Transcendental Numbers

Abstract

In 1844, Liouville showed that transcendental numbers exist, i.e., numbers that are not root in any algebraic equation:

$$f(x) = {a_n}{x^n} + {a_{n - 1}}{x^{n - 1}} + \cdot \cdot \cdot + {a_1}x + {a_0} = 0,$$

(1)

are integers. Unlike many of Liouville’s deeper ideas, the importance of this discovery was immediately recognized by his contemporaries, and it has remained one of his most celebrated results. In this chapter, I shall analyze its historical background, Liouville’s gradual approach to the question, and the methods he used. For a more comprehensive treatment of the history of transcendental numbers, see [Waldschmidt 1983].