The reletionship between continious fractions and the mobius transformations

In this study, some important specialities of continous fractions are analysed andtheir relationshıp with the Mobius Transormations are examined.In the first section, with the help of Euclid Algorithm, the way of how rationalnumbers can be written as finite continous fractions are examined. From thisapproach, it is shown that every rational number can be defined as finite continousfraction.In the second section, infinite continous fractions and periodic continous fractionsare analysed. Here, it is tried to show that any irrational number can be written asinfinite continous fraction and infinite continous fractions are also irrationalnumbers. At the same time, how the best approach be for irrational numbers is alsoexamined.In the third section, the relationship between continous fractions and the MobiusTransormations is analysed.In the fourth and the last section, the findings are summed up and the results areshown.Key Words: Euclid Algorithm, Continous Fractions, Infinite Contınous Fractions,Convergence of Continous Fractions, Periodic Continous Fractions, MobiusTransformations.