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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.
Right, but you need to specify that additional definition. Imaginary numbers are useful because they come pre-loaded with all those additional definitions about how to handle operations that use them.
I also find your hostile confusion unwarranted, given two other commenters have pointed out the same flaw in your argument that I have.
Because your arguments are just bizarre. Imaginary numbers do not have a priori definitions. Humans have to define imaginary number and define the mathematical operations on them. There is no "hostile confusion" or "flaw," there is you making the equivalent of flat-earth arguments but for mathematics. You keep claiming things that are objectively false and so obviously false it is bizarre how anyone could even make such a claim. I do not even know how to approach it, how on earth do you come to believe that complex numbers have a priori definitions and they aren't just humans defining them like any other mathematical operation? There are no pre-given definitions for complex numbers, their properties are all explicitly defined by human beings, and you can also define the properties on vectors. You at first claim that supposedly you can only do certain operations on complex numbers that you cannot on vectors, I point out this is obviously false and you can't give a single counter-example, so now you switch to claiming somehow the operations on complex numbers are all "pre-given." Makes zero sense. You have not pointed out a "flaw," you just ramble and declare victory, throwing personal attacks calling me "confused" like this is some sort of competition or something when you have not even made a single coherent point. Attacking me and downvoting all my posts isn't going to somehow going to prove that you cannot decompose any complex-valued operations into real numbers, nor is it going to prove that complex numbers somehow don't have to have their properties and operations on them postulated just like real numbers.
I'm being combative because I don't get how you don't understand our argument, and because I view claims like "You keep claiming things that are objectively false" to be hostile when they stem from a misunderstanding rather than a fault on my part.
Let me restate my main point: complex numbers can be defined as vectors with the necessary rules to define various operations, such as multiplication over them and how they relate to sqrt(-1). Those additional rules are just as important to their definition as their appearance as two real-numbered values is. Both vectors and complex numbers are defined by humans, but we have chosen to give them separate definitions, because each definition includes the rules defining these operations and relationships, and they are different between the two types of mathematical object.
And, for the record, I downvoted your posts that were hostile (not all of them) and responded in kind. It's a separate effort than trying to prove my point here.
What point is there to "prove"? Your argument now is just that we defined them differently therefore they are different. Which suggests a straw man to my original point as I never once implied or suggested that in mathematics, real and complex numbers don't have different definitions, that's not relevant to anything.
My point is that the way that you stick two real numbers together to make a complex number is important, and is unintuitive if you approach it as just two real numbers.