Lecture Description
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).
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.