Notes (By Chapters of "Basic Complex Analysis" by Marsden + Hoffman)
Lesson 1: What are Complex Numbers?
We first dive into the history and motivations for the development of the complex number system! Though we assume that the reader has some familiarity with complex numbers from outside sources, the perspective we take is that of an introductory real analysis course that is first defining what the real numbers are! We talk about the natural developments that led to their study and describe all of the modern algebraic properties of the complex numbers. Finally, we end the section by describing a fugue of the algebraic theory of the complex numbers, remarking how various algebraic properties give hints towards the much more surprising geometry that the complex numbers encapsulate!