Math is about abstractions. In math, we are always interseted in properties of things, not what the things actually are. If in any context there are properties we care about and if two things have all those properties the same, we consider them essentially same. It’s like either is just a relabeling of the other.
For example, in the context of structures that are composed of a set with an operation on that set,
>the set of integers {1,-1} with the operation * (multiplication), that is
>1*1=1, 1*(-1)=-1, (-1)*1=-1, (-1)*(-1)=1
is considered to be the same as
>the set rotational symmetries of a nonsquare rectangle with the operation ∘ (of composition of maps)
because there are just two rotational symmetries of a nonsquare rectangle, rotation by 0 and rotation by straight angle, which if I label by symbols *S*, *F* (*S* for “stays” and *F* for “flips”), satisfy
>*S*∘*S=S, S*∘*F=F, F*∘*S=F, F*∘*F=S*
one can clearly see that it’s “the same” formulas as for the set {1,-1} with operation *. We have just relabeled the element 1 with *S*, the element -1 with *F*, and the operation * with ∘.
In the context of your question, defining imaginary numbers by adding a number *i* such that *i^2*=-1 to real numbers with addition and multiplication, and defining “different imaginary numbers” by adding *x* such that *x^2=-n* for some positive *n* other than 1, one finds that either is just a relabeling of the other, so *both* will be called complex numbers.
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