Jim, The case of angles and trig functions are good for bringing out the fuzziness in most people's basic conception of physical dimension. Angles seem dimensionless if you think of them as a quotient of a length with a length (as in the definition of a radian), but they seem dimensioned if you think of how it can be useful to change the fundamental unit (e.g., degrees, radians, grads, cycles...) or if you consider that you can do a physical experiment to add two angles and observe their sum. Because the power series for Cosine only uses even powers, it has been proposed that to avoid the kind of unmotivated conversion factor (r) you put in your Sine series, the dimensions of angle could have the property of not being dimensionless, yet being a square root of dimensionless, i.e., like a square root of identity other than identity. (You would still need one factor in the Sine series, because it is the odd powers.) Can the algebra of a physical dimension be a finite cyclic group in this way? Might square roots not be dimensionally unique? Is it meaningful to ask if someday an experiment might show that mass^10=dimensionless? Is (1 meter)^Sqrt[2] defined? Is (1 meter)^Sqrt[-1] defined? I find that even people who work with dimensions all the time can have surprisingly different feelings about such questions and what the structure of the algebra of dimensions might be. As with angle and dimensionlessness, torque and energy appear to have the have the same exponents in their dimensional terms (length^2 * mass^1 * time^-2 * charge^0) yet different physical roles in which they are never added, suggesting standard algebras of dimensions do not quite capture everything of interest even for defining something as basic as when addition of two quantities is defined. These sorts of issues pop up in my book, but are not the central focus. As far as I know, no one has a fully satisfactory framework even for scalars. There is another level of richness when considering the dimensions of vectors, matrices, etc. But to the original point: including dimensions within the linear algebra brings out very clearly the difference between a vector and a co-vector. George http://georgehart.com On 1/30/2020 6:14 AM, James Propp wrote:
Trig provides another arena in which there are different types of scalars. Angles and ratios-of-lengths are scalars, but in applications of trig it makes no sense to add an angle to the sine of another angle; angles and ratios are different types of numbers.
Has anyone worked out a framework for this perspective? (I don’t think George treats trig in his book, though it’s possible that he did and I’m forgetting it.)
Such a framework would rewrite the standard power series formula for sin x as sin x = r x / 1! - r^3 x^3 / 3! + ... where r is a type-conversion coefficient.
What would the point of such a framework be (besides an idiosyncratic form of ideological purity)? Well, think of all those formulas in which some integral is shown to equal some rational coefficient times pi to the power of some rational exponent. Wouldn't it be nice to have a “dimensional” way to predict the exponent?
Jim Propp
On Wed, Jan 29, 2020 at 5:32 PM Brent Meeker via math-fun < math-fun@mailman.xmission.com> wrote:
There's a very nice book by George Hart, "Multidimensional Analysis" which brings some order and rigor to the often sloppy use of dimensioned quantities by engineers. He develops a linear algebra of dimensioned matrices.
Brent
On 1/29/2020 8:55 AM, Brad Klee wrote:
For types I say "dimensions" as in "Dimensional Analysis". A major teaching frustration is that most students can barely handle units, much less dimensions. Then they ask: "What is Electric Flux?", and can't understand the answer that it is a measurable quantity of dimension [F][L]^2 / [Q].
I would like to write out some of my thoughts about how dimensional analysis is a lot like projective geometry, but for now, do not have the time. ( And did Max Planck think about this?)
As for co-vectors, vectors, linear functionals etc., If the elements are complex numbers--as they are in quantum-- there will be type errors on mismatched addition, because norming requires complex conjugation.
--Brad
On Wed, Jan 29, 2020 at 10:11 AM Allan Wechsler <acwacw@gmail.com> wrote:
The flavor of the distinction is very similar to unit analysis from physics, which tags some scalars with "types" which behave coherently under arithmetic. In principle, you are never allowed to add two scalars of differing types.
Presumably, whenever you see a dot product, there is a vector on the right and a covector on the left, and it is a type error to add a vector and a covector. Do these types really behave consistently, or are there exceptions to the strict classification?
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