### Lecture Description

Complex Analysis, Video #23 (Complex Arithmetic, Part 23).

Main Topic: Finding the nth roots of a complex number and visualizing them as a regular polygon using Mathematica

Review that we found the 12th roots of unity in the last video and visualized them with ListPlot. Add lines between the vertices in this video to create a regular dodecagon (12-sided regular polygon). Discuss how to define the general nth roots of an arbitrary complex number, given in polar form (n is a positive integer). Write a proposed nth root in polar form and consider what the possibilities for the modulus and argument would be. Use the radical notation to indicate non-negative real nth roots of non-negative real numbers. The ambiguity in the argument of the original number gives n possibilities for the nth root...every nonzero complex number has exactly n nth roots. The nth root function is therefore multi-valued. Finish the video by making an animation using Manipulate and Locator to graph the nth roots of an arbitrary complex number (and Locator will allow us to move the starting point around)...initially accidentally raise the modulus to the 12th power, but then I catch the mistake and fix it (should raise it to the 1/12th power).

### Course Index

- The imaginary unit and how to add complex numbers
- Complex Addition and the Parallelogram Law. Use ListPlot on Mathematica to make the plot.
- Complex Number Addition and the Parallelogram Law. Use of Mathematica to create vectors.
- Complex Number Addition, Parallelogram Law, Triangle Inequality, and Manipulate on Mathematica
- Modulus of a Complex Number, Triangle Inequality, Manipulate and Locator on Mathematica
- Complex Number Subtraction in terms of Vectors, Manipulate and Locator on Mathematica
- Introduction to Multiplying Complex Numbers and Geometrically Interpreting the Product
- Complex Multiplication in terms of Moduli and Arguments. Use Mathematica to illustrate.
- Confirm the Geometry of Complex Number Multiplication with Manipulate and Locator. Principal Value.
- Complex Number Reciprocals (Multiplicative Inverses), approached Algebraically
- Complex Multiplicative Inverses, Complex Division, and Complex Conjugates
- Complex Conjugates, Complex Division, and Visualization on Mathematica.
- Introduction to the Polar Form of a Complex Number and Complex Multiplication
- Polar Form of Complex Numbers, both with "Cis" & with "e" (Euler's Formula)
- De Moivre's Formula and Trigonometric Identities (mistake at the end...see description below)
- De Moivre, Trig Identities, Sine and Cosine in Terms of Exponentials
- A Real Integral done using Complex Arithmetic (Euler's Formula)
- Check the use of Cosine as an Exponential to the Evaluation of an Integral.
- Powers of Complex Numbers (and an intro to "Table" on Mathematica).
- Using Mathematica to Visualize Powers of Complex Numbers
- Dynamic Behavior of Powers of Complex Numbers, Intro to Roots and Multi-Valued Functions
- Deriving and Graphing Complex Roots of Unity
- Graphing Complex Roots with Mathematica
- More on Visualizing Complex Roots with Mathematica
- Introduction to Basic Topology of the Complex Plane (Define an Open Disk)
- Open Sets in the Complex Plane and illustrating the definition with Mathematica

### 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.