Professor: Shlomo Libeskind
This seminar explores the development of Algebra and Number Theory through the ages, from its primitive origins in Egypt and ancient Babylon, to Greek geometric algebra as manifested in Euclid’s Elements. We also look at Chinese, Indian, and Islamic contributions (the word algebra is a Latin variant from the Arabic al-jabr) to Algebra’s progress in Medieval Europe to the frontiers of the 19th century. We explore the historical aspects, in part through viewing beautifully presented short video lectures such as “Number Theory in Euclid Elements” and “Algebra Becomes the Science of Symmetry” from the Great Courses company.
The seminar is accessible to any honors college student with a good basic knowledge of high school pre-calculus mathematics. Our focus is mathematical but does not assume previous experience with proof. We will discuss strategies for approaching proofs and solving problems and guide students toward successfully solving unfamiliar problems on their own. We will explore the following:
• How does one know how to begin a proof or a solution and how to proceed?
• Which approach is more promising and why?
• Are different solutions possible, and how do they compare?
We will emphasize that proofs and solutions to problems don’t come “out of the blue” and will discuss the thinking process leading to a proof or solution. Topics include the development of elementary algebra from antiquity to the Renaissance and topics in Number Theory from the times of Euclid and Diophantus to Gauss in the 18th century. We will use the text Number Theory and its History (by Oystein Ore, Dover $15.85) as well as handouts and internet resources. Most of the evaluation (80%) is by weekly assignments and short class presentations.