Some remarks on complex numbers

  1. I just finished reading the "Reality Bits" in a recent copy of NewScientist.
    It discusses attempts to purge mathematics of the need for complex numbers.
    Started me thinking(danger, danger) of not how to get rid of the square
    root of negative one, but, more easily, simply find out where it enters
    the mathematical picture. The Pythagoreans were probably the first who
    wondered about this;

    ##a^2 + b^2 = c^2##

    We sure know how to represent the square root of ##c^2##, it's ##c##.
    But can we write an expression for the square root of say, ##1 + x^2##?
    Try as we must, it doesn't seem to exist. But wait, if we supplement
    our real number system with an imaginary number system, it is at least
    factorable into,

    ##(1 + xi)(1 - xi) = 1 + x^2##

    That's where it enters the mathematical picture; by the simple act of
    trying to factor a sum. Anybody wish to add to this? Or even to point
    out other places it enters the mathematical picture, other than the
    factoring of sums? Have any ideas of your own, on how to "purge"
    complex numbers from your physics homework?

    I'm adding the words UBIT and U-BIT because I cant figure out how
    edit my tags.
     
    Last edited: Feb 19, 2014
  2. jcsd
  3. Why would anyone want to purge them from mathematics? They're useful theoretically and practically. If something better comes along then sure, they'll be forgotten, but I don't expect that to happen anytime soon.

    They entered the mathematical picture, historically speaking, when they became useful. Look into the cubic formula for more information. Very briefly, by manipulating square roots of negative numbers as if they "really" existed, mathematicians arrived at results that turned out to be correct. Even then, they still weren't really understood or fully trusted until people came up with a way to picture them geometrically.

    But we don't have to follow this same development though. Usually you'll see their origin explained by the reasoning "now, we still can't solve x+1=0, so we invent negatives, but then we still can't solve 2x=1, so we invent rationals, but...etc,etc." This is a nice, clean, and frankly very misleading explanation unless one makes it clear that this is not how it happened historically.
     
  4. Mentallic

    Mentallic 3,701
    Homework Helper

    Isn't [itex]\sqrt{1+x^2}[/itex] good enough?
     
  5. Probably, but only for government work. But try to factor it into
    two equal numbers...yes, "##c##" is the answer, but that's no fun.
     
  6. Perhaps there are some history buffs out there that can tell us when
    [itex]\sqrt{-1}[/itex] became "useful". "Necessity is the mother of
    invention", and all that tommyrot.
     
    Last edited: Feb 19, 2014
  7. SteamKing

    SteamKing 9,156
    Staff Emeritus
    Science Advisor
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    Don't ask a history buff, ask an electrical engineer, specifically one who works on alternating current.

    BTW, it is said that the Pythagoreans were so horrified to find that SQRT(2) was irrational, they swore themselves to secrecy about this unsettling knowledge. If they could even contemplate imaginary numbers, their heads would probably have exploded.
     
    Last edited: Feb 19, 2014
  8. Yah, who was that EE who first applied "phasors" to analysis of AC motors?
    Not Tesla, the other guy; not Edison either; he was a DC nut.
     
  9. Mentallic

    Mentallic 3,701
    Homework Helper

    Well [itex](1+ix)(1-ix)[/itex] isn't exactly factoring it into two equal numbers either. The expression isn't a perfect square in terms of elementary functions, sure, but I don't see how complex numbers changes any of that.
     
  10. Charles Proteus Steinmetz (AKA Karl August Rudolph Steinmetz).
    The patron saint of the GE motor business.
     
  11. Maybe the next best thing to finding a perfect root, is at least
    finding representable factors?
     
  12. I haven't seen the original article you referenced. But this strikes me as the idea of someone who was the unfortunate victim of bad math teaching.

    The number i is a gadget that represents a counterclockwise quarter turn of the plane. It's just a notation. i is a quarter turn to the left; i^2 is two one-quarter turns, which brings you to a point facing opposite of the way you started. i^3 goes another quarter turn, now you're pointing down. And i^4 brings you back to where you started.

    If you can conceptualize the ordered quartet (east, north, west, south), then complex numbers are just a notation that says "here's the angle to turn, and here's how far out you travel."

    Complex numbers are just a notation for rotations and stretches of the plane. This is so elementary that the invention of the notation of complex numbers was inevitable.

    It's a historical tragedy that complex numbers were discovered algebraically, and that certain complex numbers were labelled "imaginary." It's led to generations of students thinking they are unnatural; when in fact i is as natural as turning left. How could you ever abolish the notion of standing up and turning yourself in a circle, and looking near and then far? Those are the complex numbers.
     
  13. Truly fascinating...why didn't my high school algebra teacher put it that way?
     
  14. If you want the ultimate explanation of complex numbers, you should read Visual Complex Analysis by Tristan Needham. Best math book ever. He explains the origins of complex numbers and their evolution over time.

    You can get excepts from it free online if you google it and go to the book's webpage.
     
  15. SteamKing

    SteamKing 9,156
    Staff Emeritus
    Science Advisor
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    A reference to the particular article you are discussing is always appreciated.
    If everyone has a chance to read it, the discussions of it might be fuller.
     
  16. "Reality bits" in January 25-31, 2014 of NewScientist
     
  17. Personally, I don't think "i" is purge able. It's just a notation in mathematics
    following the rules:

    ##i^1## times a quantity puts the quantity onto the positive imaginary axis.

    ##i^2## times a quantity puts the quantity onto the negative real axis.

    ##i^3## times a quantity puts the quantity onto the neqative imaginary axis.

    ##i^4## times a quantity puts the quantity onto the positive real axis.

    and repeating,

    ##i^5## times a quantity puts the quantity onto the positive imaginary axis.

    ##i^6## times a quantity puts the quantity onto the negative real axis.
    .
    .
    .

    Can't think of a different way to do it, but my "thinking" or lack thereof,
    shouldn't be considered an obstacle.
     
  18. DrClaude

    DrClaude 2,327
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    I can't access the article (for some reason, there is a 30 day embargo imposed on our library for the electronic access to New Scientist), but it appears to be about the "u-bit":

    Antoniya Aleksandrova, Victoria Borish, William K. Wootters
    Real-Vector-Space Quantum Theory with a Universal Quantum Bit
    Phys. Rev. A 87, 052106 (2013)
    Arxiv preprint: http://arxiv.org/abs/1210.4535

    http://pitp.ca/videos/ubit-model-real-vector-space-quantum-theory
     
    Last edited by a moderator: Feb 20, 2014
  19. D H

    Staff: Mentor

    Your URL was incorrect, DrClaude. I fixed it.
     
  20. On another point...why does this message system keep adding
    tags that are essentially meaningless?

    IE, the tags "complex" and "remarks" and "numbers".
     
  21. That is my understanding as well. A nice fun book on the subject is "an imaginary tale" by Nahin (who was an EE prof, by the way).

    As an EE I use complex numbers all the time. If mathematicians completely drop the subject engineers will keep using (and teaching) them for a long time.

    jason
     
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