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Home » But original problem said 64,000 — so proceed numerically: - Network4

But original problem said 64,000 — so proceed numerically: - Network4

📅 May 17, 2026 👤 scraface

May 17, 2026

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📌 $ z = \pm 1 $, both real, so imaginary part is 0.

📌 Roots from $ z^4 = -1 $:

📌 Write $ -1 = e^{i\pi} $, so:

📌 z = e^{i(\pi + 2k\pi)/4} = e^{i\pi(2k+1)/4}, \quad k = 0, 1, 2, 3

📌 So the roots are:

📌 $ k=0 $: $ e^{i\pi/4} = \frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2} $

📌 $ k=1 $: $ e^{i3\pi/4} = -\frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2} $

📌 $ k=2 $: $ e^{i5\pi/4} = -\frac{\sqrt{2}}{2} - i\frac{\sqrt{2}}{2} $

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