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Intro Complex Analysis, Lec 2, Geometric Interpretations of Complex Arithmetic, Triangle Inequality
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Lecture 2
(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.
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