Properties and Applications of Continued Fractions

In this paper, this paper introduces the basic definitions of continued fractions, as well as some basic properties and many important applications. One of the most representative properties is a corollary named difference of successive convergents. It indicates the relations between two adjacent convergents. Besides, this corollary acts as the foundation of another vital corollary which shows that the nexus of the numerator and denominator of a single convergent is mutually prime. Furthermore, the continued fraction is applied to some fields of mathematics, like approximation real numbers and topology.