I generally agree with Donald's polemic, as I find the "fixes" for radians
to be more confusing (to me) than the problem they are trying to solve.
I rather like the list you provided (repeated below), and will no doubt
adapt it for use in my introductory course, as I have been sloppy in
distinguishing units from dimensions (at least I have made the attempt). I
have a question about item #4 below:
by "powers", do you mean exponents? Since I assume you don't mean to be
saying that quantities like, e.g., (velocity)^2 are dimensionless and
unitless, despite their being a power.
Joel
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When I taught the introductory course, I gave students a checklist of info
about dimensions and units. All of it is obvious, but surprisingly, many
students don't notice these things.
1. Dimensions combine by the ordinary rules of algebra. Units do also.
2. Terms which are added or subtracted must have the same dimensions and
the same units.
3. Quantities on either side of the equal sign must have the same
dimensions and the same units.
4. Powers are dimensionless and unitless. However, if the power is a
mathematical expression, quantities within that expression may have
dimensions and units.
5. dy/dx and partial(y)/partial(x) have the dimensions and the units of
y/x (look at the formula for the definition of the derivative).
6. Integral (y dx) has the simensions and the units of yx.
7. Arguments of sin, cos, tan, log, etc. must be dimensionless, but may
have units.
8. The values of sin, cos, tan, log, etc. are dimensionless and unitless.
9. The mathematical constants pi and e are dimensionless and unitless.
I also note that to write 1 kilogram = 2.2 pounds is improper, because the
equation is inhomogenous and is not coherent with respect to units and
dimensions. We should avoid equating apples and oranges unless we are only
interested in "the number of pieces of fruit". We should say, "A one
kiolgram *mass* at the earth's surface *weighs* 2.2 pounds." Even that
needs more precision of language.