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Re: [Phys-L] Why **UNITS** matter!



On 08/10/2016 12:48 PM, Richard Heckathorn wrote:

If the ending unit(s) are appropriate with the unknown desired, then
the probability is that you have done things correctly.

I would have approached that differently.

a) If the dimensions are wrong, the answer is guaranteed to be wrong.

What's even better is that this applies at every step along the
way. If at any step, the dimensions are wrong, you know that
step is wrong. When doing long complex calculations, being able
to localize an error is exceedingly valuable.

b) However, the converse does not hold. That is to say, if the
dimensions are right, that does /not/ prove the answer is right.
It's not proof; it's not even evidence.

Counterexamples abound:
-- For example, if you think the area of a circle is π times
diameter squared, you're going to get the right dimensions,
but the answer is going to be wrong by a substantial margin.
-- Given a list of measurements, the min, max, mean, and standard
deviation all have the same dimensions, but they may differ in
value by many orders of magnitude. If you confuse them, the
dimensions will not save you.
-- The energy, the Lagrangian, the Gibbs free enthalpy, and the
torque all have the same dimensions, but they are not the same
thing. Not even close. As the saying goes, there is more to
physics than dimensional analysis.
-- et cetera.

c) Units are not the same as dimensions.

In particular, in most cases you cannot look at an answer and
say that the units are "wrong" (unless the dimensions are also
wrong). That's because units can be /different/ without being
wrong. For example, 0.1 cm and 0.001 m have different units,
but they mean exactly the same thing.

If the right answer is 0.1 cm and some student writes 1 cm
instead, you have no idea (without additional information)
whether that is due to a mistake in the units or any of a
hundred other possible mistakes.

===================

I might add that there is an /algebraic structure/ to units.
That is to say, you can add/subtract/multiply/divide units as
if they were algebraic variables ... even though technically
they are not variables, since they don't vary. They live in
an abstract space all their own.

If students are having trouble with units, especially early in
the course, it is likely to be symptomatic of a deeper problem,
namely a poor grasp of basic algebra concepts.