6: Introduction to Continued Fractions

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In this chapter, we introduce continued fractions, prove their basic properties and apply these properties to solve some problems. Being a very natural object, continued fractions appear in many areas of Mathematics, sometimes in an unexpected way. The Dutch mathematician and astronomer, Christian Huygens (1629-1695), made the first practical application of the theory of "anthyphaeiretic ratios" (the old name of continued fractions) in 1687. He wrote a paper explaining how to use convergents to find the best rational approximations for gear ratios. These approximations enabled him to pick the gears with the best numbers of teeth. His work was motivated by his desire to build a mechanical planetarium. Further continued fractions attracted attention of most prominent mathematicians. Euler, Jacobi, Cauchy, Gauss and many others worked with the subject. Continued fractions find their applications in some areas of contemporary Mathematics. There are mathematicians who continue to develop the theory of continued fractions nowadays, The Australian mathematician A.J. van der Poorten is, probably, the most prominent among them.

Thanks to professor Pavel Guerzhoy from University of Hawaii for his contribution in chapter 6 on continued fraction and to

6.1: Basic Notations
6.2: Main Technical Tool
6.3: Very Good Approximation
Continued fractions provide a representation of numbers which is, in a sense, generic and canonical. It does not depend on an arbitrary choice of a base. Such a representation should be the best in a sense. In this section we quantify this naive idea.
6.4: An Application
6.5: A Formula of Gauss, a Theorem of Kuzmin and Levi and a Problem of Arnold

Contributors and Attributions

Dr. Wissam Raji, Ph.D., of the American University in Beirut. His work was selected by the Saylor Foundation’s Open Textbook Challenge for public release under a Creative Commons Attribution (CC BY) license.