### Lecture Description

Complex Analysis Video #9, (Complex Arithmetic, Part 9). Complex Multiplication and its Geometric Interpretation. Modulus and Argument. Intro to Principal Value of the Argument.

Details: Review what happens when you multiply two complex numbers. The modulus of the product is the product of the modulii of the factors. The argument of the product is the sum of the arguments of the factors. Write those two facts symbolically (using absolute value symbols for the modulii and "arg" for arguments). Experiment with the Manipulate (Locator) output to test these facts. Briefly mention the "principal value" of the argument (between -pi and pi). Show how to re-do the code using Re and Im in Mathematica). Use the Arg command in Mathematica to find the principal value of the argument.

### 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.