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Plotting complex numbers help!

  1. Mar 25, 2014 #1
    1. The problem statement, all variables and given/known data
    Let f(z) = z3-8 and g(z) = f(z-1). This information applies to questions 1-5.
    1. Express g(z) in the form g(z) = z3+az2 +bz + c
    2. Hence, solve g(z) = 0. Plot solutions on an Argand diagram.

    2. Relevant equations
    Factorisation
    i2=-1

    3. The attempt at a solution
    I have done question 1.
    g(z) = z3- 3z2 +3z - 9.
    With question 2 I think z = 3 or z = √3i or -√3i. But (assuming I am correct) I am not sure what I do next. I am sure it is the simplest solution in the world but I have looked at it for so long I have confused myself.

    Also as a side note, when plotting complex numbers is it a point or a line from the origin?
     
    Last edited: Mar 25, 2014
  2. jcsd
  3. Mar 25, 2014 #2

    HallsofIvy

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    Science Advisor

    Those are not quite correct- you don't have the real part of the last two. If you write u= z- 1, then [itex]g(z)= f(z- 1)= f(u)= u^3- 8= 0[/itex]. So, in polar form, [itex]u^3= 8= 2^3= 2^3(e^{2\pi ni})[/itex] so [itex]u= 2e^{2\pi ni/3}= 2(cos(2npi/3)+ isin(2n\pi/3))[/itex] with n= 0, 1, and 2.

    Those are the three points equally spaced around the circle with radius 2, centered on 0 with one (n= 0) the point 2 itself. In "a+ bi" terms, they are u= 2 (n= 0), [itex]u= -2+ (\sqrt{3})i[/itex] (n= 1), and [itex]u= -2- (\sqrt{3})i[/itex] (n= 2).

    Of course, z= u+ 1.

    In other words, you are trying to do a problem involving an "Argand diagram" without knowing what an "Argand diagram" is! Didn't it occur to you look it up? Each complex number, a+ bi, is plotted as the point (a, b) on an Argand diagram.
     
  4. Mar 25, 2014 #3

    Curious3141

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    Homework Helper

    He does. It just cancels out when you add the 1.

    The complex roots are wrong. The real part should be -1 for each, since they start out as -1/2 before you multiply by 2.

    And so the real parts cancel out, giving what the threadstarter got.

    EDIT: Thought better of it, while the question is ambiguous, it seems the De Moivre's solution is probably the insight they were looking for.

    In most of the places I've looked, the complex number is represented either by a line segment or an arrow pointing away from the origin. Not just a point. Of course, the line segment or arrow terminates at (a,b).
     
    Last edited: Mar 25, 2014
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