$Re(\frac{1}{z})=Re(\frac{1}{x iy})=Re(\frac{1}{x iy}*\frac{x-iy}{x-iy})=Re(\frac{x-iy}{x^{2} y^{2}})=Re(\frac{x}{x^{2} y^{2}}-\frac{iy}{x^{2} y^{2}})=\frac{x}{x^{2} y^{2}}$


$\frac{x}{x^{2} y^{2}}>1\Rightarrow x>x^{2} y^{2}\Rightarrow x^{2}-x y^{2}<0\Rightarrow x^{2}-x \frac{1}{4}-\frac{1}{4} y^{2}<0\Rightarrow (x-\frac{1}{2})^{2} y^{2}<\frac{1}{4}$

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$ax^{2} bx c=0$

$root=\frac{-b\pm \sqrt{b^{2}-4ac}}{2a}$


$z^{2} z (1-i)=0$

$a=1 , b=1 , c=1-i$


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$|ze^{\frac{\pi i}{3}}-z|=|z||e^{\frac{\pi i}{3}}-1|=|z||(cos\frac{\pi }{3} isin\frac{\pi }{3})-1|=|z||\frac{1}{2} i\frac{\sqrt{3}}{2}-1|=|z||-\frac{1}{2} i\frac{\sqrt{3}}{2}|=|z|\sqrt{(-\frac{1}{2})^{2} (\frac{\sqrt{3}}{2})^{2}}=|z|$