Let be a real number, and let be the set of positive real numbers for which(1)has (at most) finitely many solutions for and integers. Then the irrationality measure, sometimes called the Liouville-Roth constant or irrationality exponent, is defined as the threshold at which Liouville's approximation theorem kicks in and is no longer approximable by rational numbers,(2)where is the infimum. If the set is empty, then is defined to be , and is called a Liouville number. There are three possible regimes for nonempty :(3)where the transitional case can correspond to being either algebraic of degree or being transcendental. Showing that for an algebraic number is a difficult result for which Roth was awarded the Fields medal.The definition of irrationality measure is equivalent to the statement that if has irrationality measure , then is the smallest number such that the inequality(4)holds for any and all integers and with sufficiently large.The..