Graduation Year

2011

Document Type

Open Access Senior Thesis

Degree Name

Bachelor of Science

Department

Mathematics

Reader 1

Nicholas Pippenger

Reader 2

Michael Orrison

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Creative Commons License
This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 License.

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Donald Lee Wiyninger III

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.

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