Speaker: TONY JACOBS (University of North Texas)

Title: Intuitive Strategies in Transcendence Proofs

Abstract: A transcendental number is one that is not the root of any polynomial with integer coefficients. The number e was first proved to be transcendental by Charles Hermite in 1873. His proof, and many subsequent proofs, rely on strategies that are rather opaque to the uninitiated. Typical methods involve constructing large determinants of functions evaluated at sequences of points, and applying lemmas about the minimum size of polynomials of bounded degree and height evaluated at algebraic points in complex n-space. In this talk, we will eschew all such techniques.

I will be presenting several proofs of irrationality and transcendence, including the transcendence of e, where the main emphasis is to maintain a clear connection to simple ideas. We will use the Taylor Series for the exponential function, and all other methods will be completely elementary: accessible to any alert Calculus I student. We will see how these ideas can be generalized to get as far as the Hermite-Lindemann Theorem, which states that e, raised to any algebraic power, is a transcendental number.