The Liouville-Roth irrationality measure (irrationality exponent, approximation exponent, or Liouville–Roth constant) of a real number "x" is a measure of how "closely" it can be approximated by rationals. Generalizing the definition of Liouville numbers, instead of allowing any "n" in the power of "q", we find the largest possible value for "μ" such that formula_38 is satisfied by an infinite number of integer pairs ("p", "q") with "q" > 0. This maximum value of "μ" is defined to be the irrationality measure of "x". For any value "μ" less than this upper bound, the infinite set of all rationals "p"/"q" satisfying the above inequality yield an approximation of "x". Conversely, if "μ" is greater than the upper bound, then there are at most finitely many ("p", "q") with "q" > 0 that satisfy the inequality; thus, the opposite inequality holds for all larger values of "q". In other words, given the irrationality measure "μ" of a real number "x", whenever a rational approximation "x" ≅ "p"/"q", "p","q" ∈ N yields "n" + 1 exact decimal digits, we have