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Patterns from complex numbers ! !

  1. Oct 13, 2011 #1
    Patterns from complex numbers ! URGENT!

    - Use de moivre's theorem to obtain solutions for z^n=i for n=3, 4 and 5.
    - Generalize and prove your results for z^n=a+bi, where |a+bi|=1.
    - What happens when |a+bi|≠1


    Relevant Equations:

    z^n = r^n cis (n\theta)

    r = \sqrt{a^2 + b^2}

    \theta = tan^{-1}(\frac{b}{a})


    I solved the first bullet:
    Here's my solution for n=4 as an example:



    z^4 - i = 0
    ==> z^4 = i
    ==> z = i^(1/4).

    So this is equivalent to trying to find the 4 fourth roots of i.

    In polar form, we see that:

    i = cos(π/2) + i sin(π/2).

    By De Movire's Theorem:

    i^(1/4) = cos[(π/2 + 2πk)/4] + i sin[(π/2 + 2πk)/4]
    ==> i^(1/4) = cos(π/8 + πk/2) + i sin(π/8 + πk/2) for k = 0, 1, 2, and 3.

    Thus:

    k = 0 ==> z = cos(π/8) + i sin(π/8)
    k = 1 ==> z = cos(5π/8) + i sin(5π/8)
    k = 2 ==> z = cos(9π/8) + i sin(9π/8)
    k = 3 ==> z = cos(13π/8) + i sin(13π/8)

    I don't know how to apply generalization in this case..
     
  2. jcsd
  3. Oct 13, 2011 #2

    I like Serena

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    Re: Patterns from complex numbers ! URGENT!

    Welcome to PF, Matricaria! :smile:

    I prefer the form of De Moivre's theorem (or rather Euler's formula) that says:
    [tex]a + i b = r e^{i\phi}[/tex]
    where [itex]a = r \cos \phi[/itex]
    and [itex]b = r \sin \phi[/itex]


    Can you solve your equation [itex]z^n = a+ i b[/itex] with this form of the theorem?

    It also effectively gives you the answer to the third question.


    If you are really required to use De Moivre's theorem as is, you can rewrite the exponential powers into cosines and sines.
     
  4. Oct 15, 2011 #3
    Re: Patterns from complex numbers ! URGENT!

    Thank you very much for your help!
    I was supposed to use De Moivre's as it is.. So I did use the sin and cos form..
    Anyways, I have another question..

    I can't find the modulus and argument of X^4=1?
    I used De Moivre's Theorem to find the four roots which are: 1, -1, i, and -i..
    I also plotted the argand diagram but can't calculate r and theta since all four points are on the axes...
     
  5. Oct 15, 2011 #4

    HallsofIvy

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    Re: Patterns from complex numbers ! URGENT!

    Then you have serious problems understanding what the "Argand diagram" is! Both "r" and [itex]\theta[/itex], for 1, -1, i, and -i, should be trival simply by looking at the Argand diagram. Now "calculation" required!
     
  6. Oct 15, 2011 #5
    Re: Patterns from complex numbers ! URGENT!

    We learned to calculate r using Pythagoras y'know after joining the lines into a right angle, and calculate theta using tan opp/adj!
    Can you elaborate??
     
  7. Oct 15, 2011 #6

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    Re: Patterns from complex numbers ! URGENT!

    You're welcome! :smile:


    Let's try and generalize this a bit.

    If you have z = x + i y, this corresponds to a point with coordinates x and y.
    You can draw your Pythagorean triangle using this point.
    Now your "adj" is equal to x, and your "opp" is equal to y.

    In other words:

    [itex]\tan \theta = {opp \over adj} = {y \over x}[/itex],

    and [itex]r^2 = x^2 + y^2[/itex].


    Now let's take for instance your solution "i".
    What is x? What is y?
    What would therefore r be?
    And what would [itex]\theta[/itex] be?
     
  8. Oct 15, 2011 #7
    Re: Patterns from complex numbers ! URGENT!

    x=0 and y=1

    theta = 90 degrees?
     
  9. Oct 15, 2011 #8

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    Re: Patterns from complex numbers ! URGENT!

    Yep! :smile:

    And r?
     
  10. Oct 15, 2011 #9
    Re: Patterns from complex numbers ! URGENT!

    r = 1!
     
  11. Oct 15, 2011 #10

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    Re: Patterns from complex numbers ! URGENT!

    Right! :!!)

    So what about 1, -1, and -i?
    Any thoughts?
     
  12. Oct 15, 2011 #11
    Re: Patterns from complex numbers ! URGENT!

    z=-1 ---> x=-1 , y=0
    r= 1 theta= 90 degrees as well

    z=i ----> x=0 , y=1
    r= 1

    z=-i ----> x=0 , y=-1
    r=1
    !!!!!!!

    What about theta when x=0???
    Because tan1/0 is an error...
     
  13. Oct 15, 2011 #12

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    Re: Patterns from complex numbers ! URGENT!

    Yep!


    Hmm, that can't be right.
    What is the tangent of 90 degrees?
    And what is y/x in this case?


    So what's the inverse tangent of 0.9999/0.0001?



    I recommend you take a look at the definition of the unit circle:
    http://en.wikipedia.org/wiki/Unit_circle

    I think it effectively answers your questions.
     
  14. Oct 16, 2011 #13
    Re: Patterns from complex numbers ! URGENT!

    When z= -1, r= 1 and theta = 0??

    And then z = i
    x=0 and y=1
    r=1 and theta= 90

    And when z=-i
    x=0 and y=-1
    r=1 and theta = -90
    ???
    Ok.. One last thing: I was supposed to come up with a conjecture after solvin' the three equations.. A conjecture for the distance/line segment, that is..
    This is what I came up with: z^n = r*cis pi (n-2)/n
    Is it any good???

    Thank you very much for your help btw.. You're such a great tutor :)
     
  15. Oct 16, 2011 #14

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    Re: Patterns from complex numbers ! URGENT!

    Almost.
    theta = 180 degrees.



    Yep!



    Uhh... :uhh:
    I don't get it...
    Which conjecture? :confused:


    :redface:
     
  16. Oct 16, 2011 #15
    Re: Patterns from complex numbers ! URGENT!

    Use de moivre's theorem to obtain solutions for z^3-1=0
    Use graphing software to plot these roots on an argand diagram as well as a unit circle with centre origin.
    Choose a root and draw line segments from this root to the other two roots.
    Measure these line segments and comment on your results.
    Repeat the above for the quations z^4-1=0 and z^5-1=0. Comment on you results and try to formulate a conjecture.
     
  17. Oct 16, 2011 #16

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    Re: Patterns from complex numbers ! URGENT!

    Oh.
    Didn't you in your original problem have z^n = i?
    Is this the same problem?

    Anyway, you seem to have skipped a step.
    Did you measure line segments from one root to another?
    Shouldn't your conjecture be about the length of such a line segment?

    What do you mean by z^n = r*cis pi (n-2)/n then?
    What is z supposed to be?
    And what is r supposed to be?
     
  18. Oct 16, 2011 #17
    Re: Patterns from complex numbers ! URGENT!

    Yeah, it's the same problems. I was workin' the roots for z^4=1 in the last line..

    I drew all three equations (z^3=1, z^4=1, and z^5=1) on an argand diagram, and on a unit circle..
    And I solved the three equations using De Moivre's and found 3, 4, and 5 roots for the three equations, respectively..
    But, this is as far as I can go..

    How do I measure the line segments???
     
  19. Oct 16, 2011 #18

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    Re: Patterns from complex numbers ! URGENT!

    Oh. Okay.

    Well, measuring is usually something you do with a ruler...
    But you should also be able to calculate them.
    We'll get to that in a moment.

    If you have plotted the solutions of z^5=1, did you see a pattern?
    Can you describe how two neighboring roots are geometrically related?
     
  20. Oct 16, 2011 #19
    Re: Patterns from complex numbers ! URGENT!

    I'll upload the three unit circles in a minute..

    My observation was that any root for z^n=1 lies on the unit circle, and that the roots are equally spaced from one another around the circle
     
  21. Oct 16, 2011 #20
    Re: Patterns from complex numbers ! URGENT!

    Those are the unit circles for the three equations
     

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