Complex numbers would be a much easier concept to understand if they didn't call it "complex" numbers or use the term "imaginary" numbers. Both of those make it much harder to digest because now you think that it's not real. What they really are are 2 dimensional numbers, meaning they have 2 components to each number.
The same goes for the term "real" numbers. Sometimes math, and its descendants like physics and computer science, can be too cutesy with their names that make it hard for regular people to understand or accept.
I've been teaching math for 35 years, which I admit, doesn't say much, but I assure you the difficulties students have with complex numbers by and large have nothing to do with the name. It's the same with everything else: lack of preparation in arithmetic and basic algebra.
When I first learnt complex numbers in school, I couldn't understand their purpose or why they are called imaginary etc.
I checked where they are used and found examples of current, magnetism etc. But I couldn't understand, electricity is a measurable physical quantity, how can you use imaginary numbers there??
Then I searched internet and found a Stack Exchange answer telling the story and how, instead of, complex (sounds like difficult) or imaginary (sounds like not useful practically), if we had better terms for them, then it would have been easier for beginners to understand them more intuitively.
It's completely true in my case.
After learning usage of vectors, thanks to Godot, for rotation and movement, and then about matrix etc. I now feel like "complex" numbers etc. are the representation of data through multiple numbers where a single value (scalars) is not enough. I may not be right in strict mathematics terms, but at least that's how I was able to understand them.
I think it would have been clearer if they weren't called "numbers".
For example, look at matrices [1]: no one asks themselves whether they are "real" or not. We know that we have defined operations on arrays of numbers which are useful to some degree.
If complex numbers were just called "complexes", a lot of confusion would just wither away.
Also, interestingly, quaternions and octonions aren't called "numbers".
[1] Complex numbers may, of course, be seen as special kinds of matrices, but this is beside my point here.
Calculus is a mathematical model of the world, and it is wrong in the sense of the famous quote about "all models [being] wrong, but some are useful", so mathematicians used the imaginary unit almost like a hotfix for their model.
Whether that's an accurate/rigorous statement or not, it helped me as a learner.
You can, of course, express things like oscillation without i, using vectors and matrices. But the ability to use Euler's formula to express complicated sinusoids as compact exponentials with base e simply makes the analysis much easier. In the words of my prof, it's merely a mathematical structure to express things more easily that "regular calculus" isn't good at expressing.
To me, that made the apparent paradoxes of expressing e.g. phase angles and reactive power and stuff using i a lot easier to stomach.
The explanation is a bit confusing, since it basically reduces to "some equations don't have solutions, so we have to invent them", which, if you already accept, you don't need complicated equations, just x^2=-1.
One thing which would have worked better (which is briefly mentioned) is Cardano's casus irreducibilis, but that would have needed more work.
There is proof of this theorem in it that blew my mind some years ago [1]. Someday I want to finish reading the rest of the book.
> Starting with a given convex quadrilateral, construct a square, external to the quadrilateral, on each side. Van Aubel's theorem states that the two line segments between the centers of opposite squares are of equal lengths and are at right angles to one another.
Really? Of approximately one billion good, visual introductions to complex numbers, we're going to pick an article that appears to be substantially written by an LLM?
The same goes for the term "real" numbers. Sometimes math, and its descendants like physics and computer science, can be too cutesy with their names that make it hard for regular people to understand or accept.
I checked where they are used and found examples of current, magnetism etc. But I couldn't understand, electricity is a measurable physical quantity, how can you use imaginary numbers there??
Then I searched internet and found a Stack Exchange answer telling the story and how, instead of, complex (sounds like difficult) or imaginary (sounds like not useful practically), if we had better terms for them, then it would have been easier for beginners to understand them more intuitively.
It's completely true in my case.
After learning usage of vectors, thanks to Godot, for rotation and movement, and then about matrix etc. I now feel like "complex" numbers etc. are the representation of data through multiple numbers where a single value (scalars) is not enough. I may not be right in strict mathematics terms, but at least that's how I was able to understand them.
For example, look at matrices [1]: no one asks themselves whether they are "real" or not. We know that we have defined operations on arrays of numbers which are useful to some degree.
If complex numbers were just called "complexes", a lot of confusion would just wither away.
Also, interestingly, quaternions and octonions aren't called "numbers".
[1] Complex numbers may, of course, be seen as special kinds of matrices, but this is beside my point here.
Calculus is a mathematical model of the world, and it is wrong in the sense of the famous quote about "all models [being] wrong, but some are useful", so mathematicians used the imaginary unit almost like a hotfix for their model.
Whether that's an accurate/rigorous statement or not, it helped me as a learner.
You can, of course, express things like oscillation without i, using vectors and matrices. But the ability to use Euler's formula to express complicated sinusoids as compact exponentials with base e simply makes the analysis much easier. In the words of my prof, it's merely a mathematical structure to express things more easily that "regular calculus" isn't good at expressing.
To me, that made the apparent paradoxes of expressing e.g. phase angles and reactive power and stuff using i a lot easier to stomach.
One thing which would have worked better (which is briefly mentioned) is Cardano's casus irreducibilis, but that would have needed more work.
> Starting with a given convex quadrilateral, construct a square, external to the quadrilateral, on each side. Van Aubel's theorem states that the two line segments between the centers of opposite squares are of equal lengths and are at right angles to one another.
https://en.wikipedia.org/wiki/Van_Aubel%27s_theorem
It can be used to convince students that they are useful.
https://pratik.deoghare.com/complex-mars-rover.html