Re: ply What does it take to get a handle on math?

From: John Creighton (JohnCreighton__at_hotmail.com)
Date: 07/07/04 

Date: 7 Jul 2004 07:21:12 -0700

pellis@london.edu (Paul E) wrote in message news:<659defa1.0407040838.5a0b6e36@posting.google.com>...
> "Adam" <adam@bonkers.reg> wrote in message news:<KypFc.898$vBq1.438@news04.bloor.is.net.cable.rogers.com>...
> > "john" <jbooth4@mathforum.org> wrote in message
>
> And what have I learned? Apart from basic maths and significant
> (though admittedly patchy) progress in my areas of interest, most
> satisfying has been a clearer understanding of an interpretation of
> complex numbers.
 
You are thinking to hard. Give my an interpretation of the integers?
Counting is merely a mapping.

>They are normally treated in the abstract, and that's
> probably fine for mathematicians. But from the viewpoint of physics
> (and the way my mind needs something visualisable), their lack of
> apparent direct physical interpretation has, until recently, been?
> "disturbing" might be the best word.
>
> The furthest I could get on my own was to see them as something along
> the lines of the "square root of a reflection". They might be just a
> convenient trick in e.g. AC theory, but likely not in quantum
> mechanics, as without the complex character of e.g. a plane-wave
> function, it wouldn't be able to represent the uncertainty principle

It is a principle of the Fourier transform that a signal that is
narrow in time
is wide in frequency and a signal that is wide in time is narrow in
frequency.
Very similar to the Fourier transform is the expansion of a function
in terms of
sine and cosine functions. I don't think complex numbers are required
for the
uncertainty principle. I think they just simplify the math. Ponder how
many
Other bases transforms could give this trade of. Perhaps any function
of the
Form:
F(wt-x)
(note that was just a guess)

> (my reading of section 2-2 of Dicke & Wittke's well-known Intro to QM,
> Addison Wesley). So why should quantum mechanics be deeply bound up
> with numbers that appear unrelated to the measurement of physical
> quantities.
>
> Somewhere between reading that "complex numbers are not for counting",

It is interesting that there are some isomorphism's between the
integers and
the complex numbers (at least between groups and maybe between rings).
Do you
know that addition in modulo n in the integers is equivalent to
multiplication
of complex numbers with unity magnitude and separated by an angle of
360/n

> and reading about Clifford Algebras as the result of a Google enquiry
> on "versors", I came to see that in the simplest terms, perhaps the
> most physical meaning of complex numbers has to do simply with the
> description of spatial direction - somewhat in the sense that e.g.
> integers have to do with the counting of discrete objects. (Even Barry
> Mazur's recent book "Imagining Numbers" didn't seem to convey this
> geometric sense).
>
> To see this fully, one has to look into Clifford Algebras and the
> geometric product (well explained by David Hestenes' books, especially
> New Foundations for Classical Mechanics). But it's interesting that in
> spite of quaternions and spin matrices and all that, Caspar Wessel, a
> Norwegian surveyor (1745-1818) was looking for a way to describe
> direction analytically when he published a paper using the square root
> (?1) as a unit; (see the introduction to A.P. Wills' "Vector Analysis
> with an Introduction to Tensor Analysis", Dover, 1958). Yet somehow
> this simple aim got lost (abstracted) along the way, with the result
> that generations of teenagers must, like me, have puzzled over these
> entities, and mostly given up.
>
>
> Perhaps this whole post can best be summed up in some advice
> attributed to Arnold E. Ross (Dept. Math, Ohio State U.): "Think
> deeply about simple things."
>
>
>
> (I assume the rules of posting don't forbid claiming copyright, as I
> may reuse parts of this for an article) © Paul G Ellis.

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