Complex numbers first made their appearance a few hundred years ago as inconvenient "imaginary" roots of polynomials. Since then it has become clear that complex numbers are among the most fascinating and useful objects of mathematics, appearing in surprising ways throughout algebra, geometry and function theory.
One gets a hint of this in such questions as these: Why does the complex exponential involve trig functions? Why do real polynomials always have real or complex roots? How is the Mandelbrot Set defined using complex numbers? How are complex numbers used to describe magnetic fields and airplane wings? How are complex numbers used to model non-Euclidean geometry? What was the role of complex numbers in the birth of topology?
This course will explore some intriguing aspects of complex numbers, including the ideas behind the complex exponential and logarithm, applications of roots and powers of complex numbers, as well as some of the topics suggested above. The approach will include some history and biography and will work to make as many connections with the secondary curriculum as possible. Class sessions will include discussion and exploration with calculators and computer software.
The work of the course will include readings, problem sets, email discussions, and a paper. Regular class attendance is required. The main mathematical prerequisites for the course are algebra and coordinate geometry. Matrices will be reviewed and used. Some ideas from calculus will appear, but the tools will be kept as basic as possible.