Complex Numbers
We found 17 formulas for complex numbers
Variables
z
→
Complex number
a, c
→
Real part
bi, di
→
Imaginary Part
n
→
Natrural number
i
→
Imaginary unit
5.
Multiplying complex numbers
$\left(a+bi\right)\left(c+di\right)=\left(ac-bd\right)+\left(ad+bc\right)i$
6.
Dividing complex numbers
$\dfrac{a+bi}{c+di}=\dfrac{ac+bd}{c^2+d^2}+\dfrac{bc-ad}{c^2+d^2}i$
8.
Polar representation of a complex number
$a+bi=r\left(\cos \psi + i \sin \psi \right)$
10.
Product in polar representation
$z_1 \cdot z_2 = r_1\left(\cos\psi_1+i\sin\psi_1\right) \cdot r_2\left(\cos\psi_2+i\sin\psi_2\right) = \\ = r_1r_2\left(\cos\left(\psi_1+\psi_2\right)+i\sin\left(\psi_1+\psi_2\right) \right)$
11.
Conjugate in polar form
$\overline{r\left(\cos\psi+i\sin\psi\right)}=r\left[\cos(-\psi)+i\sin(-\psi\right)]$
12.
Inverse in polar form
$\dfrac{1}{r\left(\cos\psi + i \sin \psi\right)}=\dfrac{1}{r}\left[ \cos(-\psi)+i\sin(-\psi)\right]$
13.
Qotient in polar form
$\dfrac{z_1}{z_2} = \dfrac{r_1\left(\cos\psi_1+i\sin\psi_1\right)}{r_2\left(\cos\psi_2+i\sin\psi_2\right)}=\dfrac{r_1}{r_2}\left(\cos(\psi_1-\psi_2) + i\sin(\psi_1-\psi_2)\right)$
14.
Power in ploar form
$z^n= \left[r \left(\cos\psi+i\sin\psi\right)\right]^n= r^n \left[\cos(n\psi)+i\sin(n\psi)\right]$
15.
De Moivre Formula
$\left( \cos\psi + i \sin\psi\right)^n= \cos(n\psi) + i \sin(n\psi)$
16.
N-th root of complex number
$\sqrt[n]z=\sqrt[n]{r(\cos\psi+i\sin\psi)} = \sqrt[n]r \left(\cos\dfrac{\psi+2\pi k}{n} + \sin\dfrac{\psi+2\pi k}{n} \right) \\[1.2 em] k=0,1,2,\dots,n-1$