chapter  9
14 Pages

Transcendental numbers

Algebraic numbers are, by definition, roots of polynomials with integer coefficients. It is a priori not clear whether there exist numbers which are not algebraic. Such hypothetical numbers are called transcendental and this name dates at least back to Leibniz who wrote in 1704 omnem rationem transcendunt (which is Latin for transcending everything rational). As we shall show, transcendental numbers exist and, indeed, if we choose randomly a real number, then very likely this number is transcendental. The long-standing problem to give explicit examples of transcendental numbers was solved by Liouville. However, his examples could not prove the transcendence of such important numbers as e or π. The proofs of their transcendence, due to Hermite and Lindemann, are highlights of classical mathematics and the main theme of this chapter.