:PROPERTIES: :ID: 2553e0fb-12c1-42cc-8e47-54937a36e2c7 :mtime: 20231024010311 20231021010538 20231019013653 :ctime: 20231019013651 :END: #+title: complex number #+filetags: :public:project: * Definition of complex numbers \[\mathbb{C} =\{ a + b * i | a , b \in \mathbb{R} \}\] where $i^2 = -1$. * Imaginary unit We call $i$ the *imaginary unit*. \[i^{2} = -1\] \[\frac{1}{i} = -i\] * Real and imaginary part \[Re(a + b*i) = a\] \[Im(a + b*i) = b\] * Notation for complex numbers We often use the symbol $z$ to represent a complex number. We would use the notation \[z = x + iy\] * Uses of complex numbers complex numbers are used in [[id:0d2bae2f-5054-4aba-b1dc-ca704476764f][fundamental theorem of algebra]] * Polar Coordinates of Complex Numbers See more at [[id:9a9960ad-ea53-4d94-ac3b-3156da5d213e][polar representation of complex numbers]]. Let \[z = x + iy.\] ** Polar multiplication \[r_{1}e^{i\theta_{1}}r_{2}e^{i\theta_{2}} = (r_{1}r_{2})e^{i(\theta_{1} + \theta_{2})}\] \[|z_{1}z_{2}| = |z_{1}| |z_{2}|\] \[arg(z_{1}z_{2}) = arg(z_{1}) + arg(z_{2})\] ** Modulus \[r = mod(z) = |z| = \sqrt{x^{2} + y^{2}}\] ** Argument \[\theta = arg(z) = \arctan \frac{y}{x}\] Note that we must restrict $-\pi < \theta \le \pi$ The argument is NOT unique, but if we want to make the argument unique, we may restrict $-\pi < \theta \le pi$. This is called the *principal argument* * Operations on Complex Numbers ** Addition of Complex Numbers \[(a + bi) + (c + di) = (a+c) + (b + d)i\] ** Multiplication of Complex Numbers \[(a+b*i)*(c+d*i) = (ac-bd)+(ad+bc)i\] ** Complex Conjugate See [[id:f988565d-5b64-4e5e-b600-047ef4fce819][complex conjugate]] ** Division of Complex Numbers \[\frac{a+bi}{c+di} = \frac{ac+bd}{c^{2}+d^{2}} + \frac{bc-ad}{c^{2}+d^{2}}i\] \[\frac{r_{1}e^{i\theta_{1}}}{r_{2}e^{i\theta_{2}}} = \frac{r_{1}}{r_{2}}e^{i(\theta_{1}-\theta_{2})}\] \[\frac{z_{1}}{z_{2}} = \frac{z_{1}z_{2}^{*}}{|z_{2}|^{2}}\] * As a vector space the complex numbers form a 2-dimesnional [[id:5864974d-0edf-4757-9b1f-31b159c9aa7a][Vector]] space