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Z is a complex number. Find all solutions for equation z4=|z|.

 Oct 6, 2018
 #1
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\(z = r e^{i \theta},~0 \leq r, ~0 \leq \theta < 2\pi \\ z^4 = r^4 e^{i 4 \theta} = |z| = r = re^{i 2k\pi}, k \in \mathbb{Z} \\ r = 1,~ 4\theta = 2k \pi \\ \theta = 0,~\dfrac{\pi}{2},~\pi,~\dfrac{3\pi}{2} \text{ the solutions repeat after this} \\ z = 1, i, -1, -i\)

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 Oct 6, 2018
edited by Rom  Oct 6, 2018
 #2
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The fifth solution is z=0

Guest Oct 6, 2018
 #3
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Thank you!

 Oct 7, 2018

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