Download or read book Exploring Continued Fractions From the Integers to Solar Eclipses written by Andrew J. Simoson and published by American Mathematical Soc.. This book was released on 2021-04-30 with total page 480 pages. Available in PDF, EPUB and Kindle. Book excerpt: There is a nineteen-year recurrence in the apparent position of the sun and moon against the background of the stars, a pattern observed long ago by the Babylonians. In the course of those nineteen years the Earth experiences 235 lunar cycles. Suppose we calculate the ratio of Earth's period about the sun to the moon's period about Earth. That ratio has 235/19 as one of its early continued fraction convergents, which explains the apparent periodicity. Exploring Continued Fractions explains this and other recurrent phenomena—astronomical transits and conjunctions, lifecycles of cicadas, eclipses—by way of continued fraction expansions. The deeper purpose is to find patterns, solve puzzles, and discover some appealing number theory. The reader will explore several algorithms for computing continued fractions, including some new to the literature. He or she will also explore the surprisingly large portion of number theory connected to continued fractions: Pythagorean triples, Diophantine equations, the Stern-Brocot tree, and a number of combinatorial sequences. The book features a pleasantly discursive style with excursions into music (The Well-Tempered Clavier), history (the Ishango bone and Plimpton 322), classics (the shape of More's Utopia) and whimsy (dropping a black hole on Earth's surface). Andy Simoson has won both the Chauvenet Prize and Pólya Award for expository writing from the MAA and his Voltaire's Riddle was a Choice magazine Outstanding Academic Title. This book is an enjoyable ramble through some beautiful mathematics. For most of the journey the only necessary prerequisites are a minimal familiarity with mathematical reasoning and a sense of fun.
Download or read book Continued Fractions and Signal Processing written by Tomas Sauer and published by Springer Nature. This book was released on 2021-09-06 with total page 275 pages. Available in PDF, EPUB and Kindle. Book excerpt: Besides their well-known value in number theory, continued fractions are also a useful tool in modern numerical applications and computer science. The goal of the book is to revisit the almost forgotten classical theory and to contextualize it for contemporary numerical applications and signal processing, thus enabling students and scientist to apply classical mathematics on recent problems. The books tries to be mostly self-contained and to make the material accessible for all interested readers. This provides a new view from an applied perspective, combining the classical recursive techniques of continued fractions with orthogonal problems, moment problems, Prony’s problem of sparse recovery and the design of stable rational filters, which are all connected by continued fractions.
Download or read book All the Math You Missed written by Thomas A. Garrity and published by Cambridge University Press. This book was released on 2021-07 with total page 417 pages. Available in PDF, EPUB and Kindle. Book excerpt: Fill in any gaps in your knowledge with this overview of key topics in undergraduate mathematics, now with four new chapters.
Download or read book Continued Fractions written by A. M. Rockett and published by World Scientific. This book was released on 1992-08-01 with total page 202 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents the arithmetic and metrical theory of regular continued fractions and is intended to be a modern version of A. Ya. Khintchine's classic of the same title. Besides new and simpler proofs for many of the standard topics, numerous numerical examples and applications are included (the continued fraction of e, Ostrowski representations and t-expansions, period lengths of quadratic surds, the general Pell's equation, homogeneous and inhomogeneous diophantine approximation, Hall's theorem, the Lagrange and Markov spectra, asymmetric approximation, etc). Suitable for upper level undergraduate and beginning graduate students, the presentation is self-contained and the metrical results are developed as strong laws of large numbers.
Download or read book Continued Fractions written by Aleksandr I︠A︡kovlevich Khinchin and published by . This book was released on 1964 with total page 118 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Continued Fractions written by Carl Douglas Olds and published by Mathematical Association of America (MAA). This book was released on 1963 with total page 186 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Continued Fractions written by Debra K. Olson and published by . This book was released on 2005 with total page 210 pages. Available in PDF, EPUB and Kindle. Book excerpt: "English mathematician Godfrey H. Hardy (1877-1947) once said, "The mathematician's patterns, like the painter 's or the poet's must be beautiful; the ideas, like the colors or the words must fit together in a harmonious way. Beauty is the first test: there is no permanent place in this world for ugly mathematics."[3] Whether or not one agrees with this test, it is difficult to argue that the continued fraction doesn't pass it. It is, by its very form, visually beautiful and simple. While its simplicity may seem to wane in exploring it mathematically, its beauty certainly does not. Continued fractions are not only appealing in their visual form; they are also, among other applications, appealing in their ability to approximate numbers and functions. The history of their study is long and must be credited to an extensive list of inspired mathematicians. While this history is not included in this paper, it is nonetheless appreciated and honored. The aim of this paper is to provide a clear introductory look at the study of continued fractions and will focus upon their relationship with particular sets of numbers and their approximations. The chief concern of Chapter 1 is the simple continued fraction. One-to-one correspondences are established between finite simple continued fractions and rational numbers, infinite simple continued fractions and irrational numbers, and therefore between simple continued fractions and real numbers. A similar correspondence is established between periodic simple continued fractions and quadratic irrationals. The accuracy of the resulting approximations arising from these continued fractions is also addressed. Chapter 2 is concerned with the general continued fraction. Linear fractional transformations (l.f.t.'s or Mobius transformations) are employed to describe such a continued fraction and many of its properties. Some interesting applications of simple continued fractions are explored in Chapter 3"--Document.
Download or read book An Introduction to Continued Fractions written by Charles Godat Moore and published by . This book was released on 1964 with total page 112 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Continued Fractions written by Lisa Lorentzen and published by atlantis press. This book was released on 2008 with total page 321 pages. Available in PDF, EPUB and Kindle. Book excerpt: Continued Fractions consists of two volumes -- Volume 1: Convergence Theory; and Volume 2: Representation of Functions (tentative title), which is expected in 2011. Volume 1 is dedicated to the convergence and computation of continued fractions, while Volume 2 will treat representations of meromorphic functions by continued fractions. Taken together, the two volumes will present the basic continued fractions theory without requiring too much previous knowledge; some basic knowledge of complex functions will suffice. Both new and advanced graduate students of continued fractions shall get a comprehensive understanding of how these infinite structures work in a number of applications, and why they work so well. A varied buffet of possible applications to whet the appetite is presented first, before the more basic but modernized theory is given.This new edition is the result of an increasing interest in computing special functions by means of continued fractions. The methods described in detail are, in many cases, very simple, yet reliable and efficient.
Download or read book Fascinating Fractions written by Nikolaĭ Mikhaĭlovich Beskin and published by . This book was released on 1986 with total page 96 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Continued fractions written by Carl D. Olds and published by . This book was released on 1979 with total page 162 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Continued Fractions written by Andrew Mansfield Rockett and published by . This book was released on 1992 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book The Continued Fractions Found in the Unorganized Portions of Ramanujan s Notebooks written by George E. Andrews and published by American Mathematical Soc.. This book was released on 1992-09-30 with total page 84 pages. Available in PDF, EPUB and Kindle. Book excerpt: Among his thirty-three published papers, Ramanujan had only one continued fraction, the Rogers-Ramanujan continued fraction. However, his notebooks contain over 100 results on continued fractions. At the end of his second notebook are 100 pages of unorganized material, and the third notebook comprises thirty-three pages of disorganized results. In these 133 pages of material are approximately sixty theorems on continued fractions, most of them new results. In this monograph, the authors discuss and prove each of these theorems. Aimed at those interested in Ramanujan and his work, this monograph will be of special interest to those who work in continued fractions, $q$-series, special functions, theta-functions, and combinatorics. The work is likely to be of interest to those in number theory as well. The only required background is some knowledge of continued fractions and a course in complex analysis.
Download or read book Analytic Theory of Continued Fractions written by Hubert Stanley Wall and published by . This book was released on 1948 with total page 460 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Continued Fractions written by C. D. Olds and published by . This book was released on 2013 with total page 162 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Continued Fractions written by A. I. Khinchin and published by . This book was released on 1961 with total page 95 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Elementary Number Theory in Nine Chapters written by James J. Tattersall and published by Cambridge University Press. This book was released on 1999-10-14 with total page 420 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is intended to serve as a one-semester introductory course in number theory. Throughout the book a historical perspective has been adopted and emphasis is given to some of the subject's applied aspects; in particular the field of cryptography is highlighted. At the heart of the book are the major number theoretic accomplishments of Euclid, Fermat, Gauss, Legendre, and Euler, and to fully illustrate the properties of numbers and concepts developed in the text, a wealth of exercises have been included. It is assumed that the reader will have 'pencil in hand' and ready access to a calculator or computer. For students new to number theory, whatever their background, this is a stimulating and entertaining introduction to the subject.
