In this tutorial students will work together to discover and develop the basic theory of diophantine analysis. Specifically, we will begin with a careful investigation of the real numbers and develop a theory as to how well we can approximate a real number by rational numbers that are, in some sense, not very complicated. This theory leads to many avenues of investigation including such areas as continued fractions, geometry of numbers, simultaneous approximation, and generalizations to p-adic fields. Both classical theorems and current results will be explored. Evaluation based on oral and written presentations. Prerequisites: permission of instructor, no number theory background is required.