> They can be _represented_ that way (but not uniquely, e.g. c=(a+bi) vs abs(c)=r, arg(c)=Θ) or they can be treated as fundamental (defined by axioms), or as a sub-field of the quaternions, etc.
Hmm... If we are looking for the simpliest way, then I don't think that sub-field of the quaternions is simplier than tuples with definitions for operations on them: you need to define quaternions for that.
I'm not sure about definition of complex numbers through axioms. Probably it is the simplest way, because for tuples you need first to define real numbers and operations on them.
> Also, I'm not seeing why a tuple (A,B) (with order) is simpler than a set {A,B} or a bag [A,B].
I don't see it either. It is the minimal way to define complex numbers (if we have chosen this path), but why it may be simpler in general case is not obvious.
But in any case, I agree, that the whole idea of a "natural" viewpoint was not clearly stated.
Hmm... If we are looking for the simpliest way, then I don't think that sub-field of the quaternions is simplier than tuples with definitions for operations on them: you need to define quaternions for that.
I'm not sure about definition of complex numbers through axioms. Probably it is the simplest way, because for tuples you need first to define real numbers and operations on them.
> Also, I'm not seeing why a tuple (A,B) (with order) is simpler than a set {A,B} or a bag [A,B].
I don't see it either. It is the minimal way to define complex numbers (if we have chosen this path), but why it may be simpler in general case is not obvious.
But in any case, I agree, that the whole idea of a "natural" viewpoint was not clearly stated.