In part 3 we begin to learn how Rafael Bombelli was able to deal with Cardan's problem. We also get our first look at the mechanics of working with \(\sqrt{-1}\). It always surprises me how quickly most student accept \(\sqrt{-1}\).
I think in most cases this is because students are more interested in finding the answers to the homework than understanding why numbers with a ridiculous name like imaginary became part of our mathematical lexicon. I can hardly blame them, I shared the same outlook through most of high school and college. Who cares about the why when I'm only being evaluated on the how?
Not to get too preachy here, but I will briefly say that once we enter the real world with its real problems and real consequences, the why quickly becomes indispensable. When we must solve real problems - problems without an answer in the back of a book - we must be confident in our tools. To be effective, we need to understand the strengths, shortcomings, and assumptions behind the mathematics (or whatever tools) we use.
This this perspective in mind, imaginary numbers begin to look suspicious. As if they are a specifically designed "trick" to deal with certain problems. I mean really, what could these numbers possibly mean?
I think this is the correct way to approach imaginary numbers. Maintaining our fully justified skepticism, we can explore why these numbers were accepted, and what the implications are. Through remaining skeptical, we ask better questions and can truly appreciate why imaginary numbers are the solution to "real" mathematical problems.
Enjoy!