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Complex number - calculation

  1. Oct 6, 2017 #1
    • Member advised to use the homework template for posts in the homework sections of PF.
    Hi
    I was hoping some of you would give me a clue on how to solve this complex number task.
    z = (1 +(√3 /2) + i/2)^24 → x=(1 +(√3 /2), y= 1/2
    I think there must be some nice looking way to solve it.
    My way was to calculate |z| which was equal to [√(3+2√3)]/2 → cosθ = x/|z|, sinθ= y/|z|
    After using De Moivre's formula I got very awful result which was:
    z = |z|^24 * (cos(24*θ) + i sin(24*θ))
    Can you think of any better looking way to solve this ?
     
  2. jcsd
  3. Oct 6, 2017 #2

    haruspex

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    Have you tried simply squaring three times and cubing once?
     
  4. Oct 6, 2017 #3

    Ray Vickson

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    Work hard at determining ##\theta## as accurately as you can, because the result may surprise you. (Don't just give one or two decimal places; use as many places as is practical, or---even better----try for an "analytic", closed-form expression.)
     
  5. Oct 6, 2017 #4

    Mark44

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    It's not clear to me what you're trying to do.
    It appears that you want to write z in the form of x + iy. If so, x and y would not be as you show them above.
    In future posts, don't delete the homework template. Its use is required.
     
  6. Oct 7, 2017 #5

    haruspex

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    You can gain a lot of insight into the geometry by either squaring or cubing the expression then comparing the real and imaginary parts.
     
  7. Oct 7, 2017 #6
    Sorry about deleting it. I won't do so in the future. What I meant with this → was to show the represenstative form of x and iy as a part of "z"
     
  8. Oct 7, 2017 #7

    kuruman

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    Better yet, consider the quantity that you raise to the 24th power, ##z_1=\frac{\sqrt{3}}{2}+\frac{1}{2}i##. Can you find angle ##\varphi## such that ##z_1=e^{i \varphi}## ? Then raise to the 24th power.
     
  9. Oct 8, 2017 #8

    haruspex

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    Isn't that effectively what was tried in post #1? And it is 1+½√3+½i.
    Looks like TheColector had trouble finding the angle. This is a lot easier after a single squaring or cubing.
     
  10. Oct 8, 2017 #9

    kuruman

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    Sorry, I misread the parentheses. This makes the problem more interesting.
    $$1+\frac{\sqrt{3}}{2}+ \frac{1}{2}i=e^{2i\pi}+e^{i\pi/6}$$
    Can you write this as the product of two terms?

    (Edited to give less away)
     
    Last edited: Oct 8, 2017
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