On Continued Fractions and its Applications

Abstract

In this thesis, we study finite simple continued fractions, convergents, their properties and some examples on them. We use convergents and some related theorems to solve linear Diophantine equations. We also study infinite simple continued fractions, their convergents and their properties. Then, solving Pell’s equation using continued fractions is discussed. Moreover, we study the expansion of quadratic irrational numbers as periodic continued fractions and discuss some theorems. Finally, the relation between convergents and best approximations is studied and we apply continued fractions in calendar construction and piano tuning.