The study of the nature of numbers is one of the most ancient and fundamental pursuits in all of mathematics. In this course we explore the worlds of irrational and transcendental number theory. A number is algebraic if it is the solution to a nontrivial polynomial equation with integer coefficients. Numbers that are not algebraic are called transcendental. While these issues are ancient, it was not until 1844 that it was shown that transcendental numbers exist. Since then many modern techniques have been developed to shed some insight into these enigmatic numbers. These techniques beautifully weave ideas from algebra and analysis together. Here we will provide all the necessary ideas from algebraic number theory and from complex analysis. Mathematics 302 and Mathematics 313 are not pre-requisites. Format: lecture/discussion. Evaluation will be based primarily on projects, homework assignments, and exams. Prerequisites: Mathematics 301 or 305, and Mathematics 312 or 315, or permission of instructor. No enrollment limit (expected: 15).