Complex Analysis Video #10 (Complex Arithmetic, Part 10). Multiplicative Inverses of Complex Numbers approached Algebraically.

Details: Review modulus and argument properties of multiplication. What would it mean for a + b*I to have a multiplicative inverse x + y*i such that (a + b*i)*(x + y*i) = 1? Derive it using a system of equations in the unknowns x & y. Solve via substitution assuming that "a" is not zero. Find a general formula. Have Mathematica check it using Simplify and ComplexExpand. What if a = 0? Still get the same answer if b is nonzero. The only time it doesn't work is if both "a" and "b" equal zero. Find multiplicative inverse of z1 = 5 + 7*i. Use the derived formula to get it. How should we divide z1 = 5 + 7i by z3 = 2 + 3i? Same as z1*(1/z3)...could also do as z1/z3 (the answer is 31/13-i/13).

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

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.