McKenzie, Harvey C.||Beach, James W.
M.S. (Master of Science)
Department of Mathematics
The purpose of this study is to investigate a specific type of continued fractions, namely continued fractions that represent pure quadratic irrationalities. Chapter I, which is intended both as an introduction to continued fractions and as a foundation for this inquiry, is comprised of basic definitions and theorems. Chapter II begins with the development of a method for determining the continued fraction representation for any irrational number. Then, by employing “complete quotients", this method is adapted for use on pure quadratic irrationalities, that is, numbers of the form √d, where d is a positive integer, not a perfect square. Thus, the final outcome of the chapter is an effective algorithm for “expanding" any pure quadratic irrationality into a continued fraction. Chapter III, a somewhat technical chapter, gives the specifics of two computer programs, each of which utilizes the algorithm of Chapter II. The output of one of these programs is included as Appendix I; therein is contained the continued fraction expansion for every pure quadratic irrationality, √d, 1 < d ≤ 1000. Chapter IV consists entirely of theorems prompted by investigation of the table in Appendix I. Each theorem asserts that a certain expansion pattern is generated by a specific form of the pure quadratic irrationality, √d.
Bentsen, Raymond F., "Continued fraction expansions for pure quadratic irrationalities" (1968). Graduate Research Theses & Dissertations. 1951.
Northern Illinois University
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