Determine the set of all complex numbers Z for which Z, 1/Z, 1-Z have the same modules.
+6248 the word is modulus
\(|Z| = \left | \dfrac 1 Z \right | = | 1 - Z |\)
first off you should know that
\(\left | \dfrac 1 Z \right | = \dfrac {1}{| Z |} \\ \\ \text{so } |Z| = \left|\dfrac 1 Z \right| \Rightarrow |Z|=1\)
\(|1-Z|=1 \\ \\ \sqrt{(1-Z)(1-Z^*)} = 1 \\ \\ (1-Z)(1-Z^*)=1 \\ \\ 1 -Z -Z^* + Z Z^* = 1\)
\(\text{but }Z Z^* = |Z|^2 = 1 \text{ so} \\ 1-Z-Z^* + 1 = 1 \\ Z+Z^* = 1 \\ 2 Re(Z) = 1 \\ Re(Z) = \dfrac 1 2\)
\(1 = 1^2 = |Z|^2 = Re(Z)^2 + Im(Z)^2 = \left(\dfrac 1 2\right)^2 + Im(Z)^2 \\ \\ Im(Z)^2 = \dfrac 3 4 \\ \\ Im(Z) = \pm \dfrac{\sqrt{3}}{2} \\ \\ Z = \dfrac 1 2 (1 \pm i\sqrt{3})\)
