The topic of this course is the algebra and geometry of the Gaussian integers; these are the complex numbers that have integer real and imaginary parts. The reason that the Gaussian integers are worthy of a course is that they have a surprising variety of applications: to number theory, to triangle geometry, and to the Cartesian plane. In addition, the work with Gaussian integers will provide an interesting and accessible way to get a deeper understanding of -- and increased skill with -- the complex numbers in general.
The course will be a problem-based course. During class time students will learn by working problems together, solving a carefully structured set of numerical, geometrical and algebraic problems that will stretch over the whole quarter. These problems start simply and concretely, then gradually go deeper into the theory and applications of complex numbers in general and of Gaussian integers in particular.
The goal of the course is that students will emerge with a greater understanding of complex numbers: what they are, how their operations can be understood, and how to solve problems Ð including some at secondary school level -- using the Gaussian integers as tools. It is also a goal of the course that teachers, or students interested in teaching, will gain insights about ways of learning math from this experience of solving challenging problems and discussing and communicating mathematics.
The formal math background of this course does not really go beyond high school mathematics. But the course will require the willingness to tackle the more challenging problems as well as the more straightforward ones and to think about general mathematical ideas as they emerge from problems and examples.
Given the nature of the course, regular attendance will be essential. Students will work on problems and discuss them in each class session. Between classes there will be assigned some additional problems to solve and/or write-ups of work. During the quarter, there will also be occasions for formal and informal presentations. There will be no final exam, but there will be a final assignment that will include writing and a presentation. Grades will be based on all this work, including class participation.
This course will be based on one of the renowned Developing Mathematics courses taught at the Park City Mathematics Institute. There will be no textbook to purchase.