Complex Numbers
Complex number is a couple of two real numbers (x, y).
We can think about complex numbers as points in the coordinate system.
Let z is a complex number.
z = (x,y)
x is the real part of z, and y is the imaginary part of z.
Complex numbers form C  the field of complex numbers.
The field of real numbers is its subset.
If we have two complex numbers z_{1} = (x_{1}, y_{1}) and z_{2} = (x_{2}, y_{2}) then:
z_{1} ± z_{2} = (x_{1}, y_{1}) ± (x_{2}, y_{2}) = (x_{1} ± x_{2}, y_{1} ± y_{2})
z_{1}z_{2} = (x_{1}, y_{1})(x_{2}, y_{2}) = (x_{1}x_{2}  y_{1}y_{2}, x_{1}y_{2} + y_{1}x_{2})

= 

= 

, 

Another way to write z is: z = x + iy,
x
is the real part of z,
y is the imaginary part and
i
is the imaginary unit. i^{2} = 1, i = √1.
Each complex number z = x + iy has its complex conjugate z = x  iy.
 z + z = 2x  real number;
 z  z = i2y  imaginary number;
 z.z = x^{2} + y^{2} = z^{2}  real number
Each complex number (x, y) have a relevant point in the coordinate system. We can not say point A > B, because of that we can not say for two complex numbers (x_{1}, y_{1}) > (x_{2}, y_{2}) It means that complex number have no ordering.
Addition, multiplication and division of complex numbers
Complex numbers addition:
Complex numbers subtraction:
Complex numbers multiplication:
Complex numbers division:
Polar form
The polar form of a complex number is:
z = r(cos(θ) + isin(θ)) = re^{iθ}
Here, z(or r) is known as the complex modulus(it is equal ot the measure of OM) θ is known as complex argument or phase. Тhe dashed circle above represents the complex modulus z of z and the angle θ represents its complex argument.
Let we have two complex numbers z_{1} and z_{2} in polar form:
z_{1} = r_{1}(cos(θ_{1}) + i.sin(θ_{1}))
z_{2} = r_{2}(cos(θ_{2}) + i.sin(θ_{2}))
then
z_{1}⋅z_{2} = r_{1}⋅r_{2}[cos(θ_{1} + θ_{2}) + i⋅sin(θ_{1} + θ_{2})]
$\frac{z_1}{z_2}=\frac{r_1}{r_2}[cos(\theta_1\theta_2)+i\cdot sin(\theta_1\theta_2)]$
Moivre's formulaes
The powers of a complex number:
z^{n} = r^{n}(cos(nθ) + i⋅sin(nθ))
Finding the root of a complex number:
$\sqrt[n]{z}=\sqrt[n]{r}(cos(\frac{\theta+2k\pi}{n})+i\cdot sin(\frac{\theta+2k\pi}{n}))$
k = 0, 1, 2,..., n1