Definitions:
A complex number is written as a+bi where a and b are real numbers an i, called the imaginary unit, has the property that i²= -1.
The complex numbers z=a+bi and z−=a−bi are called complex conjugate of each other.
Formulas:
Equality of complex numbers
\[a + b\,i = c + d\,i \Longleftrightarrow a = c ~~and~~ b = d\]
Addition of complex numbers
\[(a + b\,i) + (c + d\,i) = (a + c) + (b + d)\,i\]
Subtraction of complex numbers
\[(a + b\,i) - (c + d,i) = (a - c) + (b - d)\,i\]
Multiplication of complex numbers
\[(a + b\,i)\cdot(c + d\,i) = (ac - bd) + (ad + bc)\,i\]
Division of complex numbers
\[\frac{a + b\,i}{c + d\,i} = \frac{a + b\,i}{c + d\,i}\cdot\frac{c - d\,i}{c - d\,i} =
\frac{ac + bd}{c^2 + d^2} + \frac{bc - ad}{c^2 + d^2} \, i\]
Polar form of complex numbers
\[a + b\,i = r\cdot(\cos\theta + i\,\sin\theta)\]
Multiplication and division of complex numbers in polar form
\[\left[r_1\left(\cos\theta_1 + i \cdot \sin\theta_1\right)\right] \cdot
\left[r_2\left(\cos\theta_2 + i \cdot \sin\theta_2\right)\right] =
r_1 \cdot r_2 \left[ \cos\left(\theta_1+\theta_2\right) + i \cdot \sin\left(\theta_1+\theta_2\right) \right]\]
\[\frac{r_1\left(\cos\theta_1 + i\,\sin\theta_1\right)}{r_2\left(\cos\theta_2 + i\,\sin\theta_2\right)}=
\frac{r_1}{r_2} \left[\cos\left(\theta_1-\theta_2\right) + i \cdot \sin\left(\theta_1-\theta_2\right)\right]\]
De Moivre's theorem
\[\left[r \left( \cos\theta + i\,\sin\theta \right) \right]^n =
r^n \left( \cos ( n\theta) + i\,\sin (n\theta) \right)\]
Roots of complex numbers
\[\left[r \left( \cos\theta + i\,\sin\theta \right) \right]^{1/n} =
r^{1/n} \left( \cos \frac{\theta + 2k\pi}{n} + i\,\sin \frac{\theta + 2k\pi}{n} \right)
~~ k=0,1,\dots, n-1\]
