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This report examines the theory of continued fractions and how their use enhances the secondary mathematics curriculum. The use of continued fractions to calculate best approximants of real numbers is justified geometrically, and famous irrational numbers are described as continued fractions. Periodic continued fractions and other applications of continued fractions are also discussed. / text

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Thesis (Ph. D.)--University of Texas at Austin, 2003. / Vita. Includes bibliographical references. Available also from UMI Company.

Continued fractions have been extensively studied in number theoretic ways. In this text we will consider continued fraction expansions with partial quotients that are in Z = f x : x 2 Zg and where = 2 cos( q ); q 3 and with 1 < < 2. These continued fractions are expressed as the composition of M obius maps in PSL(2;R), that act as isometries on H2, taken at 1. In particular the subgroups of PSL(2;R) that are studied are the Hecke groups G . The Modular group is the case for q = 3 and = 1. In the text we show that the Hecke groups are triangle groups and in this way derive their fundamental domains. From these fundamental domains we produce the v-cell (P0) that is an ideal q-gon and also tessellate H2 under G . This tessellation is called the -Farey tessellation. We investigate various known -continued fractions of a real number. In particular, we consider a geodesic in H2 cutting across the -Farey tessellation that produces a \cutting sequence" or path on a -Farey graph. These paths in turn give a rise to a derived -continued fraction expansion for the real endpoint of the geodesic. We explore the relationship between the derived -continued fraction expansion and the nearest - integer continued fraction expansion (reduced -continued fraction expansion given by Rosen, [25]). The geometric aspect of the derived -continued fraction expansion brings clarity and illuminates the algebraic process of the reduced -continued fraction expansion.

<p> This thesis examines continued fraction expansions of the square root of nonsquare positive integers of periods one to six, and shows their relationships with Newton's method of approximation. It also contains known results concerning continued fractions.</p> / Thesis / Master of Science (MSc)

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In the classical case we make use of Pells equation to compute units in the ring OF. Consider the parallel to the classical case and the quadratic field extension that creates the ring OK. We use the generalized Pell's equation to find the units in this ring since they are solutions. Through the use of continued fractions we may further characterize this ring and compute its units.

A multipoint Padë approximant, R, to a function of Stieltjes1 type is determined.The function R has numerator of degree n-l and denominator of degree n.The 2n interpolation points must belong to the region where f is analytic,and if one non-real point is amongst the interpolation points its complex-conjugated point must too.The problem is to characterize R and to find some convergence results as n tends to infinity. A certain kind of continued fraction expansion of f is used.From a characterization theorem it is shown that in each step of that expansion a new function, g, is produced; a function of the same type as f. The function g is then used,in the second step of the expansion,to show that yet a new function of the same type as f is produced. After a finite number of steps the expansion is truncated,and the last created function is replaced by the zero function.It is then shown,that in each step upwards in the expansion a rational function is created; a function of the same type as f.From this it is clear that the multipoint Padê approximant R is of the same type as f.From this it is obvious that the zeros of R interlace the poles, which belong to the region where f is not analytical.Both the zeros and the poles are simple. Since both f and R are functions of Stieltjes ' type the theory of Hardy spaces can be applied (p less than one ) to show some error formulas.When all the interpolation points coincide ( ordinary Padé approximation) the expected error formula is attained. From the error formula above it is easy to show uniform convergence in compact sets of the region where f is analytical,at least wien the interpolation points belong to a compact set of that region.Convergence is also shown for the case where the interpolation points approach the interval where f is not analytical,as long as the speed qî approach is not too great. / digitalisering@umu

In this dissertation we investigate prior definitions for p-adic continued fractions and introduce some new definitions. We introduce a continued fraction algorithm for quadratic irrationals, prove periodicity for Q₂ and Q₃, and numerically observe periodicity for Q(p) when p < 37. Various observations and calculations regarding this algorithm are discussed, including a new type of symmetry observed in many of these periods, which is different from the palindromic symmetry observed for real continued fractions and some previously defined p-adic continued fractions. Other results are proved for p-adic continued fractions of various forms. Sufficient criteria are given for a class of p-adic continued fractions of rational numbers to be finite. An algorithm is given which results in a periodic continued fraction of period length one for √D ∈ Zˣ(p), D ∈ Z, D non-square; although, different D require different parameters to be used in the algorithm. And, a connection is made between continued fractions and de Weger’s approximation lattices, so that periodic continued fractions can be generated from a periodic sequence of approximation lattices, for square roots in Zˣ(p). For simple p-adic continued fractions with rational coefficients, we discuss observations and calculations related to Browkin’s continued fraction algorithms. In the last chapter, we apply some of the definitions and techniques developed in the earlier chapters for Q(p) and Z to the t-adic function field case F(q)((t)) and F(q)[t], respectively. We introduce a continued fraction algorithm for quadratic irrationals in F(q)((t)) that always produces periodic continued fractions.