Comment by pfortuny
8 days ago
Problem is: you have chosen an orientation (x rightwards, y upwards). That makes your choice of i/-i not canonical: as is natural, because it cannot be canonical.
8 days ago
Problem is: you have chosen an orientation (x rightwards, y upwards). That makes your choice of i/-i not canonical: as is natural, because it cannot be canonical.
It is an interesting question whether it would be possible to distinguish the 2 senses of rotation in a plane that is not embedded in a 3-dimensional space where right and left are easily distinguished. The answer seems to be no.
While in a plane, if you choose 2 orthogonal vectors, from that moment on you can distinguish clockwise from counterclockwise and -i from +i, based on the order of the 2 chosen vectors.
However, from the point of view of a 3-dimensional observer that would watch this choice, it will probably look random, i.e. the senses of rotation would either match those that the 3-dimensional observer thinks as correct, or be the opposite, and within the plane there would be no way to recognize what choice has been made.
This is no big deal. Similarly, in an affine plane there is no origin, but after you choose a particular point then you have an origin to which you can bind a vector space with a system of coordinates, where the senses of rotation are established after the choice of 2 non-collinear vectors.
In an affine plane, the choice of 1 point eliminates the symmetry of translation, then the choice of 1 vector eliminates the symmetry of rotation, and then the choice of a 2nd non-collinear vector eliminates the symmetry between the 2 senses of rotation, allowing the complete determination of a system of coordinates for the 2-dimensional vector space and also the complete determination of the associated field of complex numbers.