The complexification of physics: Historical episodes and educational implications

Complex numbers were created (or discovered?) by Italian mathematicians in the 16th century as pragmatic tools to solve cubic equations, and not much attention was given to ontological questions about their "existence". However, this changed significantly in the end of the 18th century, when complex numbers were given a geometrical interpretation. Such concretization motivated physicists to use these numbers to model numerous phenomena, a process that has been called "complexification of physics" by Salomon Bochner. In this talk, different historical episodes will be presented, highlighting, in each case, how and why complex numbers became useful to physicists. Taken together, these examples provide a rather nuanced and pluralistic picture of the interplay between mathematics and physics, and its educational implications.