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