(0:00) Comments on ungraded exercises to do. (0:30) This lecture will focus on the geometry of complex arithmetic. (1:01) Pre-class and post-class lecture Mathematica notebooks. (1:57) The complex numbers form a field, with field properties of addition and multiplication, such as the commutative and associative properties. (5:03) The natural one-to-one correspondences (associations) between the set of complex numbers, points in the plane, and vectors in the plane. (14:07) Polar coordinates and PolarPlot on Mathematica. (23:19) Exercise to find the polar coordinates of a product in terms of the polar coordinates of the factors. (36:04) Modulus and argument and the general fact relating these things for the product of two complex numbers. (38:27) Mathematica demonstration for this fact. (41:22) Defining the Euclidean (distance) metric using the modulus and its geometric interpretations. (43:50) Parallelogram law for addition and the triangle inequality. (50:12) Complex conjugate and its relation to complex division (in particular, to interpreting 1/z geometrically). (57:48) Complex addition as a mapping that is a translation of the complex plane.
Based on "Fundamentals of Complex Analysis, with Applications to Engineering and Science", by E.B. Saff and A.D. Snider (3rd Edition). "Visual Complex Analysis", by Tristan Needham, is also referred to a lot. Mathematica is often used, especially to visualize complex analytic (conformal) mappings.