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What's the point of imagery numbers?

  1. Sep 25, 2010 #1

    Femme_physics

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    You can't really do anything with them since they're not real. Did mathematicians come up with it to give students some junk study material to bulk up math?
     
  2. jcsd
  3. Sep 25, 2010 #2

    Mentallic

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    lol yes definitely this. :biggrin:

    By the way, it is imaginary - not imagery - numbers

    I've heard that complex numbers help in quite a lot of aspects in the real world - although I haven't personally experienced them as of yet. They are imaginary numbers, but their use solves real problems.

    We can use imaginary numbers to convert multiples of cos and sin into powers of cos and sin instead. For example cos(2x)=2cos2x-1. Of course there are other ways besides using complex numbers to give the same result, but complex numbers can make things much easier. Same goes with some integration, and many others.

    Ages ago people believed that negative numbers were useless too, they had no physical significance because mass, distance, volume etc. are all positive. They had no use for them, but as you know we have found many uses for them. Thinking about it, most real world problems can be solved without negative numbers but we use these negative numbers to make things more simple. Rather than saying 2km forward and 3km backward, we use positive to describe forward and negative to describe backwards.

    But then again I could just be wrong and negative numbers were invented by mathematicians just like the complex numbers were to bombard students with more useless stuff to fill their time with :tongue:
     
  4. Sep 25, 2010 #3

    Femme_physics

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    Ahem... yes, I know it's imaginary and not imagery...:P the reason I typed imagery in is called indiscretion... and I can't edit title...

    Anyway, it's just my first run-up with them...and it seems likely they have no use... just wanted to find out if that's the case...which it seems to be!
     
  5. Sep 25, 2010 #4
    "What's the point of imagery numbers?"

    What's the point of 'zero'?
     
  6. Sep 25, 2010 #5
    There is an association which invents new stuff just to annoy student.... ah... just kidding.

    Complex numbers are incredibly important in maths and physics. With complex numbers many integrals are easily solved. Complex numbers and the Fourier transform help to solve differential equations which are basis of just everything in the real world.
    The equalizer on your stereo that shows the strength of the individual sound frequencies uses complex numbers to do the calculation. In physics all of quantum mechanics is based on complex numbers so they are the reason why you have all the electronic high tech equipment. In fact, you will (have?) learn(ed) at school that electric circuits are best handled with complex numbers.

    BTW: Search for complex numbers (and maybe "fluid dynamics") on this forum. There has been an extensive thread before.
     
  7. Sep 25, 2010 #6

    Femme_physics

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    The point of zero is to indicate nothing. But...that still doesn't answer my question.
     
  8. Sep 25, 2010 #7

    Hurkyl

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    You're confusing two pairs of homonyms -- the mathematical terms "real" and "imaginary" have pretty much nothing to do with the English words "real" and "imaginary".


    Complex numbers were originally invented because the solution method to find the real roots of the cubic equation involved taking square roots of negative numbers.

    Subsequently, they have been found to have better algebraic, geometric, and analytical properties than the real numbers. Therefore, even if you are only asking questions about real numbers, they still often come into play because they make it easier to answer the question.



    Since people like physics, I'll point out examples of their utility in that field:
    • The impedance of an AC circuit is a complex number
    • Signals are often measured as complex numbers
    • Complex numbers are useful for studying two-dimensional ideal fluid flow
    • Physical states and operators of quantum mechanics are firmly entrenched in the realm of complex linear algebra
     
  9. Sep 25, 2010 #8

    HallsofIvy

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    Historically, a major impetus for developing complex numbers ("imaginary numbers" are a subset) was Cardano's formula for solving cubic equations. It involves taking the third root of a square root of combinations of the coefficients. There are cases in which a cubic equation has real roots but using Cardano's formula to find those roots requires taking the square root of a negative number. The imaginary numbers cancel out in the end but are required in the formula.
     
  10. Sep 25, 2010 #9
    But don't worry, Dory. Currently the student-occupation-agency is working on Surreal Numbers
    http://en.wikipedia.org/wiki/Surreal_number
    which are even more complicating. So as soon as you grasp complex numbers there is more to come :-D
     
  11. Sep 25, 2010 #10
  12. Sep 25, 2010 #11
  13. Sep 25, 2010 #12
    strictly speaking a number only has magnitude.
    A number with magnitude and direction is actually an ordered pair of numbers.
    There are many fields that use ordered pairs of numbers.
    The field of complex numbers is just one.

    After you learn about complex numbers then you can move on to quaternions. They are used in rotations or something like that.

    (I am using the word 'field' loosely here)
     
  14. Sep 25, 2010 #13

    Hurkyl

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    Strictly speaking, this statement is only true if you are restricting your usage of the term "number" to refer to sets like the non-negative real numbers, or similar objects. (assuming I have interpreted your use of "only has" correctly.

    Many prefer to use the a meaning appropriate to the situation, rather than adopt some dogmatic principle.



    Just because you can analyze doesn't mean you should analyze -- don't forget the importance of synthesis.

    (Also, don't confuse the analysis with the original object of study!)
     
  15. Sep 25, 2010 #14

    LCKurtz

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    They are just as real as the "real" numbers are. You might find

    "There's nothing imaginary about complex numbers" interesting. See:

    http://math.asu.edu/~kurtz/complex.html
     
  16. Sep 26, 2010 #15

    Mentallic

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    Imaginary numbers are useful, but this doesn't change the fact that they're still imaginary.
     
  17. Sep 26, 2010 #16
    The point he is trying to make is that the real numbers themselves are imaginary. Numbers (of any kind) are abstract concepts. The reason why people might not consider the integers as abstract is because they use them all the time. It doesn't make them any less 'imaginary' than the 'imaginary' numbers. This is why a lot of people simply refer to the imaginary numbers as Complex numbers.
     
  18. Sep 26, 2010 #17
    I’ll answer your question if you tell me for f(x)=x^2 + 1 what values f(x)= 0
     
  19. Sep 27, 2010 #18
    negative square root of 1

    sorry, instantly had an answer for once o_o
     
  20. Sep 27, 2010 #19

    Mentallic

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    negative square root of 1 is [tex]-\sqrt{1}[/tex] which isn't the same as square root of negative 1 :tongue:
     
  21. Sep 27, 2010 #20
    -_- w.e
     
  22. Sep 27, 2010 #21

    Hurkyl

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    "None!" would be the answer given both by someone who refuses to acknowledge the complex numbers, and by someone who interpreted the domain of f as being real numbers.

    (of course, the latter would acknowledge that extending the domain to the complex numbers is useful, and there there would be two roots)
     
  23. Sep 27, 2010 #22

    epenguin

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    It is a pity if a student is introduced to them or spends much time with them without the applications, I can understand Dory's reaction in that case. Temporary I hope. I found myself saying on PF only yesterday "But if you ask about how to solve them algebraically... well you first need to be fairly at ease handling imaginary and complex numbers. These are not that difficult and quite fun if anything in maths is."

    They were first introduced as HallsofIvy says to solve cubic equations. It did seem like a crazy, purely formal idea (genius if there ever was) at first, and actually for a long time mathematicians held their noses at them. They had to use them as they couldn't get the results otherwise, but they thought they would eventually be able to replace them with some better idea, or at least there would just be an equivalent, whether less convenient or not, formulation with just real numbers. However I have read it has been proved that if we cannot solve cubics without them it is not because no one has been smart enough to see how, but that it is actually impossible. I don't know how advanced that proof is.

    Another example of how this unreal concept turns up in real things, the best way to figure out the general problem of in how many ways can you change a dollar uses complex numbers.

    One place you can learn about them is here https://www.physicsforums.com/showpost.php?p=2895403&postcount=14 I remember the author wrote (rough quotation from memory) "they make us feel we have been shown behind the scenes" .
     
  24. Sep 27, 2010 #23
    What is an apple? What is 'it'? What is 'is'? now we're getting profound
     
    Last edited: Sep 27, 2010
  25. Sep 27, 2010 #24
    And then my answer to them would be, wouldn’t it be useful to be able to solve this problem?
     
  26. Sep 27, 2010 #25
    The problem is solved, however, by saying there are no solutions. That is in fact the answer to the problem if we're dealing with real numbers. It's kinda like, what number is bigger than all the integers? You'll probably say "None!" but I can just easily say the +infinity in the extended reals is bigger than all of them. Your answer, however, is correct if we are dealing with only real numbers.

    The real interest of complex number was not to solve x^2 + 1 but rather, as someone else pointed in the thread, to explain some of the strange things that happened with Cardano's formula. Sometimes, the formula involved square roots of negative numbers when the roots were real, and thus to understand what was going complex numbers were developed. There's actually a good book on this, "Square root of -1: An Imaginary Tale."
     
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