transcendental. Since a number α is transcendental if and only if P(α) ≠ 0 for every non-zero polynomial P with integer coefficients, this problem can be approached by trying to find lower bounds of the form where the right hand side is some positive function depending on some measure A of the size of the coefficients of P, and its degree d, and such that these lower bounds apply to all P ≠ 0. Such a bound is called a transcendence measure. The case of d = 1 is that of "classical" diophantine approximation asking for lower bounds for . The methods of transcendence theory and diophantine approximation have much in common: they both use the auxiliary function concept. Major results The Gelfond–Schneider theorem was the major advance in transcendence theory in the period 1900–1950. In the 1960s the method of Alan Baker on linear forms in logarithms of algebraic numbers reanimated transcendence theory, with applications to numerous classical problems and diophantine equations. Mahler's classification Kurt Mahler in 1932 partitioned the transcendental numbers into 3 classes, called S, T, and U. Definition of these classes draws on an extension of the idea of a Liouville number (cited above). Measure of irrationality of a real number One way to define a Liouville number is to consider how small a given real number x makes linear polynomials |qx − p| without making them exactly 0. Here p, q are integers with |p|, |q| bounded by a positive integer H. Let be the minimum non-zero absolute value these polynomials take and take: ω(x, 1) is often called the measure of irrationality of a real number x. For rational numbers, ω(x, 1) = 0 and is at least 1 for irrational real numbers. A Liouville number is defined to have infinite measure of irrationality. Roth's theorem says that irrational real algebraic numbers have measure of irrationality 1. Measure of transcendence of a complex number Next consider the values of polynomials at a complex number x, when these polynomials have integer coefficients, degree at most n, and height at most H, with n, H being positive integers. Let be the minimum non-zero absolute value such polynomials take at and take: Suppose this is infinite for some minimum positive integer n. A complex number x in this case is called a U number of degree n. Now we can define ω(x) is often called the measure