### Lecture Description

Complex Analysis Video #4 (Complex Arithmetic, Part 4). Complex Addition and the Parallelogram Law. Intro to the Triangle Inequality and use of "Manipulate" on Mathematica.

More detail: Geometric interpretation of complex number addition in terms of vector addition (both in terms of the parallelogram law and in terms of triangles). Emphasize that complex numbers can either be thought of as points or as vectors. Triangle inequality verbally described based on the picture. Introduction to the Mathematica command "Manipulate" to make an animation to illustrate complex number/vector addition when the real part of one of the addends changes.

### 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.