Forums

Laplace Transform vs Fourier transform

Started by fisico32 October 27, 2009
On Oct 30, 7:29&#2013266080;pm, Richard Dobson <richarddob...@blueyonder.co.uk>
wrote:
> Jerry Avins wrote: > > glen herrmannsfeldt wrote: > > > &#2013266080; ... > > >> Which in the end describes the interdependence between the real and > >> imaginary parts of the dielectric constant and/or index of refraction. > > >> Yes it is all phase shifts, but just saying that leaves a lot out. > > > There's no doubt that "real and imaginary" is a good way to describe the > > phenomena. We gain understanding with the compact, high-level, > > abstraction, but we lose some understanding by taking the mathematical > > description to be the actual phenomenon. > > Isn't this kind of stating the obvious? Come to it, even "real numbers" > represented in a computer are really combinations of integers under the > hood (and below that, merely aggregations of binary states),
exactly! we are getting closer to the physical issue. it's not so much that numbers are being manipulated in the computer, but it's binary states according to some combinatorial logic that is implemented with MOSFETs acting as switches which is a consequence of physical interactions happening inside of these devices.
> so they are > no more real than imaginary numbers. All numbers are ultimately > mythological;
oh, i dunno if i should do this at 2 in the morning. okay, numbers themselves are a human construct, but i am convinced that another intelligent life being (if SETI ever stumbles upon one) that is sophisticated enough to transmit messages that we happen upon will also have the concept of numbers, and will, in fact discover and formalize the same objective facts as derivable theorems (in whatever symbology/convention they used) that we do (maybe they would prove Fermat's Last Theorem "before" we had). so numbers have a broader existence than just a human construct. indeed some birds and chimps think in terms of small integers. but, the issue of reality is this: whether or not "numbers", real or imaginary, exist as a real thing or just in the imagination of a person or computer is the phenomena we call "physical quantity". those things are real. physical quantity is a real thing that really exists whether there were intelligent beings aware of it (and quantifying it with these made-up numbers). now the http://nrich.maths.org/5961 link posted by Rich was interesting in comparing to the concept of negative numbers, whether they be truly "real" or not. when they first started popping up was to represent debt as a "negative fortune" so that if it added to a fortune of equal size it adds to zero - the two parties don't owe each other a thing. but, it is argued, that money and debt are human social constructs. material itself (and the value of a service) which are traded between peoples still are non-negatively valued. are *differences* between commensurable physical quantities, themselves physical quantities? then potential energy is a physical quantity that can have negative value, so then negative numbers are real. but maybe the "difference" between two physical quantities is just a human construct, and not directly existing. how about net force. pick any single dimension (so the vectors are scalers) and for a body with two equal and opposite forces applied, it does not accelerate and the net force is zero. picking which direction is a convention, but some positive force got something added to it, and the result is zero. what would you call that other force? non-real? now, what we believe to be true about many various kinds of quantities or real stuff in physical reality is that when objects or particles or "things" are collected or aggregated, the quantitative physical property that they have in common adds according to the model or meaning of addition we have in arithmetic. so, whether or not certain kinds of numbers are "real" in that there are real things out there that have those numbers as properties, what *is* real in physical reality are physical quantities; mass, charge, energy, and physical interactions (fields, forces, probabilities of existence, etc.). and some of these real things add in their effect on other real things. finally, if that does not persuade someone that negative quantities do not exist for real in physical reality, then consider charged particles. whether they be electrons and positrons (-1 and +1) or the quarks (-1/3 and 2/3) and some of these things combine and >>you get no charge<<. so it looks like that not only negative integers are real things, so are positive and negative rational numbers real. at least they describe or model physical quantities in reality accurately. now picking which is "negative" and which is "positive" might be a convention and the aliens on the planet Zog might have been smart enough to adopt a convention that those muck less massive little things "flying about" the nucleus are the positive ones, making what we would call protons negatively charged. at least they would have to learn that the electron flow is in the opposite direction of the current they solved for with Kirchoff's (or Zog's) laws. but either way, there are physical quantities that can only be represented as quantitatively negative. negative numbers are real. now do the same for complex or imaginary numbers. every fundamental particle property or quantitative interaction you have (including QM, because the rubber does not yet hit the road until you compute expectation values) is real. every quantitative law of interaction that you will find in the physics book can be expressed as a more complicated set of equations involving only real quantities. you will never put a pair of leads on a voltage at a particular instance of time and have the meter tell you that it is "3+4i volts". or that 3+4i electrons passed a defined boundary in a given time period.
> the rest is simply counting, and very few things in the > physical world can be measured ~exactly~.
only things that exist only as integer quantities. nearly everything (but not everything, masses of particles is an example) can be boiled down to integer quantities. but this still says nothing about if physical quantities that really exist, exist in imaginary quantities. so far, *every* argument that has insisted some do have been sophistic. Jerry has recognized that. r b-j
it's like the Wicked Bible.  spurious addition or omission of the word
"not" has the tendency to alter the meaning of a sentence.  please
move the second occurrence of "not" in:

> finally, if that does not persuade someone that negative quantities do > not exist for real in physical reality, then consider charged > particles.
to this sentence:
>&#2013266080;at least they would [not] have to > learn that the electron flow is in the opposite direction of the > current they solved for with Kirchoff's (or Zog's) laws.
r b-j
Jerry Avins <jya@ieee.org> wrote:
(snip, I wrote)
 
>> Another place where complex numbers seem more than just an abstraction >> is the evanescent wave:
>> http://en.wikipedia.org/wiki/Evanescent_wave
> I think you confound the explanation for the explained phenomenon.
Maybe. I completely agree that for problems where complex numbers are used to represent phase, for example voltages and currents, that they are just to simplify the math, and that voltages and currents are real. I is slightly less obvious in the case of lumped impedances, where one can still separate out the real (R) from the imaginary (L, C). One can write the solution to the differential equation as a complex exponential or a decaying exponential multiplied by a sinusoid. With less ideal L, R, C, and even more in the case of spatially distributed L, R, C, and then to the case of an EM wave going through a lossy dielectric it is much less obvious. The EM wave is interacting with the electrons according to some differential equations. The solutions to those equations have complex exponential in them related to the exp(iwt) term in the EM wave. They still do even if the EM wave is cos(wt). Even more, consider a material whose properties change continuously in space. First, the analytical solution gets much harder, but then you have to switch over somewhere from the cos(kx) solution to the exp(-ax) solution. Even more, the solution can be different in different directions. In the evanescent wave case, it is a decaying exponential along one axis, but not the other axis. Calling a voltage or current complex is somewhat strange. Calling an impedance, dielectric constant, or index of refraction complex much less strange. For one, voltages and currents and electric fields are measured (more or less) directly. Impedances and dielectric constants by their effect on currents, voltages, and electric fields. Voltages and currents have phase, but impedances don't. -- glen
On Oct 29, 4:46&#2013266080;pm, robert bristow-johnson <r...@audioimagination.com>
wrote:
> On Oct 29, 7:21&#2013266080;pm, John Monro <johnmo...@optusnet.com.au> wrote: > > > > > > > robert bristow-johnson wrote: > > > just to add to what brent and Jerry and Hardy said... > > > > On Oct 27, 9:39 pm, "fisico32" <marcoscipio...@gmail.com> wrote: > > >> everyone is familiar with the Fourier transform and its importance. > > >> Its independent variable, the angular frequency w, is a measurable > > >> physical quantity. > > > > there's a part of that angular frequency that is mathematical, but not > > > physical. &#2013266080;how do you physically measure negative frequency? &#2013266080;and > > > negative frequency is in the F.T. > > > (snip) > > > > r b-j > > > Configure a CRO for XY mode. > > Connect the I signal to the X channel and the Q signal to > > the Y channel. > > The spot traces an anti-clockwise circular path for positive > > frequency signals; clockwise for negative frequency. > > Count the number of circles per second and attach the > > appropriate sign. > > that's fine and a legit human interpretation or abstraction of what is > happening with two real and physically related signals are observed on > an oscilloscope in XY mode. > > you still measure the Y signal at any instance in time as a real > value.
Same difference. Saying that a pair of numbers is complex describes how certain arithmetic operators act on the pair. If two real quantities follow the same arithmetic closure rules under a given system constraint, then calling them 2 reals, or 1 real and 1 imaginary component of a complex pair is just a choice of terminology. Lot's of idealized simple physical system have pairs of real measurable quantities that follow the rules being complex, if you stay within the constraints. Pull one of the wires on the 'scope and you break constraint and thus the arithmetic closure, same as if you suddenly change the rules for complex addition when playing math games. Then you can call them 2 pumpkins, or whatever. Choose your rules before playing either type of game. IMHO. YMMV. -- rhn A.T nicholson d.0.t C-o-M
robert bristow-johnson wrote:
..
> > how about net force. pick any single dimension (so the vectors are > scalers) and for a body with two equal and opposite forces applied, it > does not accelerate and the net force is zero. picking which > direction is a convention, but some positive force got something added > to it, and the result is zero. what would you call that other force? > non-real? >
That's an interesting one. Consider the classic "tug-of-war" exercise, with two equally matched teams. Sure, assuming they pull equally, the net motion of the rope is zero, but it is hardly zero "force" - there is massive tension in the rope, and in the limit it will break, with a corresponding quasi-explosive release of energy. A positive number and negative number can "collide" and result in a zero, but such things generally do not happen in the physical world, as other laws come into play, not least those dealing with conservation. There is some game being played here with the difference between a quantity and a direction. Wrap that rope round a tree, so that the two halves are now virtually coincident, and both (identical) forces become "positive", add constructively, and the tree gets pulled down (or not). Is a negative force one pulling "in the opposite direction", or one that is pushing? ..
> might be a convention and the aliens on the planet Zog might have been > smart enough to adopt a convention that those muck less massive little > things "flying about" the nucleus are the positive ones, making what > we would call protons negatively charged. at least they would have to > learn that the electron flow is in the opposite direction of the > current they solved for with Kirchoff's (or Zog's) laws. but either > way, there are physical quantities that can only be represented as > quantitatively negative. >
The real world offers plenty of examples of opposites, for which we need a vocabulary. I have had to consider this very recently, in designing and documenting programs to do Ambisonic panning. By arithmetic convention, the "positive" direction of rotation is anti-clockwise (using sine and cosine functions, needless to say); but this is counter-intuitive for the typical non-mathematical user for whom the natural analogy is the motion of clock hands. So in the end I decided to invert things so that a positive argument means clockwise rotation. It's a tough call - I don't want to mis-educate users, but I want the program to be as easy and intuitive to use as possible, too. Even the number line has (by convention) a direction - positive to the right. A negative number is simply a "leftwards" one. So on the complex plane starting at X=1, the angle ~increases~ such that the point becomes increasingly ~negative~, towards x = -1. The names are a convention - we have to call these quasi-opposite (or complementrary) states ~something~! An interesting question would be, how would those aliens see "the world" if their equivalent to Newton had happened upon quantum mechanics first, so that they would regard any notion of a continuous quantity as an illusion, or at least a counter-intuitive mystery? Richard Dobson
Jerry Avins <jya@ieee.org> writes:
> [...] > 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.
Jerry, First of all, Peano's axioms have nothing to do with complex, or even real, numbers. They are another way to get, eventually, to the ring of integers (other than the standard group-based development) and integral domains. And once again I must point out that your intimations are, at a minimum, at odds with the field of mathematics (namely, abstract algebra) that has proven extremely useful (re: the realization that no formulas exist for roots of equations greater than order 4). Under this system of mathematics, the complex numbers are NOT just a notational convenience. Rather, they are a field without which the solution of arbitrary algebraic equations over the reals cannot be determined. You have argued many times that they are just a "notational convenience." I posit that the extremely rich utility in viewing algebraic systems through the lens of two operations (addition and multiplication), i.e., as in the basic ring definition, makes this assertion patently false. -- Randy Yates % "The dreamer, the unwoken fool - Digital Signal Labs % in dreams, no pain will kiss the brow..." mailto://yates@ieee.org % http://www.digitalsignallabs.com % 'Eldorado Overture', *Eldorado*, ELO
glen herrmannsfeldt wrote:
> Jerry Avins <jya@ieee.org> wrote: > (snip, I wrote) > >>> Another place where complex numbers seem more than just an abstraction >>> is the evanescent wave: > >>> http://en.wikipedia.org/wiki/Evanescent_wave > >> I think you confound the explanation for the explained phenomenon. > > Maybe. > > I completely agree that for problems where complex numbers are > used to represent phase, for example voltages and currents, that > they are just to simplify the math, and that voltages and currents > are real. I is slightly less obvious in the case of lumped > impedances, where one can still separate out the real (R) from > the imaginary (L, C). One can write the solution to the differential > equation as a complex exponential or a decaying exponential multiplied > by a sinusoid. > > With less ideal L, R, C, and even more in the case of spatially > distributed L, R, C, and then to the case of an EM wave going > through a lossy dielectric it is much less obvious. The EM wave > is interacting with the electrons according to some differential > equations. The solutions to those equations have complex exponential > in them related to the exp(iwt) term in the EM wave. They still > do even if the EM wave is cos(wt). > > Even more, consider a material whose properties change continuously > in space. First, the analytical solution gets much harder, but > then you have to switch over somewhere from the cos(kx) solution > to the exp(-ax) solution. Even more, the solution can be different > in different directions. In the evanescent wave case, it is a decaying > exponential along one axis, but not the other axis. > > Calling a voltage or current complex is somewhat strange. > Calling an impedance, dielectric constant, or index of refraction > complex much less strange. For one, voltages and currents and > electric fields are measured (more or less) directly. > Impedances and dielectric constants by their effect on currents, > voltages, and electric fields. Voltages and currents have phase, > but impedances don't.
Maybe. I'm not (yet) convinced, but it's too deep for a quick comeback. I'll let it have time to coagulate, then see what shape it takes. Thanks! Jerry -- Engineering is the art of making what you want from things you can get. &#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;
Randy Yates wrote:
> Jerry Avins <jya@ieee.org> writes: >> [...] >> 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. > > Jerry, > > First of all, Peano's axioms have nothing to do with complex, or even > real, numbers. They are another way to get, eventually, to the ring of > integers (other than the standard group-based development) and integral > domains. > > And once again I must point out that your intimations are, at a minimum, > at odds with the field of mathematics (namely, abstract algebra) that > has proven extremely useful (re: the realization that no formulas exist > for roots of equations greater than order 4). Under this system of > mathematics, the complex numbers are NOT just a notational > convenience. Rather, they are a field without which the solution of > arbitrary algebraic equations over the reals cannot be determined. > > You have argued many times that they are just a "notational > convenience." I posit that the extremely rich utility in viewing > algebraic systems through the lens of two operations (addition and > multiplication), i.e., as in the basic ring definition, makes this > assertion patently false.
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 -- Engineering is the art of making what you want from things you can get. &#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;
Jerry Avins <jya@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." 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. -- Randy Yates % "Ticket to the moon, flight leaves here today Digital Signal Labs % from Satellite 2" mailto://yates@ieee.org % 'Ticket To The Moon' http://www.digitalsignallabs.com % *Time*, Electric Light Orchestra
On 10/31/2009 12:07 AM, robert bristow-johnson wrote:
> On Oct 30, 7:29 pm, Richard Dobson<richarddob...@blueyonder.co.uk> > wrote: >> Jerry Avins wrote: >>> glen herrmannsfeldt wrote: >>> ... >>>> Which in the end describes the interdependence between the real and >>>> imaginary parts of the dielectric constant and/or index of refraction. >>>> Yes it is all phase shifts, but just saying that leaves a lot out. >>> There's no doubt that "real and imaginary" is a good way to describe the >>> phenomena. We gain understanding with the compact, high-level, >>> abstraction, but we lose some understanding by taking the mathematical >>> description to be the actual phenomenon. >> Isn't this kind of stating the obvious? Come to it, even "real numbers" >> represented in a computer are really combinations of integers under the >> hood (and below that, merely aggregations of binary states), > > exactly! we are getting closer to the physical issue. it's not so > much that numbers are being manipulated in the computer, but it's > binary states according to some combinatorial logic that is > implemented with MOSFETs acting as switches which is a consequence of > physical interactions happening inside of these devices. > >> so they are >> no more real than imaginary numbers. All numbers are ultimately >> mythological; > > oh, i dunno if i should do this at 2 in the morning. > > okay, numbers themselves are a human construct, but i am convinced > that another intelligent life being (if SETI ever stumbles upon one) > that is sophisticated enough to transmit messages that we happen upon > will also have the concept of numbers, and will, in fact discover and > formalize the same objective facts as derivable theorems (in whatever > symbology/convention they used) that we do (maybe they would prove > Fermat's Last Theorem "before" we had). so numbers have a broader > existence than just a human construct. indeed some birds and chimps > think in terms of small integers. > > but, the issue of reality is this: whether or not "numbers", real or > imaginary, exist as a real thing or just in the imagination of a > person or computer is the phenomena we call "physical quantity". > those things are real. physical quantity is a real thing that really > exists whether there were intelligent beings aware of it (and > quantifying it with these made-up numbers). > > now the http://nrich.maths.org/5961 link posted by Rich was > interesting in comparing to the concept of negative numbers, whether > they be truly "real" or not. when they first started popping up was > to represent debt as a "negative fortune" so that if it added to a > fortune of equal size it adds to zero - the two parties don't owe each > other a thing. but, it is argued, that money and debt are human > social constructs. material itself (and the value of a service) which > are traded between peoples still are non-negatively valued. > > are *differences* between commensurable physical quantities, > themselves physical quantities? then potential energy is a physical > quantity that can have negative value, so then negative numbers are > real. but maybe the "difference" between two physical quantities is > just a human construct, and not directly existing. > > how about net force. pick any single dimension (so the vectors are > scalers) and for a body with two equal and opposite forces applied, it > does not accelerate and the net force is zero. picking which > direction is a convention, but some positive force got something added > to it, and the result is zero. what would you call that other force? > non-real? > > now, what we believe to be true about many various kinds of quantities > or real stuff in physical reality is that when objects or particles or > "things" are collected or aggregated, the quantitative physical > property that they have in common adds according to the model or > meaning of addition we have in arithmetic. so, whether or not certain > kinds of numbers are "real" in that there are real things out there > that have those numbers as properties, what *is* real in physical > reality are physical quantities; mass, charge, energy, and physical > interactions (fields, forces, probabilities of existence, etc.). and > some of these real things add in their effect on other real things. > > finally, if that does not persuade someone that negative quantities do > not exist for real in physical reality, then consider charged > particles. whether they be electrons and positrons (-1 and +1) or the > quarks (-1/3 and 2/3) and some of these things combine and>>you get > no charge<<. so it looks like that not only negative integers are > real things, so are positive and negative rational numbers real. at > least they describe or model physical quantities in reality > accurately. now picking which is "negative" and which is "positive" > might be a convention and the aliens on the planet Zog might have been > smart enough to adopt a convention that those muck less massive little > things "flying about" the nucleus are the positive ones, making what > we would call protons negatively charged. at least they would have to > learn that the electron flow is in the opposite direction of the > current they solved for with Kirchoff's (or Zog's) laws. but either > way, there are physical quantities that can only be represented as > quantitatively negative. > > negative numbers are real.
There's also the issue of being able to place an appropriate reference. For example, if one wishes to chart the level of a stream, one may start with a stake placed in the stream with zero at the level of the stream on the day the data collection starts. If the stream goes down from that point, a negative number can be recorded without loss of information or consequence as long as one knows what the reference is. We do this routinely all the time with all sorts of things (e.g., temperature scales other than Kelvin). Sure it's an abstraction, where the reference is can be arbitrary, but does it make the "negative" quantities any less meaningful to the physical reality?
> now do the same for complex or imaginary numbers. every fundamental > particle property or quantitative interaction you have (including QM, > because the rubber does not yet hit the road until you compute > expectation values) is real. every quantitative law of interaction > that you will find in the physics book can be expressed as a more > complicated set of equations involving only real quantities. you will > never put a pair of leads on a voltage at a particular instance of > time and have the meter tell you that it is "3+4i volts". or that > 3+4i electrons passed a defined boundary in a given time period. > >> the rest is simply counting, and very few things in the >> physical world can be measured ~exactly~. > > only things that exist only as integer quantities. nearly everything > (but not everything, masses of particles is an example) can be boiled > down to integer quantities. but this still says nothing about if > physical quantities that really exist, exist in imaginary quantities.
As I think you allude, whether something is integer or "real" might boil down to scaling, just like the negative number issue has a lot to do with where the zero reference was. Your particle charge example is apropos: if quarks were -1 and 2, then positrons and electrons would be +3 and -3. No more rational values. If everything was based on the appropriate quanta, perhaps rational numbers wouldn't be needed and only integers would get us by just fine for nearly everything, perhaps everything. They'd be big, but it'd work. So one could argue that rational numbers are just as hokey as negative numbers or complex numbers or anything along those lines.
> so far, *every* argument that has insisted some do have been > sophistic. Jerry has recognized that. > > r b-j
Not sure I agree with that point. -- Eric Jacobsen Minister of Algorithms Abineau Communications http://www.abineau.com