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Physical Meaning of Complex Numbers

  1. Oct 3, 2009 #1

    I've used complex numbers before for contour integration, circuit theory, and analysis; the only problem is, I have no idea how to think of complex numbers, physically - no teacher, book, or website seems to offer me any adequate solution.

    Can anyone please help me ? I hate doing problems in which I cannot fully understand!
  2. jcsd
  3. Oct 3, 2009 #2


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    Complex numbers are a part of mathematics, just as are real numbers. Physical things can be modelled and described using complex numbers or real numbers, but the numbers themselves are not physical.

    To give a simple example, pi is the ratio of the circumference of a circle to its diameter, but the circle and its parts are mathematical constructs, not physical objects.
  4. Oct 3, 2009 #3


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    I don't know if the following will help : The electrical impedance can be represented by a complex number.
  5. Oct 5, 2009 #4
    Like mathman said, you asked a question that doesn't make sense. Numbers don't have a physical meaning. They can be represented quite nicely in real life, but the representations aren't the numbers themselves.

    If you want a physical representation of a complex number, think of it like an arrow, anchored at the origin, having a length and a direction in the plane. Alternatively, if you like electronics, you can think of a pure sine wave as a complex number, with its real part being the amplitude of the wave and the complex part being its frequency. (Although, you have to allow for waves with negative frequencies too).

    Using complex numbers in the math for electronics is just a shorthand, by the way. You don't *need* to use them. They are just handy because you deal with a lot of expressions like e^(wjt), and when you take their integrals or derivatives, you simply multiply or divide by w to get your answer.
  6. Oct 5, 2009 #5


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    Tic-Tacs, I wish I understood what you said. I can make no better sense of complex numbers than i*i=-1 and then trying to show this using a Real axis at right angle to an Imaginary axis does not help, being that I cannot place the second axis into physical mechanical meaning.

    Can less advanced cyclic relationships be ways of applying complex numbers? Seasons, moon phases, vibrating strings? Vibrations through solids or liquids?
  7. Oct 5, 2009 #6

    If you're talking about a scalar quantity such as position, length, volume, temperature, pressure, or potential energy, there is no meaning. These quantities are understood to be real-valued.

    If you're talking about impedence or some electrical quantity that uses complex numbers, understand it's just a short hand. You could just as easily break it up into resistance and reactance and do your math using only real numbers.
  8. Oct 5, 2009 #7
    you probably need some time to think about what complex numbers are and what it means to have physical meaning. It's a deep question that you have to struggle with for yourself and probably no short answer on this board can make you satisfied. And don't be down on yourself for struggling to understand this because for hundreds of years, even the greatest mathematicians had the hardest time coming to terms with complex numbers and negative numbers as well though these days teachers just zip through them as to make students feel like they're stupid if it doesn't really click.

    Something that really got me started on this similar question was something my math professor said once as a joke when he started to run into complex numbers in some fourier analysis. "People have a problem with complex numbers claiming they're not real. But when you think about it, real numbers aren't real either."

    Also, I recommend a book called "Negative Math." It goes into the long history of mathematicians trying to understand the meaning of negative numbers and the complex numbers that necessarily follow.
  9. Oct 5, 2009 #8


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    Complex numbers are generally used to represent the mathematics of combining waves. You can have to light waves with intensity 1 that sum to an intensity of 4. Or, you can have two light waves with intensity 1 that sum to an intensity of zero! Obviously, you can't make sense of these two sittuations if you represent waves simply as real numbers.

    Like waves, complex numbers have an amplitude and a phase. How they sum depends on whether the complex numbers are in-phase or out-of-phase.
  10. Oct 6, 2009 #9
  11. Sep 22, 2010 #10
    Bottom line for another real world example of complex numbers is here


    which was found by entering the two character strings, exact phrases, exactly as shown below into google

    "complex numbers" "land surveying"

    oh how I love parsing and and operators :).

    I am out of time right now today; however, I still by and large have the same problem in my gut so to speak. Problem being how to get a hug around the concept of complex numbers, a physical example. Even though, I saw "the light" when I read the example on this thread regarding two light waves equaling 0 as well as four or whatever it said.

    Anywhoooo, I have a math book which states that one of the first "practical uses" for complex numbers was land surveying.
  12. Sep 22, 2010 #11
    Complex numbers (and also quaternions) naturally appear in the geometric algebra (Clifford algebra [tex]Cl(0,3)[/tex]). For instance the basic 3-vector [tex]e_1e_2e_3[/tex] is of square -1.
  13. Sep 22, 2010 #12
    There's a reason that [itex]i[/tex] is called an imaginary number...

    It only exists in our minds (it's imaginary!)
  14. Sep 22, 2010 #13
    But real numbers aren't real either, are they?
  15. Sep 22, 2010 #14


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    I am surprised you have managed to do circuit theory and remain without a physical idea of complex numbers. Anyway, it is a distant memory but I first learned complex numbers on a train journey at tender years from this book


    which I think a lot of people remember with affection. The next time you order any other books I see it can be got for less than $1 ! from Amazon.
  16. Sep 22, 2010 #15


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    Hurrah for year old threads!

    A number has exactly the physical meaning that a theory of physics says it does, no more, no less; this is pretty much tautological.

    Since the opening poster is no longer paying attention, I'm going to close this thread before it devolves into the usual argument with someone equivocating mathematical terms like "real" or "natural" with their English homonyms.
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