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Actually I agree with you. That’s a good point. To clarify, the reason I like my approach is not because it’s comprehensive, but because it’s better than the tr

by dsacco 9y ago

Actually I agree with you. That’s a good point. To clarify, the reason I like my approach is not because it’s comprehensive, but because it’s better than the traditional exposition for letting students build initial intuition. In my experience people dislike learning about complex numbers because the notation of z = a + bi feels arbitrary to them, and there’s not enough context in which to motivate them. To your point, representing them as quotients of vectors is definitely better for overall understanding. But for a quick and dirty approach that still involves some theory for building intuition of why complex numbers exist and are useful, representing them as ordered pairs is better than what’s usually done, which is the introduction of complex numbers in computational courses instead of theoretical ones. Ordered pairs are extremely intuitive because they can be plotted on two axes, and topological reasoning in general is fairly intuitive in lower dimensions. Were I not confined to a small comment I’d definitely go with your approach, though. I’m just optimizing for solving a student’s “what even is?” initial intimidation, not for really working with them comprehensively.

jacobolus 9y ago
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.

by dsacco 9y ago

jacobolus 9y ago

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.