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It's also not the case that torque really has the same units as energy.
There's the little matter of a cross product. I don't see how units in a
cross product end up the same as from a dot product or scalar
multiplication.
I still have not heard how the dot product of a force and a displacement
can have the _same_ dimensions as the cross product of the two. We
differentiate between the multiplicative product and the quotient, why not
the dot and cross products?
Dewey, you've quoted yourself. I thought I put that to
rest (in between, so it didn't get quoted). If that
bothers you (and it shouldn't) then think about the
magnitude of torque instead. That's a perfectly valid
physical quantity, a respectable scalar.
By the way, I have done some work with a new/old kind of
algebra which we call geometric algebra (and which was
originally called Clifford algebra) in which we perform
operations you might consider outrageous. One operation
allowed in this formalism is the addition of scalars and
vectors, but the operation is only physically meaningful
if the vector and scalar have the same dimensions. One
such quantity is the sum of displacement and time interval,
the quantity which represents the separation between two
events in spacetime. (The time must be scaled by a factor
of c for this quantity to be physically meaningful.) The
method has a lot to recommend it, I think.
The bottom line is this: energy and torque have the same
dimensions. They can be given the same units, too, but
there is probably a lot to be said for distinguishing
them by using cosmetically different units.