Randy Yates wrote:
Jerry Avins <jya at (no spam) ieee.org> writes:
[...]
I wouldn't argue even about promoting "notational convenience" to
"notational necessity". I argue that the notation and the phenomenon
described by it are not the same thing. As Korzybski wrote, "The map
is not the territory it represents". Granted, he went on, "but if
correct, it has a similar structure to the territory, which accounts
for its usefulness". Still, I think it is important to distinguish
between "similar" and "same".

Jerry,

This is not a matter of mistaking the map for reality, but rather of
arguing whether or not the map accurately portrays reality.

Since neither one of us knows with absolute certainty and complete
characterization what the "phenomenom" (i.e., some aspect of reality)
is, neither one of us can argue that it "is" or "is not" correctly
portrayed by a certain "map."

That goes only one way. It is certainly possible in some instances
that a particular map misrepresents the area it purports to describe.

However, the longer, more extensively, and more repeatedly a "map"
demonstrates itself by experience and experiment to accurately
represent, and even to predict, that phenomenom - that aspect of
reality - the less likely it is that the map is wrong.

I believe that is the case with complex numbers. Arguing against their
existence is, in my opinion, similar to arguing against the statement
that you have four grandchildren (e.g.) on the basis that the concept of
"4" is a map and not necessarily representative of reality.

If course complex numbers exist .. on paper.

That doesn't mean that the physical phenomena that we use them to
describe are themselves complex

or can be understood only as complex numbers.

If course complex numbers exist .. on paper.

You know that's not what I meant. At the point we fail to communicate,
or refuse to hear one another, we kill the discussion.

That doesn't mean that the physical phenomena that we use them to
describe are themselves complex

My point is, it doesn't mean they aren't, either. And interpreting them
as such gives us so much more understanding than any other method that
it's utter foolishness to abandon that understanding, all to hold the
door open for some mythical alternate reality that no one has yet seen.

or can be understood only as complex numbers.

Of course this is true.

Randy Yates wrote:
[...]
My point is, it doesn't mean they aren't, either. And interpreting them
as such gives us so much more understanding than any other method that
it's utter foolishness to abandon that understanding, all to hold the
door open for some mythical alternate reality that no one has yet seen.

Interpreting them as complex is a powerful analysis. Rejecting other
interpretations narrows our view. We don't need to choose one. We can
profit from the broader view.

Jerry Avins <jya at (no spam) ieee.org> writes:

Randy Yates wrote:
[...]
My point is, it doesn't mean they aren't, either. And interpreting them
as such gives us so much more understanding than any other method that
it's utter foolishness to abandon that understanding, all to hold the
door open for some mythical alternate reality that no one has yet seen.
Interpreting them as complex is a powerful analysis. Rejecting other
interpretations narrows our view. We don't need to choose one. We can
profit from the broader view.

Name one (profit) from doing so.

OK, back to the ~original question,

why is the LT defined for postitive values of "t" ?
At one point I thought this had something to do w/ turning on the
signal at time "0"

Jerry Avins wrote:

   ...

Yes. I used to program my IMSAI Nova from the front panel toggles.

IMSAI and Nova 1200

On Nov 1, 12:22 am, Jerry Avins <j... at (no spam) ieee.org> wrote:
Randy Yates wrote:
Jerry Avins <j... at (no spam) ieee.org> writes:
Randy Yates wrote:
[...]
snip
I suggest that one has a much better understanding of what a Fourier
transform is having done this by hand with sines and cosines than by
turning the crank on pairs of complex exponentials. I don't suggest that
it is more efficient.

Jerry
--
Engineering is the art of making what you want from things you can get.
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

OK, back to the ~original question, why is the LT defined for
postitive values of "t" ?
At one point I thought this had something to do w/ turning on the
signal at time "0"

"Jerry Avins" <jya at (no spam) ieee.org> wrote in message
news:X1GGm.448$X77.147 at (no spam) newsfe24.iad...
glen herrmannsfeldt wrote:
Jerry Avins <jya at (no spam) ieee.org> wrote:
(snip, someone wrote)

This is because the magnitude of the imaginary component of a complex
signal is a real signal.
Of course. All things measurable are real. "Complex" numbers are merely
a clever and useful bookkeeping scheme for manipulating related pairs of
real quantities.
Or pairs of real numbers are a convenient way of measuring complex
quantities. Impedance and index of refraction are both complex, and for
similar
reasons. We can separately measure resistance and reactance, or
the real and imaginary parts of the index of refraction. (The imaginary
part comes from absorption.) Both are due to
the interaction of electrons with atoms, and with each other.

Often the available materials are fairly close to ideal, such
that we can separate the quantities. Resistors do have inductance,
inductors (except superconductors) do have resistance.
Complex numbers are a clever and useful way to represent those quantities
in cases where phase shift has meaning.

Compact notations expand out ability to comprehend (same root as in
prehensile) complicated things. Vector analysis, quaternions, complex
numbers, matrix algebra .. without them, we'd be hard pressed to express,
let alone understand, some phenomena as concepts. Nevertheless, being
built up from simpler stuff (Shall we go back to Peano's axioms?) thay
constitute the HLLs of math. Ultimately, all running code executes
assembly language.

Jerry
--
Engineering is the art of making what you want from things you can get.

¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
Well, I'd say machine code rather than assembly, though there's usually a
1:1 mapping.

--Phil

Yes. I used to program my IMSAI Nova from the front panel toggles.

Jerry
--

Jerry Avins wrote:

...

Yes. I used to program my IMSAI Nova from the front panel toggles.

IMSAI and Nova 1200

Jerry
--
Engineering is the art of making what you want from things you can get.
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
A friend of mine had an IMSAI.