Negative Frequencies

Randy Yates wrote:
> 
  ...
> 
> I get the impression that Jerry (and others?) think that there is nothing
> really gained in using the complex over the reals. I vehemently disagree. Jerry's
> argument has been, over and over, that whatever you can do with the complex,
> you can do with the reals - it just may take a few more operations. It's
> precisely those "few more operations" that make the complex (along with
> the operations of addition and multiplication) a significantly different
> mathematical beast.

  ...

Randy,

Your impression is unfounded. That whatever you can do with the complex,
you can do with the reals refutes the argument that that it must be
fundamental because it's necessary. (Anyhow, I thought this discussion
was primarily about negative frequencies, even though I did drag complex
frequencies into it at one point.) Calculating with only real numbers is
almost always harder and more error prone that using complex numbers,
and not merely by a few more operations. This discussion (at least my
part in it) has been about what is necessary and what is "real".

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

Jerry Avins wrote:
> 
> Randy Yates wrote:
> >
>   ...
> >
> > I get the impression that Jerry (and others?) think that there is nothing
> > really gained in using the complex over the reals. I vehemently disagree. Jerry's
> > argument has been, over and over, that whatever you can do with the complex,
> > you can do with the reals - it just may take a few more operations. It's
> > precisely those "few more operations" that make the complex (along with
> > the operations of addition and multiplication) a significantly different
> > mathematical beast.
> 
>   ...
> 
> Randy,
> 
> Your impression is unfounded. That whatever you can do with the complex,
> you can do with the reals refutes the argument that that it must be
> fundamental because it's necessary. 

My parser is having a hard time extracting the meaning from this sentence.

> (Anyhow, I thought this discussion
> was primarily about negative frequencies, even though I did drag complex
> frequencies into it at one point.) Calculating with only real numbers is
> almost always harder and more error prone that using complex numbers,
> and not merely by a few more operations.

But I'm not talking about differences in "difficulty" and whatnot. They
are different things, fundamentally. That is why, e.g., the solution to
a polynomial with real coefficients is, in general, not real but rather
complex - because there are things you can do with this arithmetic system
that you cannot do with the reals.
-- 
%  Randy Yates                  % "...the answer lies within your soul
%% Fuquay-Varina, NC            %       'cause no one knows which side 
%%% 919-577-9882                %                   the coin will fall."
%%%% <yates@ieee.org>           %  'Big Wheels', *Out of the Blue*, ELO   
http://home.earthlink.net/~yatescr

Randy Yates wrote:
> 
> Jerry Avins wrote:
> >
> > Randy Yates wrote:
> > >
> >   ...
> > >
> > > I get the impression that Jerry (and others?) think that there is nothing
> > > really gained in using the complex over the reals. I vehemently disagree. Jerry's
> > > argument has been, over and over, that whatever you can do with the complex,
> > > you can do with the reals - it just may take a few more operations. It's
> > > precisely those "few more operations" that make the complex (along with
> > > the operations of addition and multiplication) a significantly different
> > > mathematical beast.
> >
> >   ...
> >
> > Randy,
> >
> > Your impression is unfounded. That whatever you can do with the complex,
> > you can do with the reals refutes the argument that that it must be
> > fundamental because it's necessary. [That is extra! ^^^^]
> 
> My parser is having a hard time extracting the meaning from this sentence.

The statement, "whatever you can do with the complex, you can do with
the reals" refutes ...

I'm sorry for being opaque. "I know what I mean, why don't you?" :-)
 
> > (Anyhow, I thought this discussion
> > was primarily about negative frequencies, even though I did drag complex
> > frequencies into it at one point.) Calculating with only real numbers is
> > almost always harder and more error prone that using complex numbers,
> > and not merely by a few more operations.
> 
> But I'm not talking about differences in "difficulty" and whatnot. They
> are different things, fundamentally. That is why, e.g., the solution to
> a polynomial with real coefficients is, in general, not real but rather
> complex - because there are things you can do with this arithmetic system
> that you cannot do with the reals.

True, but Fourier transforms are not among those things.

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

Randy Yates wrote:
> 
> Jerry Avins wrote:
> >
> > Randy Yates wrote:
> > >
> >   ...
> > >
> > > I get the impression that Jerry (and others?) think that there is nothing
> > > really gained in using the complex over the reals. I vehemently disagree.
Jerry's
> > > argument has been, over and over, that whatever you can do with the complex,
> > > you can do with the reals - it just may take a few more operations. It's
> > > precisely those "few more operations" that make the complex (along with
> > > the operations of addition and multiplication) a significantly different
> > > mathematical beast.
> >
> >   ...
> >
> > Randy,
> >
> > Your impression is unfounded. That whatever you can do with the complex,
> > you can do with the reals refutes the argument that that it must be
> > fundamental because it's necessary. [That is extra! ^^^^]
> 
> My parser is having a hard time extracting the meaning from this sentence.

The statement, "whatever you can do with the complex, you can do with
the reals" refutes the argument that that it must be fundamental because
it's necessary.

The scope of "anything" there is limited to Fourier transforms and the
like; complex exponential replacements for trigonometric functions.

I'm sorry for being opaque. "I know what I mean, why don't you?" :-)
 
> > (Anyhow, I thought this discussion
> > was primarily about negative frequencies, even though I did drag complex
> > frequencies into it at one point.) Calculating with only real numbers is
> > almost always harder and more error prone that using complex numbers,
> > and not merely by a few more operations.
> 
> But I'm not talking about differences in "difficulty" and whatnot. They
> are different things, fundamentally. That is why, e.g., the solution to
> a polynomial with real coefficients is, in general, not real but rather
> complex - because there are things you can do with this arithmetic system
> that you cannot do with the reals.

True, but Fourier transforms are not among those things.

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

Piergiorgio Sartor <piergiorgio.sartor@nexgo.REMOVE.THIS.de> wrote in message news:<9trtu-f56.ln1@lazy.lzy>...
> Rune Allnor wrote:
> 
>  Mathematics is supposed to be based on just a few axioms. I have a vague
>  recollection there are some eleven (or thereabouts) such axioms. 
> 
> Does anyone know where to find those axioms?
>
That is totally wrong..Every theory in mathematics begins from some
basic axioms,but no,there are not only eleven axioms for the whole
mathematical structure.
Of course if you want to find the axioms for a specific theory in
mathematics,that would be easier..
i.e for the Euclidean Geometry,or the Riemann Geometry etc etc..

"Randy Yates" <yates@ieee.org> wrote in message
news:3F1C8ADD.4796E506@ieee.org...

(snip)

> I get the impression that Jerry (and others?) think that there is nothing
> really gained in using the complex over the reals. I vehemently disagree.
Jerry's
> argument has been, over and over, that whatever you can do with the
complex,
> you can do with the reals - it just may take a few more operations. It's
> precisely those "few more operations" that make the complex (along with
> the operations of addition and multiplication) a significantly different
> mathematical beast.

There are a number of cases where the math is significantly simpler when
expressed in complex notation.  Cauchy's theorem and the Kramers-Kronig
relations are ones I happen to think of first.

It seems obvious to me, though, that if you replace each complex number with
an ordered pair of real numbers, and supply the appropriate operations to
those ordered pairs such that you get the same results, that you have shown
that you can do everything with (ordered pairs of) real numbers.  It might
look ugly, but it should work.

Rulers only measure real lengths, clocks measure real time, voltmeters
measure real volts.  But some physical quantities can be considered complex
with useful results.  Dielectric constant and index of refraction, where the
imaginary part is related to absorption.  (Consider the imaginary part of
inductance and capacitance as the resistance and conductance, respectively.)
But again, it is just convenience.  We can always write them separately.

-- glen

Xefteris Stefanos wrote:

> That is totally wrong..Every theory in mathematics begins from some
> basic axioms,but no,there are not only eleven axioms for the whole
> mathematical structure.

But the natural numbers themself are axioms... :-)

bye,

-- 
   Piergiorgio Sartor

Jerry Avins wrote:

> The point please? I wrote -- anyway, intended to write, that for any a,
> there is one and only one cos(a). Do you contradict that?

The equation cos(a)=1 has infinite solutions, not
only one.

So, for each "a" there is one cos(a), but for each
cos(a) there are infinite "a"...

Usually the form "one and only one" means that the
expression is invertible (one by one), which, of course
it is not true for the cos() operation.

So, saying (or writing) "for any a there is one and
only one cos(a)" can lead to some confusion in the
reader (or listener).
Since I'm sure there is no mistake in the sentence,
than probably this needs just some clarification.

> P.S. "Contradict" means "speak against". When writing, should we use
> "contrascribe"?

I guess the latin root (contra-dicere) has lost its
original meaning, to reach a much broader one, but
I can support "contrascribe" too!

It's interesting that, in some countries, it is common
to use something like "see you again", when leaving
someone, also at the phone. In some other countries,
there is a different form, like "hear you again", for
the phone and "see you again" in case of face to face.

bye,

-- 
   Piergiorgio Sartor

Piergiorgio Sartor wrote:
> [...]
> Usually the form "one and only one" means that the
> expression is invertible (one by one), which, of course
> it is not true for the cos() operation.

I've never heard "one and only one" in a mathematical
context, so I don't know how it would be defined, but
if you are using this synonymously with "one-to-one" 
then this is incorrect. A mapping is invertible if and
only if it is both one-to-one and onto (also known and
"injective" and "surjective," respectively). If a mapping
b:S-->T is "invertible" then there exists a mapping b^{-1} such
that b * b^{-1}(a) = a for all a \in S, where "*" denotes 
composition of mappings. 
-- 
%  Randy Yates                  % "...the answer lies within your soul
%% Fuquay-Varina, NC            %       'cause no one knows which side 
%%% 919-577-9882                %                   the coin will fall."
%%%% <yates@ieee.org>           %  'Big Wheels', *Out of the Blue*, ELO   
http://home.earthlink.net/~yatescr

Glen Herrmannsfeldt wrote:
> [...]
> It seems obvious to me, though, that if you replace each complex number with
> an ordered pair of real numbers, and supply the appropriate operations to
> those ordered pairs such that you get the same results, that you have shown
> that you can do everything with (ordered pairs of) real numbers. 

Then why do we need complex numbers to solve polynomials with real
coefficients??? ...
-- 
%  Randy Yates                  % "...the answer lies within your soul
%% Fuquay-Varina, NC            %       'cause no one knows which side 
%%% 919-577-9882                %                   the coin will fall."
%%%% <yates@ieee.org>           %  'Big Wheels', *Out of the Blue*, ELO   
http://home.earthlink.net/~yatescr