# 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:

_{1}= z

_{2}<=> x

_{1}= x

_{2}

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})

$\frac{z_1}{z_2}=\frac{(x_1, y_1)}{(x_2, y_2)}=\big(\frac{x_1x_2+y_1y_2}{x_2^2+y_2^2}, \frac{x_2y_1-x_1y_2}{x_2^2+y_2^2}\big)$

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:

^{2}+ d

^{2})

#### Polar form

The polar form of a complex number is:

^{iθ}or

z = r(cos(θ) + isin(θ)) = re

^{iθ}

Here, |z|(or r) is known as the complex modulus

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 formulas

The powers of a complex number:

z^{n} = r^{n}(cos(nθ) + i⋅sin(nθ))

Finding the nth 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,..., n-1