Transcendence of e and π

Abstract

The main purpose of this chapter is to prove that the number π is transcendental, thereby completing the proof of the impossibility of squaring the circle (Problem III of the Introduction). We first give the proof that e is a transcendental number, which is somewhat easier. This is of considerable interest in its own right, and its proof introduces many of the ideas which will be used in the proof for π. With the aid of some more algebra — the theory of symmetric polynomials — we can then modify the proof for e to give the proof for π.