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# Complex sum

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What is the value of the sum $$\sum z$$ where z ranges over all 7 solutions (real and nonreal) of the equation z^7 = -1?

Jul 12, 2022

#1
+193
-1

Use roots of unity!

z^7 = 1 x (-1)^7

$$z_k = -1 \cdot e^{\frac{2k\pi}{n}i}$$

The sum of the roots of unity is 0 by vieta's formula. So -1(0) = 0.

Jul 12, 2022