Continued fraction and its application

Continued fractions are important tools in mathematics to express numbers and solutions of Diophantine equations for thousands of years. But, especially since 18th century, continued fractions have become an important notion in various areas.Nowaday the notion of continued fraction which is very important facility in approximation theory has many remarkable properties related to the Euclidean algorithm for integers. In this thesis, the notion of (simple) continued fraction and its properties are studied in detail. Besides, different examples using continued fraction are given. The first section provides an introduction to the notion of continued fraction. Some definitions are given and the representation of real numbers by continued fractions(rational and irrational numbers respectively) are clarified. In the second section, basic properties of simple continued fractions are pointed out. The relationship between finite, infinite, periodic continued fractions and real numbers are presented. In the last section, in addition to some applications of continued fractions, the solutions of linear Diophantine equations with continued fractions are elaborated. Besides, as a special variation of Diophantine equations, the solutions of Pell Equations with continued fractions are included.