A number x is algebraic if there exists n a natural number and integers a0, a1,... , an such that: an xn + an-1xn-1 + ... + a1 x + a0 = 0. In other words, a number is algebraic if it is the root of a polynomial with integer coefficients. A number is transcendental if it is not algebraic. Cantor showed that there are more real numbers than algebraic numbers. Thus, there must exist transcendental numbers and in fact, they dominate the real line. Unfortunately, Cantor was unable to give a specific example of a transcendental number. Liouville was able to construct an infinite class of transcendental numbers as well as give a method for determining if a given number is transcendental. Unfortunately, he was unable to show the existence of a "significant" transcendental number. This was later rectified by Hermite and Lindemann who respectively showed that e and π are transcendental.
In this talk, we will prove several "elementary" results from transcendental number theory including Cantors' Counting Argument, Liouville's Construction, and the transcendence of e. Time permitting, we will also discuss another famous result, the Gelfond-Schneider Theorem.
No prior knowledge of number theory will be assumed. The level of the proofs involved will be kept to a minimum. Hence, these proofs should be accessible to an undergraduate who has successfully completed Calculus II and Math Reasoning.