Continued fractions are of current interest in mathematics. In a recent publication, Jack E. Graver describes a method for computing terms in the Calkin-Wilf sequence, a list of the positive rationals introduced by Neil Calkin and Herbert S. Wilf in 2000. This paper explores an original method which uses continued fractions to evaluate and locate terms in the Calkin-Wilf sequence, as well as its natural extension to include all of the rational numbers. A generalization of the Calkin-Wilf tree leads to a characterization of rational numbers by continued fractions with integer coefficients. Finally, the meaning of infinite continued fractions and irrational numbers is studied using the structure of the Calkin-Wilf tree. We characterize the irrational numbers which have periodic continued fractions by developing a matrix representation of the setup, and we explain why irrational square root numbers have periodic continued fractions with palindromic coefficients.

• This report represents the work of one or more WPI undergraduate students submitted to the faculty as evidence of completion of a degree requirement. WPI routinely publishes these reports on its website without editorial or peer review.

Creator

• Gobler, Ben

Publisher

• Worcester Polytechnic Institute

Identifier

• 67366
• E-project-050222-213749

Keyword

• Calkin-Wilf Tree
• Euclidean algorithm
• Continued fractions

Advisor

• Servatius, Herman
• Servatius, Brigitte

Year

• 2022

Date created

• 2022-05-02

Resource type

• Major Qualifying Project

Major

• Mathematical Sciences

Rights statement

• In Copyright

License

• All rights reserved