alt Hacker NewsWhen 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.
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I particularly like this book by Steven Pemberton. https://homepages.cwi.nl/~steven/Numbers.pdf
Imaginary numbers never made sense to me in high school. But this book gradually builds it up in a way that made the conclusion positively intuïtive. Highly recommended!
To me their purpose only came clearly recently. They're just a way to rotate smoothly one dimension into another. And control the speed of the rotation.
x into y. Potential energy to kinetic energy. Charge into current.
Keeping this definition but going higher in the dimensions: x into y, y into z and z into x. Going meta, the x into y rotation defines a plane; And a plane going into another direction is a volume. (x into y) into z.
What made it click, what made it beautiful is Geometric Algebra:
I've come to understand it as something with both a magnitude and a phase. When you have a unity sine wave, it has two points per period where the magnitude is zero but those points have different phases.
I really was surprised on this when learning about these in Welch labs [1] and 3b1b.
[1] https://www.youtube.com/watch?v=T647CGsuOVU&list=PLiaHhY2iBX...
> I now feel like "complex" numbers etc. are the representation of data through multiple numbers where a single value (scalars) is not enough.
Ok, now find a single term that you think would adequately describe that concept.
I had a calculus prof who described it thus:
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.
Agreed, the naming system is atrocious. The one that really pisses me off at this point is that "real" numbers get to be called that. Imaginary numbers with rational coefficients are SIGNIFICANTLY more "real" than a converging infinite series, imo
So much of mathematics is just this.
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.