Liouville's theorem

The theory of transcendental numbers was originated by Liouville in his famous memoir of 1844 in which he obtained, for the first time, a class, très-étendue, as it was described in the title of the paper, of numbers that satisfy no algebraic equation with integer coefficients. Some isolated problems pertaining to the subject, however, had been formulated long before this date, and the closely related study of irrational numbers had constituted a major focus of attention for at least a century preceding. Indeed, by 1744, Euler had already established the irrationality of e, and, by 1761, Lambert had confirmed the irrationality of π. Moreover, the early studies of continued fractions had revealed several basic features concerning the approximation of irrational numbers by rationals. It was known, for instance, that for any irrational α there exists an infinite sequence of rationals p/q (q > 0) such that |α–p/q| < 1/q2, and it was known also that the continued fraction of a quadratic irrational is ultimately periodic, whence there exists c = c(α) > 0 such that |α–p/q| > c/q2 for all rationals p/q(q > 0). Liouville observed that a result of the latter kind holds more generally, and that there exists in fact a limit to the accuracy with which any algebraic number, not itself rational, can be approximated by rationals. It was this observation that provided the first practical criterion whereby transcendental numbers could be constructed.