I teach a precalculus course almost every semester, and over these semesters I've found various things that work quite well. For example, when talking about polynomials and rational functions, in particular "zeroes" and "vertical asymptotes", I introduce them as the same thing, only the asymptotes are "points at infinity". This (projective plane) model helps the students in understanding what multiplicity of a zero/asymptote really means. I later use the same projective plane model to show how all conic sections (circle, ellipse, parabola, hyperbola, lines) are related. I get very nice feedback on this, and the students seem to really enjoy it.
I have not, however, been able to find similar motivating examples for introducing complex numbers. I know there must be similar (pictorial!) arguments to engage the students and pique their curiosity, but I haven't found it yet. Simply saying "all polynomials have a zero over the complex numbers" doesn't really do it for them (again, the more pictures involved, the better).
Are there "neat" and "cool" ways of talking about complex numbers for the first time, that are understandable by beginning precalculus students, but also interesting enough to capture their attention and provoke thought?
Best Answer
Edan Maor provides an answer on this post, which gives a nice explanation:
Another idea comes from Byron Schmuland answer on this Math Overflow thread: