keywords: magnitude, conjugate, de Moivre, Euler

Complex Number Identities and Formulas

Complex Number Identities and Formulas (info)
Basic Definitions
imaginary number $i=\sqrt{-1} \$
electrical engineers' imaginary number $j=\sqrt{-1}\$
(info) conjugate of a complex number $\text{if}\ z=a+ib,\ \text{for}\ a,\ b \in {\mathbb R},\ \text{then} \ \bar{z}=a-ib$
(info) magnitude of a complex number $\| z \| = \sqrt{ z \bar{z} }$
(info) magnitude of a complex number $\| z \| = \sqrt{\left(Re(z)\right)^2+\left(Im(z)\right)^2}$
(info) magnitude of a complex number $\| a+ib \| = \sqrt{a^2+b^2},\ \text{for}\ a,b\in {\mathbb R}$
(info) magnitude of a complex number $\| r e^{i \theta} \| = r,\ \text{for}\ r,\theta\in {\mathbb R}$
Complex Number Operations
addition $(a+ib)+(c+id)=(a+c) + i (b+d) \$
multiplication $(a+ib) (c+id)=(ac-bd) + i (ad+bc) \$
multiplication in polar form $\left( r_1 (\cos \theta_1 + i \sin \theta_1) \right) \left( r_2 (\cos \theta_2 + i \sin \theta_2) \right)= r_1 r_2 \left( \cos (\theta_1+\theta_2)+i \sin (\theta_1-\theta_2) \right)\$
division $\frac{a+ib} {c+id}=\frac{ac+bd} {c^2+d^2}+ i \frac{bc-ad} {c^2+d^2} \$
division in polar form $\frac{ r_1 (\cos \theta_1 + i \sin \theta_1)}{ r_2 (\cos \theta_2 + i \sin \theta_2) }= \frac{r_1}{ r_2} \left( \cos (\theta_1-\theta_2)+i \sin (\theta_1+\theta_2) \right)\$
exponentiation $i^n =\left\{ \begin{array}{ll}1,& \text{when }n\equiv 0\mod 4 \\ i,& \text{when }n\equiv 1\mod 4 \\-1,& \text{when }n\equiv 2\mod 4 \\-i,& \text{when }n\equiv 3\mod 4 \end{array} \right. \$
Euler's Formula and Related Equalities (info)
(info) Euler's formula $e^{iw_0t}=\cos w_0t+i\sin w_0t \$
A really cute formula $e^{i\pi}=-1 \$
Cosine function in terms of complex exponentials $\cos\theta=\frac{e^{i\theta}+e^{-i\theta}}{2}$
Sine function in terms of complex exponentials $\sin\theta=\frac{e^{i\theta}-e^{-i\theta}}{2i}$
Other Formulas
De Moivre's theorem $\left(\cos x+i\sin x\right)^n=\cos\left(nx\right)+i\sin\left(nx\right).\,$
Root of a complex number $\left( r (\cos x+i\sin x) \right)^{\frac{1}{n}}=r^{\frac{1}{n}} \cos\left(\frac{x+2 k \pi}{n}\right) +i\sin\left(\frac{x+2 k \pi}{n} \right), k=0,1,\ldots, n-1.\,$

Alumni Liaison

Basic linear algebra uncovers and clarifies very important geometry and algebra.

Dr. Paul Garrett