Problem: Find all complex numbers $z$ such that $|z|=\frac{1}{|z|}=|1-z|$.
Basically I have an idea how to solve this and I get $x=\frac12$ but how should I express it mathematically? Should I go and find $y$ also?
[Math] Find all complex numbers $z$ such that $|z|=\frac{1}{|z|}=|1-z|$
complex numbers
Best Answer
$$|z|=\frac1{|z|}\Rightarrow |z|=1,$$ $$|z|=|1-z|\Rightarrow \sqrt{x^2+y^2}=\sqrt{(1-x)^2+y^2}\Rightarrow x=\frac12,$$ where $z=x+iy$. Next, we will determine the imaginary part $$\sqrt{\left(\frac12\right)^2+y^2}=1\Rightarrow y=\pm \frac{\sqrt{3}}2.$$ Therefore solutions are $$z_1=\frac12+i\frac{\sqrt{3}}2,$$ $$z_2=\frac12-i\frac{\sqrt{3}}2.$$