Complex numbers... Help needed!

Question

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Let $z$ be a complex number so that

$z+\frac{1}{z}=1$

Find z, then calculate $z^3$ .

edited by Mathematix1981 Mar 3, 2025
edited by Mathematix1981 Mar 3, 2025
edited by Mathematix1981 Mar 3, 2025

CPhill · Answer 1 · 2025-03-04T04:50:35

z + 1/z = 1

z^2 + 1 = z

z^2 - z = -1 complete the square on z

z^2 - z + 1/4 = -1 + 1/4

(z - 1/2)^2 = - 3/4

z - 1/2 = +/- sqrt [ -3] /2

z = [1 - sqrt (3) i ] / 2 or z = [ 1 + sqrt (3) i ] / 2

z^2 = [ 1 - 2sqrt(3)i - 3] / 4

z^2 = [ -1/2 -sqrt(3)i / 2]

z^2 = - [ 1 + sqrt (3) i ] / 2

z^3 = z * z^2

z^3 = - [ 1 - sqrt (3) i] / 2 * [1 +sqrt(3)i] / 2 = - [1 + 3] / 4 = -1

And

z^2 = [ 1 + 2sqrt (3)i -3 ] / 4 = [ -1 + sqrt (3)i ] / 2

z^3 = z * z^2 = [ 1 + sqrt (3) i ] / 2 * [ -1 + sqrt (3) i ] / 2 =

[-1 - 3 ] / 4 = -1