Relations between continued fractions and ideals

In the first chapter, finite continued fractions and infinite continued fractions are studied. It is shown that every finite simple continued fraction is a ratioal number and that a rational number is expressed as a finite simple continued fraction. It is shown that every infinite simple continued fraction is represented as an irrational number. On the other hand, every irrational number is expressed as an infinite continued fractions.In the second chapter, periodic infinite continued fractions are studied. It is shown that every periodic infinite continued fractions repsesent a quadratic irrational number and also pure periodic infinite continued fractions and the simple infinite contiued fraction exponsion of are studied.In the third chapter, relations between integer solutions Pell?s equations and infinitecontinued fractions are investigated.In the fourth chapter, relations between integer continued fractions and ideals are investigated.