Question Number 185371 by Mastermind last updated on 20/Jan/23
$$\left(\mathrm{1}−\mathrm{z}\right)\left(\mathrm{1}−\overset{−} {\mathrm{z}}\right)\:=\:? \\ $$$$\mathrm{Where}\:\mathrm{z}\:\mathrm{is}\:\mathrm{the}\:\mathrm{complex}\:\mathrm{number} \\ $$$$ \\ $$$$. \\ $$
Answered by Frix last updated on 20/Jan/23
$${z}={a}+{b}\mathrm{i}\wedge\bar {{z}}={a}−{b}\mathrm{i} \\ $$$$\Rightarrow \\ $$$$\left(\mathrm{1}−{z}\right)\left(\mathrm{1}−\bar {{z}}\right)={a}^{\mathrm{2}} −\mathrm{2}{a}+{b}^{\mathrm{2}} +\mathrm{1}=\left({a}−\mathrm{1}\right)^{\mathrm{2}} +{b}^{\mathrm{2}} = \\ $$$$=\left(\mathrm{real}\:\left({z}\right)\:−\mathrm{1}\right)^{\mathrm{2}} +\left(\mathrm{imag}\:\left({z}\right)\right)^{\mathrm{2}} \\ $$