How to understand complex numbers?

Question

Mathematics

I can't figure out what a complex number is.
I perceive it as just such a structure

class {
int x, y;
}

For example, in Unity there is a class Vector3 and 2 - are these complex numbers?
In the examples that I looked at, complex numbers are drawn on the graphs exactly as in my example, and thus, the direction is set.
Help me please!

Answer 1

Maybe I'm wrong from the point of view of an advanced mathematician, but I represent complex numbers as two-dimensional numbers. Roughly speaking, for ordinary numbers, we can draw a straight line, where what is on the left is less than what is on the right. But for complex numbers, a plane is already needed, and the usual more / less do not work there. It is necessary to introduce new definitions for the new "greater" and the new "less".
Why is all this necessary? They make it easier to trigonometry and count all sorts of tricky things. Faster on paper. I don't know how to teach a commuter and programming languages ​​to perceive this matter. Perhaps this is just an abstraction, and there are no accelerations in calculations, only visually more pleasant for those people who are used to counting in complex numbers.
There is even more tinny, where the number is of the form a + bi + cj, well, or, for special aesthetes, when the number is the sum of unlimited total, living, katyh, and others.

Answer 2

Complex numbers are different from the usual 2d vector. A 2d vector is just an ordered pair of real numbers. And although a complex number is a 2d vector (in this sense), some more rules are set for it. For example, what can you say about multiplying 2d-by itself (well, that is, squaring)? Almost nothing. Because, in general, this operation is not defined for it (although no one interferes with redefining it, but it will no longer be a 2d vector). But for a complex number, squaring is very clear what it is.
If you follow your logic of representing a complex number as a structure, then a class with redefined operations is more suitable here: addition and subtraction (this is the same for a 2d vector), multiplication, division, degree, root, logarithm ...... plus we must not forget about the exponential form (although from the point of view of a 2d vector, this is just a length and orientation).
PS I read about different mathematicians, about how they work with mathematical abstractions. There are those who fantasize, imagine all this, and there are those who use an operational approach: "I don't know what it is, but I know how to work with it." The former open up new mathematical horizons, while the latter put new theories on solid scientific footing and write thick monographs. It is clear that these are extremes and usually every mathematician somehow visualizes for himself what he is doing ....

Answer 3

Open any math textbook for grades 10-11 and read what complex numbers are.
A complex number has the form a + bi , where a and b are the real part and i is the imaginary unit (the square of this unit is -1 ).
What you wrote is not a structure.

Answer 4

Everything is very simple

Answer 5

So it's on YouTube. Here, in more than detail, the geometric meaning of a complex number was presented.
https://www.youtube.com/watch?v=b3adw5igSzI
And more (there are Russian subtitles in English)
https://www.youtube.com/watch?v=T647CGsuOVU