The problem I have with just an “ordered pair” pedagogy is that it confuses (in students’ minds) the concepts of Euclidean point, displacement vector between po

The problem I have with just an “ordered pair” pedagogy is that it confuses (in students’ minds) the concepts of Euclidean point, displacement vector between points, and “complex number” (which I think of in most contexts as a similarity transformation of Euclidean vectors). In an arbitrary Euclidean vector space, the coordinates chosen are somewhat arbitrary, except for being typically orthogonal and of the same length (a sometimes arbitrary chosen or sometimes meaningful “unit”). But similarity transformations are in some sense “unitless” (insofar as they don’t involve meters or inches or whatever, because they are quotients of vectors and the units cancel): the value 1 is very meaningful and not arbitrary at all (it represents the identity transformation), the value –1 represents a reflection of vectors through the origin, the value i is a unit bivector (in this case representing a specific quarter-turn rotation), and all values of unit magnitude (e.g. √3/2 + i/2) represent pure rotations. To teach someone who has done some analytic geometry in Cartesian style (which is itself sort of jumbled conceptually, because students are taught that the grid is the plane instead of being an arbitrary coordinate system laid down over the plane), explaining this takes quite a bit of nuance and clarity on the part of the instructor. And even many mathematicians and technical professionals elide these distinct concepts all the time. For example, they represent points (e.g. points on plane curves), vectors, and transformations all as complex numbers. Or they represent points as “position vectors” without always making it obvious that the point per se also involves adding the displacement to an arbitrarily chosen origin.