Book Description

"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.