. I define length as *that which is measured by
comparison to a straight stick*. Units are completely arbitrary human
inventions. Dimensions are dictated to us by Nature (and THAT to me
seems a very deep issue, but I'm nowhere near prepared to explain just
HOW they're dictated to us). Just think of all the units we have for
*that which is measured by comparison to a straight stick*: inches,
feet, yards, meters, kilometers, light years, parsecs, etc. Think of
all the units we have for *that which is measured by comparison with a
known quantity*: gram, kilogram, ounce, dram, etc. A physically valid
mathematical expression (e.g. K = 0.5 m v^2 or v = w r) relates the
dimensions of the various quantities rather than their units.

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. Otherwise,
it's incorrect to assign units of radians. I argue, therefore, that it
would be incorrect to say that 2 kg/1 kg = 2 rad. Likewise, when
comparing two lengths that are not associated with an angle, such as
the lengths of two sticks, it's inappropriate to assign radians as the
unit. So it's incorrect to write 3 m/6 m = 0.5 rad.

Finally, it seems to me that all quantities with the same *dimension*
are on equal footing. There's no distinction between the *length* of
the arc of a circle and the *length* of your computer keyboard. They
can both be measured with a straight stick (provided you "unroll" the
arc first, without stretching it, of course). All things that can be
measured with a stick are equal.

Cheers,
Joe Heafner

-----
If Mr. Sterling is anything close to realistic, it's no wonder our
government is the atrocity that it has become.