Introduction to complex numbers

Hello!

Finding out that many familiar programmers do not remember complex numbers or remember them very badly, I decided to make a small cheat sheet using formulas.


')
And students can learn something new;)
// Anyone interested interested under cat.

So, this complex numbers are numbers that can be written as

$$

$$x + iy$$

Where x, y are real numbers (that is, numbers that are familiar to everyone), and i is a number for which
equality is fulfilled

$$

$$i ^ 2 = -1$$

By the way, -i squared also gives -1.
So the statement that if the discriminant is negative, then there is no root is a lie.
Or rather, it is performed on a set of real numbers.



Ie we can write:

$$

$$z = x + yi$$

x is called the real part, y is the imaginary.

This is an algebraic form of a complex number.

There is also a trigonometric form for recording a complex number z:

$$

$$z = r (cos ϕ + i sin ϕ)$$

With the introduction, perhaps, everything.

We turn to the most interesting - operations on complex numbers!
To begin, consider addition.

We have two such complex numbers:

$$

$$z1 = 1 + 2i, z2 = 3 + 5i$$

How to fold them?
Very simple: add up the real and imaginary parts.
We get the number:

$$

$$z3 = 4 + 7i$$

It's simple, isn't it?
Subtraction is the same as adding.
You just need to subtract from the real part of 1 number the real part of 2 numbers,
and then do the same with the imaginary part.
Get the number

$$

$$z3 = -2-3i$$

Multiplication is done like this:

$$

$$z3.x = z1.x * z2.x-z1.y * z2.y$$

$$

$$z3.y = z1.x * z2.y + z1.y * z2.x$$

Let me remind you, x is the real part, y is imaginary.
The division is done like this:

$$

$$z3.x = (z1.x * z2.x + z1.y * z2.y) / (z2.x * z2.x + z2.y * z2.y)$$

$$

$$z3.y = (z1.y * z2.x - z1x * z2.y) / (z2.x * z2.x + z2.y * z2.y)$$

By the way, support for complex numbers is in the standard Python library:

z1=1+2j z2=3+5j z3=z1+z2 print(z3) #4+7i 

Instead of i, j is used.
By the way, this is because Python adopted the convention of electrical engineers who have
letter i stands for electric current.
Ask your questions, if any, in the comments.
I hope you learned something new.

UPD: In the comments asked to talk about the practical application.
So complex numbers have found wide practical application in aviation.
(wing lift) and electricity.
As you can see, a very necessary thing;)