Continued fraction

Abstract

In this thesis we will deal with continued fractions, an expression which allow us to represent different mathematical objects through an iterative process (which can be finite or not). We will begin Chapter 1 with a brief historic background about how continued fractions were developed since Ancient Greek to the 20th century. It will follow the general definition of continued fractions, as well as some basic properties and theorems, and finally some observations about specific cases will be commented. Chapter 2 will focus on the simplest type of continued fractions, the ones using integer numbers. Here we will explain the most common application of continued fractions in the field of approximations of real numbers and how they are used to solve some particular equations that are harder to handle with the traditional methods. Up to this point we will only have studied continued fractions of real numbers, and therefore in Chapter 3 we will generalize these results to the complex plane, developing some new methods to aboard continued fractions of complex numbers. Most of the previous theorems will also hold in this case, but for others some aditional requirements will be needed. Finally, Chapter 4 shows some examples about the practical uses of continued fractions outside mathematics, in different fields such as astronomy, hemerology and music. We will discover here how to build a planetarium, create a calendar or even why it is not possible to tune a piano in a precise sense.