Formally defining complex numbers without recursion b/w ordered pairs and (assuming valid) extension of R.

I do not understand the addition of imaginary and real numbers, and hence any further operations between the two numbers. I understand that, if we have the situation such that x*x+1 = 0, it would make sense that x = sqrt(-1) or -sqrt(-1). Since there are no real numbers which when squared produce a number less than zero, it is understandable that this number must lie on a different number line. By this understanding, I am happy to take (sqrt(-1))^n, where n is some integer. However, to have the two number lines interact outside of x*x +1 = 0, I do not agree with. I do also find it reasonable to have x*x +a = 0 , where a is any positive number, to be valid.

I have also looked at the ordered pairs definition of complex numbers, and while they would definitely be perfectly consistent, the way multiplication is defined still relies on the identity for bare real numbers x*x +1 = 0. I would please ask of at least one of two things. Prove to me that it is completely consistent to add and multiply complex numbers together and that the ordered pairs definition of complex numbers can rely on i^2=-1 without needing to rely on and assumption that the algebra on real numbers is the same for complex numbers.

I was reading complex visual analysis and realised I had taken complex numbers and their operations for granted, and I hate that. I understand that if we assume complex numbers behave like normal numbers we can perform the same operations on them, like (a+bi)(c+di)=(ac-bd)+i(ad+bc). Also, that their absolute values are multiplied (or their magnitude) and their angles added together. From there you could derive the properties you want for defining complex numbers as R^2 or RxR. But with that kind of logic, I should just take the extensions of the complex numbers like quaternion, octonions, sedenions etc. and do whatever I want with them.

From my understanding, if we take the ordered pair definition, then at the same time it does not make sense to come to an ordered pair definition without already being confident that the (possibly inconsistent) complex numbers are consistent.

I have also re-evaluated my understanding of what a number is, and my understanding is that it is a point, on a line, from a place we define as the origin. And if I move away from the origin, I am displaced some distance, which changes in magnitude directly proportional to that distance.

I do not wish to simply accept that complex numbers work because someone told me so, as to me this would be perhaps the same as joining a cult. You slowly accept a new way of thinking, and choose not to think about that it might be invalid or inconsistent because it’s too hard

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Nevin Manimala

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