This may just be a difference in terminology, but I'm not familiar with mathematicians calling the norm of a complex number its "size." To me, if you call the norm the size, then you could say that one complex number is larger than another, which I believe is nonsensical.
Of course, you could just define a relation < on the set of complex numbers by "x < y if and only if norm(x) < norm(y)". This relation is obviously trichotomous, and it could be considered an ordering of the complex numbers. In my experience, however, this assumption is generally not held or used by mathematicians.
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This may just be a difference in terminology, but I'm not familiar with mathematicians calling the norm of a complex number its "size." To me, if you call the norm the size, then you could say that one complex number is larger than another, which I believe is nonsensical.
Of course, you could just define a relation < on the set of complex numbers by "x < y if and only if norm(x) < norm(y)". This relation is obviously trichotomous, and it could be considered an ordering of the complex numbers. In my experience, however, this assumption is generally not held or used by mathematicians.