#algebra 

Комплексные числа - числа вида $\LARGE a+bi$, где $\LARGE a,b$ - действительные числа, а $\LARGE i$ - мнимая единица, такая, что 
$$\LARGE i^2=-1$$
Если $\LARGE a=0$, то число называется **чисто мнимым** (pure imaginary number).
Комплексные числа - НЕ СУММА.

## Свойства
Addition, substraction, and multiplication of complex numbers is to be done as if the two or more complex numbers you work with are simply binomials.

To divide compex numbers, multiply the fraction by a **complex conjugate** (if $\LARGE z=a+bi$, then its complex conjugate is $\LARGE z'=a-bi$)

Modulus of a complex number:
$$\LARGE |z|=\sqrt{a^2+b^2}$$
i.e. the length of the [[Вектор|vector]] $\LARGE \vec{z} (a,b)$


## Complex plane
To graph real numbers, one needs just the number line. However, since complex numbers have a real part and an imaginary one, one needs two axes to graph complex numbers:

![[Pasted image 20240917181037.png]]
![[Pasted image 20240917181248.png]]
### Trigonometric form
![[Pasted image 20240917202115.png]]
A complex number $\LARGE z=a+bi$ has the trigonometric form:
$$\LARGE z=|z|(\cos{\theta}+i\sin{\theta})$$
where $\LARGE \tan{\theta} = \frac{b}{a}$ - **argument of z**
$$\LARGE argz=\arctan{\frac{b}{a}}$$
![[Pasted image 20240917184456.png]]


> [!Writing complex numbers in the trigonometric form]-
> ![[Pasted image 20240917182602.png]]

Multiplication and division of complex numbers in trigonometric form:
$$\LARGE z_1 z_2 = r_1 r_2 (\cos(\theta_1+\theta_2)+i\sin({\theta_1+\theta_2}))$$

$$\LARGE \frac{z_1}{z_2}=\frac{r_1}{r_2}(\cos(\theta_1-\theta_2)+i\sin(\theta_1-\theta_2))$$
Proof is left as an exercise


### Exponential form
Euler obtained the Euler's formula for representing complex numbers:
$$\LARGE e^{i\theta}=\cos{\theta}+i\sin{\theta}$$
(proved using [[Ряд Тейлора|Taylor series]])
Therefore:
$$\LARGE z=|z|\cdot e^{i\theta}$$
## [[De Moivre's theorem]]

## Complex roots

^213d80

An **nth root** of a complex number $\LARGE z$ is any complex number $\LARGE w$ that $\LARGE w^n=z$

![[Pasted image 20240917190403.png]]
![[Pasted image 20240917190414.png]]


## Поле комплексных чисел
**[[Алгебраическое поле|Поле]] комплексных чисел** - всякое поле $\LARGE \mathbb{C}$, обладающее следующими свойствами:
1) содержит в качестве подполя $\LARGE \mathbb{R}$
2) содержит такой элемент $\LARGE i$, что $\LARGE i^2=-1$
3) минимально среди полей с этими свойствами (т.е. если $\LARGE K \subset \mathbb{C}$ - какое-либо подполе, содержащее $\LARGE \mathbb{R}$ и $\LARGE i$, то $\LARGE K=\mathbb{C}$)

>[!Вывод поля комплексных чисел]-
>![[Pasted image 20241118210614.png]]
>![[Pasted image 20241118210626.png]]
>![[Pasted image 20241118210638.png]]

[[Автоморфизм]] комплексного поля - комплексное сопряженное:

$\LARGE f: \mathbb{C} \rightarrow \mathbb{C}', z \rightarrow \bar{z}, \space\space\space\space\space(a, b) \in \mathbb{R}$