Re: [Phys-l] Types of scalars

[Jack Uretsky <jlu@hep.anl.gov>]

I'm afriad that your distinctions, shich are unnecessary for 
compu8tational purposes will be confusing to your students, like ewuiring 
them to walk three times withershins around their chair before doing any 
calculation ("Now which way did he say was 'withershins'?"). 
Mathemeticians dispose of your distinctions quite simply by specifying 
sets from which the real numbers are taken (you seem to exclude complex 
numbers from your set of scalars).
	Your caegories 1 and 2 appear to correspond respectively to the 
following sets: (1)the positive real axis (the values of all points lying 
on the positive real axis) and, correspondingly (2) the real line.  Your 
third category does not seem to present a mathematical distinctionm but, 
rather a physical one; that is, for every subset of positive numbers, you 
are free to choose any number to be the smallest number in the subset. 
This freedom is the freedom the John D. has called "gauge invariance".
	You are playing with a branch of mathematics called "Analysis", 
and I suggest that you start by learning about "Zorn's Lemma", to see how 
the mathematicians discuss such concepts.
					Regards,
						Jack

"Trust me.  I have a lot of experience at this."
		General Custer's unremembered message to his men,
		just before leading them into the Little Big Horn Valley




On Sun, 2 Jan 2011, Scott Hill wrote:

> I've been reflecting on fundamental mathematics that I gloss over in
> my introductory physics classes, and it occurred to me that there are
> actually three types of scalars that we use in introductory physics:
> 1) Never-negative scalars, like mass or length or distance.
> 2) Scalars which can be negative, and the zero is significant: like
> charge or flux or current, or changes in temperature, or vector
> components.

> 3) Scalars which can be negative, and the zero is not significant:
> like temperature (in Celsius) and potential energy.
>
> I am thinking it would be convenient to make this distinction to my
> introductory students, because they behave in different ways (for
> example, Category-3 scalars are never multiplied).  Has anyone seen
> these distinctions made before?  I think all physicists are aware of
> these categories, but I don't know that anyone ever explained them to
> me.
>
> More importantly, are there NAMES for these three categories?
>
> They can be broken up even further if one has a mind:
> 1a) Quantities like mass or length or duration, which can be added and
> subtracted directly.
> 1b) Vector magnitudes like speed, which probably can't be added
> without going back to the original vector.
> 2a) Fluxes like current or work: the sign indicates direction of flow.
> 2b) Surpluses, like charge (or bank accounts): the sign indicates
> presence or absence.
> 2c) Changes in quantities, like ∆T: the sign indicates increase or decrease.
> 3) I can't think of any subcategories of this one.
>
> Too many categories may be more confusing than helpful, however.
>
>
> /
> :@-) Scott
> \
> _______________________________________________
> Forum for Physics Educators
> Phys-l@carnot.physics.buffalo.edu
> https://carnot.physics.buffalo.edu/mailman/listinfo/phys-l