Modern number theorists continue to research questions asked by the ancient Greeks. Even though the questions remain the same, the method in which they are investigated has changed dramatically over the past two thousand years. In this short course, we will consider some of these classic questions in number theory as we develop one of the mathematical objects used in modern Number Theory.

Day 1: We will review the rational and real numbers with a focus on how these sets of numbers are constructed. In our investigation we will work towards an understanding of how the real numbers are a “natural” extension of the rational numbers, and consider alternate ways of extending the rational numbers.

Recommended: Review the definitions of the rational and real numbers systems

Day 2: We will develop an alternate way of extending the rational numbers, this gives rise to a new set of numbers: the p-adic numbers. Just as it is the case with the real numbers, the p-adic numbers contain the rational numbers. This alternate extension of the rational numbers gives rise to a new way of measuring distance, and we will explore how distance works in this new setting

Recommended: Read up on Modular Arithmetic

Day 3: We continue our investigation of the p-adic numbers with a focus on tying it back to the number theory. Additional applications will be considered if time permits.

Basic Internet searches will be sufficient for reading about the rational numbers, real numbers, modular arithmetic, and (for the motivated student) field completions.