Continued fractions

Following is my thesis submitted in partial satisfaction of the requirements for the Master's Degree in Mathematics, Option 2. I focused on arithmetic continued fractions, stating and proving many of the important theorems while paying attention to maintaining a style of easy readability with many worked examples. I discuss the use of continued fractions to find rational approximations to irrational numbers. I wrote some computer programs that will illustrate some of the theorems. The second half of the paper contains a discussion of Farey numbers and their use in approximating irrational numbers. I compare a continued fraction algorithm with a Farey process algorithm. I include computer programs for the Farey process in the appendix. The continued fraction expansions of some famous irrational numbers are discussed and an original proof for the expansion of e is included in the appendix. Finally, a short overview of the presentation of this topic to an academic high school math class is offered, together with examples, theorems, and answered exercises.