I am having genuine difficulty imagining in your head how you think you made a point here. It seems you’re claiming that given if two vectors have the same symbols between them, they should have identical output, such as (a,b) * (c,d) should have the same mathematical definition as (a+bi) * (c+di), or complex numbers are not reducible to real numbers.

You realize mathematical symbols are just conventions, right? They were not handed down to us from Zeus almighty. They are entirely human creations. I can happily define the meaning of (a,b) * (c,d) to be (ac-bd,ad+bc) and now it is mathematically well-defined and gives identical results.

Negative numbers are just real numbers with a symbol attached. Yes, that’s literally true. In computer code we only ever deal with 0s and 1s. We come up with a convention to represent negative numbers, they are still ultimately zeros and ones but we just say “zeros and ones in this form represent a negative number,” usually just by having the most significant bit 1. They are no physical negative numbers floating out there in the world like in a Platonic sense. What we call “negative” is contextual. It depends upon how we frame a problem and how we interpret a situation. You can lose money at a casino and say your earnings are now negative, or you can say your losses are now positive. Zeus isn’t going to strike you down for saying one over the other. There is nothing physically dictating what convention you use. You just use which convention you find most intuitive and mathematically convenient given the problem you’re trying to describe.

Yes, when we are talking about how computers work, we are talking about how numbers actually manifest in objective, physical reality. They are not some magical substance floating out there in the Platonic realm. Whenever we actually go to implement complex numbers or even negative in the real world, whenever we try to construct a physical system that replicates their behavior and can perform calculations on a physical level, we always just use unsigned real numbers (or natural numbers), and then later establish signage and complexity as conventions combined with a set of operations on how they should behave.

I’m not sure your point about fractional numbers. If you mean literally a/b, yes, there is software that treats a/b as just two natural numbers stitched together, but it’s actually a bit mathematically complicated to always keep things in fractional form, so that’s incredibly rare and you’d only see it in very specialized math software. Usually it’s represented with a floating point number. In a digital computer that number is an approximation as it’s ultimately digital, but I wouldn’t say that means only digital numbers are physical, because we can also construct analogue computers that can do useful computations and are not digital. Unless we discover that space is quantized and thus they were digital all along, then I do think it is meaningful to treat real numbers as, well, physically real, because we can physically implement them.

uh… broski… you do realize a vector of two real numbers can be rotated… right? Please give me a single example for a supposed impossible operation to do on a vector of two real numbers that you can do on complex numbers. I can just define v² where v is a vector (a,b) as (a,b)²=(a²-b²,2ab). Okay, now I’ve succeeded in reproducing your supposedly mathematically impossible operation. Give me another one.

A complex number is just two real numbers stitched together. It’s used in many areas, such as the Fourier transform which is common in computer science is often represented with complex numbers because it deals with waves and waves are two-dimensional, and so rather than needing two different equations you can represent it with a single equation where the two-dimensional behavior occurs on the complex-plane.

In principle you can always just split a complex number into two real numbers and carry on the calculation that way. In fact, if we couldn’t, then no one would use complex numbers, because computers can’t process imaginary numbers directly. Every computer program that deals with complex numbers, behind the scenes, is decomposing it into two real-valued floating point numbers.