Continued Fractions: A New Form
Wiyninger, Donald Lee · 2011
While the traditional form of continued fractions is well-documented, a new form, designed to approximate real numbers between 1 and 2, is less well-studied. This report first describes prior research into the new form, describing the form and giving an algorithm for generating approximations for a given real number. It then describes a rational function giving the rational number represented by the continued fraction made from a given tuple of integers and shows that no real number has a unique continued fraction. Next, it describes the set of real numbers that are hardest to approximate; that is, given a positive integer n, it describes the real number α that maximizes the value |α− Tn|, where Tn is the closest continued fraction to α generated from a tuple of length n. Finally, it lays out plans for future work.