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RE: supplementary S.I. units




On Fri, 25 Sep 1998, John Mallinckrodt wrote:

2. Terms which are added or subtracted must have the same dimensions and
the same units.

I think I know what you mean by this, but I hesitate to say that one
cannot add 1 inch to 1 foot.

I never do, or at least never write such things in an equation, though
I've seen some older books, and even recent elementary books which do.
Like: 1 inch + 1 foot = 13 inches = 0.03 yard (approx).

3. Quantities on either side of the equal sign must have the same
dimensions and the same units.

Does this mean that it is not proper to write "1 inch = 2.54 centimeters"
or "1 m = 100 cm"?

I shouldn't toss things onto the keyboard in haste, just before a class.
That whole set of "rules" applies only to *physical* equations, those
which have physical content and represent physical relations or laws. I
normally preface this list with a discussion pointing out that equations
we use fall into several classes with different purposes. Conversion
equations are a special case, and are not in the same class as equations
with physics content. A third class is made up of equations with strictly
mathematical content which have no necessary relation to physics (and no
units and dimensions, either). However you present it, there's some value
in realizing that not all equations we write are in the same category. My
wife, the mathematician, reminds me that even in pure math there are
several meanings for the "=" sign (four, I think), and it's also important
to realize that fact: that not all "equals" are "equal".

To students, I'll usually say, "To write 12 = 1 is surely absurd.
Equations like 12 = 8 + 4 are in this category, relations between pure
numbers which have only mathematical content and don't necessarily tell us
anything about the physical world. We must re-interpret the numbers as
something one can measure before this can be applied to physics. However,
the equation 12 inches = 1 foot has meaning, for it is a conversion
relation between different measures of length, and the units are essential
to its meaning. But it still says nothing about the real world or about
physics. To write F = ma says something about physics, but only if one
has agreed upon a consistent, coherent set of units to which it applies,
and, of course, defined F, m, and a precisely in terms of things one can
measure. These are physical equations. *Then*, I'll discuss those "rules"
for units and dimensions in physical equations. The rules don't apply to
conversion equations, as you have noted.

-- Donald

......................................................................
Dr. Donald E. Simanek Office: 717-893-2079
Professor of Physics FAX: 717-893-2048
Lock Haven University of Pennsylvania, Lock Haven, PA. 17745
dsimanek@eagle.lhup.edu http://www.lhup.edu/~dsimanek
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