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Epsilon Pi's ideas on imaginary numbers

  1. Sep 27, 2004 #1
    It is known that it was Descartes the one that gave the symbol i the connotation of imaginary; in electrical engineering there is the concept of apparent power(MVA)
    S = P + i Q
    where
    P(MW)=generation or consumed power
    and
    Q(MVAR) = reactive power
    and they both can be measured, so they exist, they are not "imaginary".

    In fact the symbol i, can be taken as a symbol to differentiate two different realities that in general cannot be reduced the one to the other, as P is quite different from Q, even in its physical behavior, and a reason of those rules explained in this thread too for its mathematical operations, as we must not confuse pearls and apples, when working with it.

    Regards
    EP


     
  2. jcsd
  3. Sep 27, 2004 #2
    In the physics of waves (electromagnetics, sounds, strings, quantum mechanics etc.), cos(x) and sin(x) are ever present. Using Euler's equation allows us to work with [tex]e^{ix}[/tex] instead of cos(x). The special properties of [tex]e^x[/tex] simplify problems and theories so much that frankly I don't even want to know or attempt to know what they would look like using only cos(x).

    So if you have a cos(x), you can add isin(x) to it to make [tex]e^{ix}[/tex], do a whole bunch of math with this [tex]e^{ix}[/tex], and then retrieve a physical, meaningful answer by dropping the isin(x) again, leaving only the rational number cos(x).
     
  4. Sep 27, 2004 #3
    It is also known that it was Leonhard Euler with its well-known Euler relation
    i(theta)
    e = cosine(theta)+ i sine(theta)
    found in 1745, when studying infinite series the one who finally gave status to complex numbers. As a matter of fact it took 100 years for the complex plane to be accepted, and even such a powerful and brilliant mind as that of Gauss was reluctant to accept it, so it is not strange that they are not used in its full power yet, except in EE.
    Regards
    EP
     
  5. Sep 27, 2004 #4

    mathwonk

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    I am curious as to the evidence of Gauss' reluctance to accept complex numbers, especially since he is credited with the first proof, at the age of 22, of the fundamental theorem of algebra, using them. Indeed he is sometimes credited with having helped place complex numbers on firm ground.
     
  6. Sep 28, 2004 #5
    I am sorry, I did not say complex numbers, but the complex plane, and I talked about reluctancy about accepting it, in fact:

    "Wessel's memoir appeared in the Proceedings of the Copenhagen Academy for 1799, and is exceedingly clear and complete, even in comparison with modern works. He also considers the sphere, and gives a quaternion theory from which he develops a complete spherical trigonometry. In 1804 the Abbé Buée independently came upon the same idea which Wallis had suggested, that <math>\pm\sqrt{-1}<math> should represent a unit line, and its negative, perpendicular to the real axis. Buée's paper was not published until 1806, in which year Jean-Robert Argand also issued a pamphlet on the same subject. It is to Argand's essay that the scientific foundation for the graphic representation of complex numbers is now generally referred. Nevertheless, in 1831 Gauss found the theory quite unknown, and in 1832 published his chief memoir on the subject, thus bringing it prominently before the mathematical world. "

    And my point was that it was completly accepted(the complex plane, not complex numbers) at the end of XIX century, with the works of Oliver Heaviside, Maxwell and Steinmentz when they found applications for it, putting in this way the mathematical foundation of EE on firm ground.

    Regards
    EP

     
  7. Sep 28, 2004 #6

    matt grime

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    "In fact the symbol i, can be taken as a symbol to differentiate two different realities that in general cannot be reduced the one to the other, as P is quite different from Q, even in its physical behavior, and a reason of those rules explained in this thread too for its mathematical operations, as we must not confuse pearls and apples, when working with it." - epsilon pi

    but this is maths, if we want to do philosophy of physics about what is an apparent reality then we'd be doing that instead. the sooner mathematics removes this kind of attempt to explain itself the better.
     
  8. Sep 28, 2004 #7
    sorry, I thought maths were just a tool or a language we use to represent reality "out there" to make easier and more realistic its applications.

    Regards
    EP
     
  9. Sep 29, 2004 #8

    matt grime

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    well, it isn't. grothendieck duality don't do a whole lot to decribe what's out 'there', though someone may have other ideas as to what it describes. maths is not a tool, it is a beautiful abstract subject. it may be used for practical things, but that is not how one should think about it anymore since it will rapidly only serve to confuse people. look at the standard confusions that arise from learning about imaginary numbers. even the name is a poor choice, though we can't do anything about that.
     
  10. Sep 29, 2004 #9

    Tom Mattson

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    That's simply false. In fact, I have never met an electrical engineer who knew as much about complex analysis as a mathematician or physicist. Engineers typically don't use contour integration or conformal mapping to solve problems in electrodynamics, but physicists learn it as undergraduates. Use of the Euler relation does not even come close to using complex variables "in its full power".
     
  11. Sep 29, 2004 #10

    matt grime

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    there is one electrical engineer who might know more than a mathematician, but he is the exception: Raul Bott. Apart from that, I've never heard of an EE'er even knowing what the germs of continuations of analytic functions are.
     
  12. Sep 29, 2004 #11
    Well Mr Mattson, I really was wondering if you are willing to have a serious discussion and my apologies if I seem so direct. As you most probably know because of our preview discussions, my point is not quantity, who knows more or less; if you have read what I promised you and you claimed then you will know what is my point.
    I know maths as a pure science has many possibilities, but should we not make it simple at the moment of applications?
    My claim is the same: can you put the fundamental equations of physics under a unified framework? As far as I know you can't even with the best you have. And this can be done for sure with Euler relation and the basic unit system derived from it, this is my real point.

    Best regards
    EP

     
  13. Sep 29, 2004 #12
    confusion is not a result, precisely because, there is not anymore a unified framework when coping fundamental matters? why are you really so sure we can't do anything?
    Regards
    EP
     
  14. Sep 29, 2004 #13

    matt grime

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    who said we can't do anything. we can do a lot. i will strenuously maintain that the best way to do mathematics is to do so in the abstract where an object *is* its properties: the reals are an ordered complete field, the complex numbers are are degree two fied extension of R, the kernel of a surjective homomoprhism is the obeject such that every map which factors as zero factors through the kernel.... and so on.

    thinking that i *is* the imaginary electromotive force or whatever it was you said it is gets you nowhere since it doesn't even imply that the complex numbers are a division algebra.
     
  15. Sep 29, 2004 #14
    In the same way we can make pure philosophy, of course, you have all the right to make pure mathematics, but for sure even from the point of view of pure maths, there are another ways of presenting the matter, as it is not at all a closed field, is it?

    sorry I did not get what you meant; i is just a symbol to differentiate two different entities that must not be reduced one to the other, a reason of why we have those different rules for mathematical operations as far as I know, and was taught.
    Regards
    EP
     
  16. Sep 29, 2004 #15

    matt grime

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    And where in you alternate view point did you define what i is, and its properties? You didn't. You just said it was a symbol to differentiate between different realities that cannot be reduced to one another. That is flowery prose, but acutally almost an entirely vacuous sentence, and meaningless in terms of mathematics. And as I keep saying, this is my opinion on the best way to teach and learn mathematics. I am a mathematician and a teacher of mathematics.
     
  17. Sep 29, 2004 #16
    Are not the properties defined by the way mathematical
    operations are done with it? Is it not a property of complex numbers to remain with the same form with differentiation and integration except by a rotation of +/- 90 degrees? Is not this another result of the properties of them?
    The sentence that you call flowery prose, has specific applications when defining the complex power in EE, that as a matter of fact can be measured.
    And I am sorry if I am contradicting your opinion!
    Thank you so much for your time!
    Regards
    EP

     
  18. Sep 29, 2004 #17

    matt grime

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    "Is it not a property of complex numbers to remain with the same form with differentiation and integration except by a rotation of +/- 90 degrees? Is not this another result of the properties of them?"


    erm, that is meaningless as far as i can tell.

    "has specific applications when defining the complex power in EE, that as a matter of fact can be measured."

    but does not actually define the complex numbers. it describes a physical phenomenon that can be accruately represented with complex numbers. that isn't the same thing at all.
     
  19. Sep 29, 2004 #18

    Tom Mattson

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    I was wondering the same thing about you. You made several false statements in our previous discussions, and exhibited a number of severe misconceptions. I made the necessary corrections and referred you to a source or two, but you wouldn't listen.

    This remark serves no other purpose than to dodge what I am saying. You claim that only electrical engineers use complex numbers to their full power. You are wrong. Physicists use the Euler relation in their equations (indeed, the differential equations of EE are equations of physics), and in addition to that we use the other techniques I mentioned.

    It should be made as simple as possible, but no simpler.

    By the way, this is the second time you have introduced an undefined idea into this thread. First, when you say that EE's use the complex number system "in its full power", you give no indication of what that means. I assumed it meant that EE's use the theory of complex analysis in its entirety, and I pointed out that that is false. You now say that it isn't about quantity, so one can only speculate as to what you mean.

    Now, you introduce your undefined notion of "simple". Most people--and I suspect you are included in this--mean that "simple" should mean "elementary" or "easy to understand". But most scientists and mathematicians use the word "simpler" to mean "relies on fewer axioms".

    And here is undefined concept #3. What does it mean to be "unified"?

    You have made it perfectly clear in our previous discussion that you don't understand very much about the equations of physics. During our entire interaction, you steadfastly held onto the false idea that QM and SR offer competing ideas on space and time, and that those of QM should be preferred over those of SR (or at least that the picture of spacetime in SR should be modified by that of QM).
     
    Last edited: Sep 29, 2004
  20. Sep 29, 2004 #19

    Tom Mattson

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    I split this off from the thread i, as the discussion is off topic, to say the least.
     
  21. Sep 29, 2004 #20
    Hi all,

    Mathematics can exist in an abstract space all its own. It need only refer to other forms within this space for consistancy and completeness.

    Physical reality is another story. Only those mathematical tools that can explain the perceved reality (qualitatively and quantitatively) are valueble is this area.

    Complex numbers, for instance, when used to explain reality are valueble since the results map onto the perceived reality.

    The operations in the complex plane give real valued answers to real physical problems. The might also give answers that have no physical reality. These latter answers do not map. This is the confusion. Mathematics can give two answers to the same question, where one maps and one doesn't.

    juju
     
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