
Chapter 24
Transcendental Numbers
Our discussion of the three geometric problems of antiquity—trisecting the angle,
duplicating the cube, and squaring the circle—left one key fact unproved. To com-
plete the proof of the impossibility of squaring the circle by a ruler-and-compass
construction, crowning three thousand years of mathematical effort, we must prove
that π is transcendental over Q. (In this chapter the word ‘transcendental’ will be
understood to mean transcendental over Q.) The proof we give is analytic, which
should not really be surprising since π is best defined analytically. The techniques
involve symmetric polynomials, integration, differentiation, ...