Badly Approximable

Every irrational number has an approximation constant defined by

where is the nearest integer to and is the infimum limit. The quantity measures how well is approximated by the rational number .

is said to be badly approximable if . An irrational number is badly approximable iff the terms of its continued fraction are bounded. Since quadratic surds have periodic continued fractions, they are badly approximable. The maximum possible value of is , attained for example at , where is the golden ratio.