:PROPERTIES: :ID: f988565d-5b64-4e5e-b600-047ef4fce819 :mtime: 20231021013143 :ctime: 20231021013142 :END: #+title: complex conjugate #+filetags: :public:project: * Definition of complex conjugate Let $z$ be a [[id:2553e0fb-12c1-42cc-8e47-54937a36e2c7][complex number]] such that $z = x + iy$, then \[z^{*} = x - iy\] * Algebraic Properties of complex conjugate \[|z| = |z^{*}|\] \[arg(z^{*}) = -arg(z)\] \[z + z^{**} = 2Re(z)\] \[z - z^{**} = 2iIm(z)\] \[(z^{**})^{**} = z\] \[zz^{*} = |z|^{2}\]