# Complex Numbers

A **complex number** is a couple of two real numbers (x, y).

We can think about complex numbers like points on the coordinate plane.

Let z be a complex number,

i.e. z = (x, y)

x is the **real** part of z, and y is the **imaginary** part of z.

Complex numbers are denoted by $\mathbb{C}$

The set of real numbers is its subset. Real numbers written as complex are $(x, 0), \ \ x \in \mathbb{R}$

Each complex number (x, y) have a relevant point on the coordinate plane. We can not write point A > B, because of that we can not
write for two complex numbers (x_{1}, y_{1}) > (x_{2}, y_{2})

**Complex numbers have no ordering.**

Let z_{1} = (x_{1}, y_{1}) and z_{2} = (x_{2}, y_{2}) be two complex numbers then:

$z_1 = z_2 \Leftrightarrow x_1 = x_2$ and $y_1 = y_2$

$z_1 \pm z_2 = (x_1, y_1) \pm (x_2, y_2) = (x_1 \pm x_2, y_1 \pm y_2)$

$z_1z_2 = (x_1, y_1)\times (x_2, y_2) = (x_1x_2 - y_1y_2, x_1y_2 + y_1x_2)$

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

**Another way to write** the complex number z = (x, y) is: z = a + bi,

a
is the real part of z,

b is the imaginary part and

i
is the imaginary unit. i^{2} = -1, i = √-1.

Each complex number z = a + bi has its complex conjugate z = a - bi.

- z + z = 2a - a real number;
- z - z = 2bi - an imaginary number;
- z ⋅ z = a
^{2}+ b^{2}= |z|^{2}- a real number

#### Addition, multiplication and division of complex numbers

Let (a + bi) and (c + di) be two complex numbers.

Complex numbers *addition*:

Complex numbers *subtraction*:

Reals are added with reals and imaginary with imaginary.

Complex numbers *multiplication*:

Complex numbers *division*:

$\frac{a + bi}{c + di}=\frac{(ac + bd)+(bc - ad)i)}{c^2+d^2}$

Rule | Equivalent | Exponent |
---|---|---|

$i^1 = i$ | $i^{4n + 1} = i$ | Multiple of 4 + 1 ${4n + 1, \ n \in \mathbb{Z}} = {1; 5; 9...}$ |

$i^2 = -1$ | $i^{4n + 2} = -1$ | Multiple of 4 + 2 ${4n + 2, \ n \in \mathbb{Z}} = {2; 6; 10...}$ |

$i^3 = -i$ | $i^{4n + 3} = -i$ | Multiple of 4 + 3 ${4n + 3, \ n \in \mathbb{Z}} = {3; 7; 11...}$ |

$i^4 = 1$ | $i^{4n} = 1$ | Multiple of 4 ${4n, \ n \in \mathbb{Z}} = {4; 8; 12...}$ |

#### 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