Date of Award
5-2011
Document Type
Open Access Senior Thesis
Degree Name
Bachelor of Science
Department
Mathematics
First Thesis Advisor
Nicholas Pippenger
Second Thesis Advisor
Michael Orrison
Rights Information
Donald Lee Wiyninger III
Terms of Use & License Information
This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 License.
Abstract
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 $\alpha$ that maximizes the value $|\alpha - T_n|$, where $T_n$ is the closest continued fraction to $\alpha$ generated from a tuple of length $n$. Finally, it lays out plans for future work.
Recommended Citation
Wiyninger, Donald Lee III, "Continued Fractions: A New Form" (2011). HMC Senior Theses. Paper 14.
http://scholarship.claremont.edu/hmc_theses/14