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Determine the set of all complex numbers Z for which Z, 1/Z, 1-Z have the same modules.

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Determine the set of all complex numbers Z for which Z, 1/Z, 1-Z have the same modules.

Sep 19, 2018

#1
+4779
+1

the word is modulus

$$|Z| = \left | \dfrac 1 Z \right | = | 1 - Z |$$

first off you should know that

$$\left | \dfrac 1 Z \right | = \dfrac {1}{| Z |} \\ \\ \text{so } |Z| = \left|\dfrac 1 Z \right| \Rightarrow |Z|=1$$

$$|1-Z|=1 \\ \\ \sqrt{(1-Z)(1-Z^*)} = 1 \\ \\ (1-Z)(1-Z^*)=1 \\ \\ 1 -Z -Z^* + Z Z^* = 1$$

$$\text{but }Z Z^* = |Z|^2 = 1 \text{ so} \\ 1-Z-Z^* + 1 = 1 \\ Z+Z^* = 1 \\ 2 Re(Z) = 1 \\ Re(Z) = \dfrac 1 2$$

$$1 = 1^2 = |Z|^2 = Re(Z)^2 + Im(Z)^2 = \left(\dfrac 1 2\right)^2 + Im(Z)^2 \\ \\ Im(Z)^2 = \dfrac 3 4 \\ \\ Im(Z) = \pm \dfrac{\sqrt{3}}{2} \\ \\ Z = \dfrac 1 2 (1 \pm i\sqrt{3})$$

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Sep 19, 2018
#2
+1

Thank you.

Guest Sep 20, 2018