In article <ipv1ok$3r1$2@dont-email.me>,
glen herrmannsfeldt <gah@ugcs.caltech.edu> wrote:
> steve <bungalow_steve@yahoo.com> wrote:
>
> (big snip)
>
> > the problem with our "proper education system" is if you don't
> > understand it, you are sent on to the next class with the assumption
> > you did
>
> What I finally figured out, way later than I might have, is that
> the first time you see something you aren't really expected to
> understand it. The second time you are. If somehow you didn't
> notice the first time, then too bad. That includes the assumption
> that every teacher along the way taught everything that was supposed
> to be taught.
>
> > and if you didn't get a A in every test you never understood it
>
> -- glen
This cartoon seems especially appropriate today:
http://xkcd.com/895/
Cheers,
KP
Reply by Randy Yates●May 6, 20112011-05-06
On 05/04/2011 11:15 PM, Clay wrote:
>
>>
>>> We pretty much accept (at least by physicists) that all measurements
>>> are real valued. So buried in this experimental fact is an idea that
>>> we don't have to have complex numbers, but Hadamard says that may be
>>> much harder to work with.
>>
>> I see this as a non-sequitur. All measurements are real-valued, thus
>> complex numbers are not required to establish any truths?
>> --
>>
>
> What if the truths we are looking for must exist in the physical
> world.
>
> QM gets by just fine as a physical theory even if it requires all
> observables to be real valued. If we allow only mathematical entities,
> then we can define one that has to be complex valued to exist e.g. a
> number that when squared equals -1. But we can't seem to go out in
> nature and find it laying around. That is the conundrum.
One can't go out into nature and find real numbers, either. Or
integers, for that matter. "Hey, lookie here honey: a 42!"
And we've probably already left the frying pan and jumped into
the fire when we began talking about "truths," "existance,"
and "reality," since these are very hard to quantify.
--
Randy Yates % "So now it's getting late,
Digital Signal Labs % and those who hesitate
mailto://yates@ieee.org % got no one..."
http://www.digitalsignallabs.com % 'Waterfall', *Face The Music*, ELO
Reply by glen herrmannsfeldt●May 5, 20112011-05-05
steve <bungalow_steve@yahoo.com> wrote:
(big snip)
> the problem with our "proper education system" is if you don't
> understand it, you are sent on to the next class with the assumption
> you did
What I finally figured out, way later than I might have, is that
the first time you see something you aren't really expected to
understand it. The second time you are. If somehow you didn't
notice the first time, then too bad. That includes the assumption
that every teacher along the way taught everything that was supposed
to be taught.
> and if you didn't get a A in every test you never understood it
-- glen
Reply by steve●May 5, 20112011-05-05
On May 5, 12:13�am, "steveu" <steveu@n_o_s_p_a_m.coppice.org> wrote:
> >On Fri, 1 Apr 2011 10:32:29 -0700 (PDT), robert bristow-johnson
> ><r...@audioimagination.com> wrote:
>
> > � � � � � � � � � [Snipped by Lyons]
>
> >>i think a better term would be "comprehensive numbers". �these numbers
> >>have everything they need, a real and imaginary component, either
> >>component can be negative or not, an integer or not, rational or
> >>irrational.
> >Hi Robert,
> > � yes, "complex numbers" is an unfortunate term.
>
> >The great electrical engineer Charles Proteus Steinmetz
> >(who pioneered the analysis of transformers and
> >alternating current in the late 1800s) preferred to
> >use the phrase "general numbers" for what we today
> >call "complex numbers."
>
> >See Ya',
> >[-Rick-]
>
> Maybe he had a proper education. If you meet them early enough in your
> schooling, complex numbers are the most natural thing in the world, from
> which reals, integers and cardinals are progressive levels of debasement.
>
> Steve- Hide quoted text -
>
> - Show quoted text -
the problem with our "proper education system" is if you don't
understand it, you are sent on to the next class with the assumption
you did
and if you didn't get a A in every test you never understood it
Reply by Jerry Avins●May 5, 20112011-05-05
On May 5, 12:13�am, "steveu" <steveu@n_o_s_p_a_m.coppice.org> wrote:
> >On Fri, 1 Apr 2011 10:32:29 -0700 (PDT), robert bristow-johnson
> ><r...@audioimagination.com> wrote:
>
> > � � � � � � � � � [Snipped by Lyons]
>
> >>i think a better term would be "comprehensive numbers". �these numbers
> >>have everything they need, a real and imaginary component, either
> >>component can be negative or not, an integer or not, rational or
> >>irrational.
> >Hi Robert,
> > � yes, "complex numbers" is an unfortunate term.
>
> >The great electrical engineer Charles Proteus Steinmetz
> >(who pioneered the analysis of transformers and
> >alternating current in the late 1800s) preferred to
> >use the phrase "general numbers" for what we today
> >call "complex numbers."
>
> >See Ya',
> >[-Rick-]
>
> Maybe he had a proper education. If you meet them early enough in your
> schooling, complex numbers are the most natural thing in the world, from
> which reals, integers and cardinals are progressive levels of debasement.
Complex numbers take a while to get used to. Not everybody -- that
includes el-hi math teachers -- is there yet. (The same is true about
arbitrary number bases.) They are concise, elegant, and useful. The
question we're discussing here, I thought, is whether they represent
physical phenomena in the same way that integers represent piles of
coconuts or sand on a beach.
Jerry
--
Engineering is the art of making what you want from things you can get.
Reply by steveu●May 5, 20112011-05-05
>On Fri, 1 Apr 2011 10:32:29 -0700 (PDT), robert bristow-johnson
><rbj@audioimagination.com> wrote:
>
> [Snipped by Lyons]
>
>>i think a better term would be "comprehensive numbers". these numbers
>>have everything they need, a real and imaginary component, either
>>component can be negative or not, an integer or not, rational or
>>irrational.
>Hi Robert,
> yes, "complex numbers" is an unfortunate term.
>
>The great electrical engineer Charles Proteus Steinmetz
>(who pioneered the analysis of transformers and
>alternating current in the late 1800s) preferred to
>use the phrase "general numbers" for what we today
>call "complex numbers."
>
>See Ya',
>[-Rick-]
Maybe he had a proper education. If you meet them early enough in your
schooling, complex numbers are the most natural thing in the world, from
which reals, integers and cardinals are progressive levels of debasement.
Steve
Reply by Clay●May 5, 20112011-05-05
>
> > We pretty much accept (at least by physicists) that all measurements
> > are real valued. So buried in this experimental fact is an idea that
> > we don't have to have complex numbers, but Hadamard says that may be
> > much harder to work with.
>
> I see this as a non-sequitur. All measurements are real-valued, thus
> complex numbers are not required to establish any truths?
> --
>
What if the truths we are looking for must exist in the physical
world.
QM gets by just fine as a physical theory even if it requires all
observables to be real valued. If we allow only mathematical entities,
then we can define one that has to be complex valued to exist e.g. a
number that when squared equals -1. But we can't seem to go out in
nature and find it laying around. That is the conundrum.
Clay
Reply by Randy Yates●May 4, 20112011-05-04
On 05/04/2011 10:24 AM, Clay wrote:
> [...]
> Hello Randy,
>
> There is no question that complex numbers enrich the world of
> mathematics.
>
> The real question (pardon the pun) is do we have to use complex
> numbers to represent reality?
Hi Clay,
Yes, this is the question. I might rephrase and ask, "Are we required
to use complex numbers and/or abstract algebra to arrive at certain
truths?"
> A good physical theory is quantum mechanics (QM) which uses complex
> numbers at its core. The interesting thing is all measurable things
> are real valued. And QM certainly goes along this line of thinking.
>
> We have the famous quote from the French mathematician Jacques
> Hadamard:
>
> "It has been written that the shortest and best way between two truths
> of the real domain often passes through the imaginary one."
>
> This may be found in various translations from the original French:
> "On a pu �crire depuis que la voie la plus courte et la meilleure
> entre deux v�rit�s du domaine r�el passe souvent par le domaine
> imaginaire."
>
> Researchers have not found where this has been written other than by
> Hadamard himself!
That is interesting, but I'm not sure it's all that relevent. Hadamard
was certainly a great mathematician, and I should carefully weigh
anything he cared enough to write down regarding mathematics. But "the
shortest way" and "the best way" are not the same as "the only way."
> Now the question is Do we need to modify Hadamard's statement? I.e.,
> are there some truths in the real domain that must be connected via
> the complex one? I.e, is the connection one of convenience or
> necessity?
I would remove the constraint "in the real domain." That is, I would
ask more simply (and generally), "Are there some truths that can
only be arrived at by assuming the complex numbers exist?"
> We pretty much accept (at least by physicists) that all measurements
> are real valued. So buried in this experimental fact is an idea that
> we don't have to have complex numbers, but Hadamard says that may be
> much harder to work with.
I see this as a non-sequitur. All measurements are real-valued, thus
complex numbers are not required to establish any truths?
--
Randy Yates % "So now it's getting late,
Digital Signal Labs % and those who hesitate
mailto://yates@ieee.org % got no one..."
http://www.digitalsignallabs.com % 'Waterfall', *Face The Music*, ELO
Reply by Clay●May 4, 20112011-05-04
>
> I believe for index of refraction that most physicists would
> consider it complex. �(Read about ellipsometry, for example.)
>
> Since the QM wave function is not normally measured directly,
> I wouldn't try to claim that as a complex measurement.
>
> -- glen
I'm quite familiar with ellipsometry as I had to consider it when
making holograms using metallic mirrors with randomly polarized
light[1]. Apart from the convenience of complex notation is it
necessary? To me a complex index of refraction is simply refraction
and attenuation. Each of which may be viewed in terms of real values.
I'd rather not calculate these without complex notation because the
simplification is great, but no matter the amount of simplication, is
it necessay in that it is not possible to do otherwise? And we have to
be sure to separate impossible from impractical.
In the QM applications, the complex numbers allow you to add up
mutliple paths to effect interference. The phase shifts themselves are
unobservable - it is the relative shifts between two or more paths
that become observable. Again the simplification afforded by complex
notation is useful, but is it necessary?
Clay
[1] I was working with small randomly polarized HeNe lasers to make
holograms using aluminized mirrors. I measured the index to be 1+4.45i
for Al at 632.8nm. This is easily done by finding the incident/
reflection angles where the relative phase shift between the plane
parallel and plane perpendicular cases are 90 deg, 45 deg, 30
degrees, etc and then fitting Fresnel's equation. The reason for using
these angles is you then can start with circularly polarized light and
then set up a 1, 2, 3 or more mirror (each mirror is adjusted to have
the same angle of incidence) bounce situation to create linearly
polarized light. Using a linear polarizer as an analyzer, one can
quickly adjust the mirrors and find these magic angles. I wrote a
program, HoloCad, that used this information to optimize the
interference at the photographic plate between the object and
reference light.
Reply by glen herrmannsfeldt●May 4, 20112011-05-04
Clay <clay@claysturner.com> wrote:
(big snip)
>> I believe, if I understand you correctly, the umbrella point you're
>> making through all this is that you can choose to define (and work
>> with) things in different ways. That I agree 100 percent with.
>> However, if you are going another step and maintaining that
>> the only difference between different respresentations, or
>> notions, of the various arithmetic systems is the ease of
>> use (e.g., complex arithmetic makes it easier to write
>> a product), then I would disagree.
(not so big snip)
> There is no question that complex numbers enrich the world of
> mathematics.
> The real question (pardon the pun) is do we have to use complex
> numbers to represent reality?
For the common case of using exp(iwt) in place of sine and cosine,
for example in computing voltages (which aren't complex) I completely
agree that we don't. It makes the math easier, but in the end
we only want the real part. In the usual case, the expression is
(about) twice as complicated, puting in exp instead of both
sine and cosine.
> A good physical theory is quantum mechanics (QM) which uses complex
> numbers at its core. The interesting thing is all measurable things
> are real valued. And QM certainly goes along this line of thinking.
For QM, it is slightly less obvious. In many cases it isn't the
real part, but the magnitude. If you want to avoid complex
numbers, the expressions will be more than twice as big.
(That is, more than just replacing exp with sine and cosine.)
> We have the famous quote from the French mathematician Jacques
> Hadamard:
> "It has been written that the shortest and best way between two truths
> of the real domain often passes through the imaginary one."
> This may be found in various translations from the original French:
> "On a pu �crire depuis que la voie la plus courte et la meilleure
> entre deux v�rit�s du domaine r�el passe souvent par le domaine
> imaginaire."
As I have written a few times, it seems to me that index of
refraction does make more sense as a complex value. Note, for
example, that when you go to/from total internal reflection
the exponent goes from complex to imaginary back to complex.
Unlike voltage, for example, index of refraction naturally
belongs in an exponential. (I believe that there are some cases
where voltage is in an exponent, but not the usual ones.)
That if you try to rewrite complex index of refraction expressions
using only sine and cosine they get much more than twice as big,
maybe more than four times.
> Researchers have not found where this has been written other than by
> Hadamard himself!
> Now the question is Do we need to modify Hadamard's statement? I.e.,
> are there some truths in the real domain that must be connected via
> the complex one? I.e, is the connection one of convenience or
> necessity?
> We pretty much accept (at least by physicists) that all measurements
> are real valued. So buried in this experimental fact is an idea that
> we don't have to have complex numbers, but Hadamard says that may be
> much harder to work with.
I believe for index of refraction that most physicists would
consider it complex. (Read about ellipsometry, for example.)
Since the QM wave function is not normally measured directly,
I wouldn't try to claim that as a complex measurement.
-- glen