John Brzezicki, Dr. Victor Ginzburg and Dr. Ivona Grzegorczyk
University of California Santa Cruz Mathematics
We study continued fractions as mathematical representations of real numbers. We describe the procedures for generating continued fractions geometrically and algebraically, including their relationship with the Euclidean algorithm for finding the greatest common divisor. A summary of the history of the apparatus is given from ancient Babylon to the 19th century, as well as applications from the fields of number theory, knot theory, applied mathematics, and engineering. We give a description of computations on continued fractions and the taking of reciprocals. We define the convergents of a continued fraction as certain rational numbers and study their properties. We prove the uniqueness of the continued fraction representation of any rational or real number. The continued fraction representation of several commonly used irrational numbers are given, along with proofs. We provide two computer programs written in Python in order to evaluate the decimal expansions and the rational convergents of a given finite simple continued fraction.