:PROPERTIES: :ID: 9a9960ad-ea53-4d94-ac3b-3156da5d213e :mtime: 20231021014457 :ctime: 20231021014406 :END: #+title: polar representation of complex numbers #+filetags: :public:project: * Polar representation of complex numbers We may represent [[id:2553e0fb-12c1-42cc-8e47-54937a36e2c7][complex number]]s using polar coordinates. ** Euler's Identity \[re^{i\theta} = r\cos\theta = ri\sin\theta\] * Operations over polar representations ** Polar multiplication \[r_{1}e^{i\theta_{1}}r_{2}e^{i\theta_{2}} = (r_{1}r_{2})e^{i(\theta_{1} + \theta_{2})}\] ** Modulus \[|re^{i\theta}| = r\] ** Argument \[arg(re^{i\theta}) = \theta + 2\pi n\] be warned that the argument is not unique. ** Polar Complex Conjugate See [[id:f988565d-5b64-4e5e-b600-047ef4fce819][complex conjugate]] \[(re^{i\theta})^{*} = re^{-i\theta}\] ** Polar Division of Complex Numbers \[\frac{r_{1}e^{i\theta_{1}}}{r_{2}e^{i\theta_{2}}} = \frac{r_{1}}{r_{2}}e^{i(\theta_{1}-\theta_{2})}\]