   Introduction to Basic Topology of the Complex Plane (Define an Open Disk) by
Video Lecture 25 of 26
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### Lecture Description

Complex Analysis, Video #25 (Complex Arithmetic, Part 25)

Main Topics: numerically check 5th roots from previous video. Define an open disk and visualize it using RegionPlot on Mathematica.

Do a quick numerical check of the 5th roots of -1 + sqrt(3)*I from the previous video (use "N" on Mathematica). Use the percent sign % to refer back to the preceeding output (as a list) and raise every number in the list to the 5th power. Definition of an "open disk" of radius positive radius "rho", centered at a given complex number z0. Use set-builder notation to define, as well as the modulus of the difference z - z0, which represents the distance between the points representing z and z0 in the complex plane. Also write in terms of rectangular coordinates x and y by using the distance formula (Pythagorean Theorem). To be an open disk, we must use a strict inequality so that we do NOT include the boundary points of the disk. Use RegionPlot to graph a disk of radius 2 centered at the point 3 + 4*i. RegionPlot graphs it as if the boundary circle is included, but it is not. We can use Show, Graphics, Dashed, and Circle to emphasize that the boundary is not included.

### Course Description

This is a mini crash course providing all you need to know to understand complex numbers, and study Complex Analysis. Mathematica is used to help visualize the complex plane.