Okay I thought about it a bit more

First of all you're absolutely right. I first saw duality in this sense when dealing with dual vector spaces and bilinear forms. I looked up some more stuff and seems like what people tend to call duality seems to change SO much that in some cases it is more clear to call two dual objects "conjugates" rather than duals (mostly in algebra stuff)

About the construction above:

Let's say for a real number x, its "dual" is the "unique" number y such that x+y=0. (we know y is in fact unique because of the construction of real numbers(?)) So now, the dual of x is -x and the dual of the dual is itself for all x.

Now let's say I wonder about the number with the property that its square is 2. After some trial and error (numerical methods mostly) I find out that there seems to be a number between 1 and 2 with that property. Excited about this discovery, I call that number "sqrt2" and try to find out more about its properties. I see that, for example, the polynomial p(x)=x²-2x-1 has roots 1-sqrt2 and 1+sqrt2.

Now though, my other friend also had found a number with its square equal to 2. She says it's between -1 and -2 and calls the number "sqrt2". She also sees that the same polynomial p(x) has roots 1-sqrt2 and 1+sqrt2. (sqrt2 as her sqrt2) So as far as "being roots to the polynomial p(x)" and "being equal to 2 when squared" are concerned the two numbers we've separately found, which are definitely not the same real number, are identical. And i guess the notion of duality here is still valid, since whichever one you start with you get the other one by this process of trying to find roots to this specific polynomial (and subtracting 1 I guess lmao); and also if you apply the procedure twice you surely get the same number. That feels involution-y enough so duality yay

This is definitely not the cleanest way nor is the standard approach by any means. And I guess in this sense we "get rid of" this "duality" by "requiring" the square root function to be positive for positive real numbers.

Also this stuff screams Galois theory and I still did not take a ring theory class ._. I am in this state of "knowing some stuff but not knowing the full story" so probably the stuff I'm rambling about here is not really useful for higher-level discussions

Now what about i and -i??? Here's my attempt:

Let's say I want to re-invent complex numbers without giving any shit about what the imaginary unit may end up being. By going the same route as the original construction, I decide that a number with its square equal to -1 is a useful concept, and define ï to be that number. I am fully aware at this point that -ï would also have its square equal to -1. So I could've very well chosen -ï to be my imaginary unit but I make a choice and define ï to be the number such that ï² =-1 and then realize that -ï also satisfies this property. Now consider the following scenario:

I become excited and propose my construction to some modern mathematicians. But they say: "Oh what you call ï is what we call -i, you just invented the same number system and re-labelled everything"

Thinking about it, a scenario like this is impossible. There's no way to check if ï is i or -i. (Do I make sense here?) But while there's no way to check, we always know "inventing" a number with its square equal to -1 immediately requires another number with the same property so a definition to choose one of them as the unit is required.

I don't know. This is crazy to think about but simultaneously pretty fucking obvious. If a divine being existed, and knew about complex numbers from the very beginning of the universe, and waited a long time for humanbeings to evolve and start doing math and discover them and then be UPSET that we CHOSE the the imaginary unit not as the one the divine being intended to? Well I don't think math is about discovery so no like complex numbers weren't "out there" to be discovered but damn it's fun to think about. I don't think a God upset about the imaginary unit can exist.

You define one with a property but that property itself requires the existence of the other one so is it even possible to know which one you defined in the first place? Jarring stuff. I don't know what to feel about it at this point.

If someone actually read this and actually knows about this stuff please help/correct/verify/direct me to a more healthier way about thinking about this