Math Department (Thesis/2008-09)

Continued Fractions

Under the direction of: Susan Niefield

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A continued fraction is an expression of the form $${a_{1} + {1 \over {a_{2} + {\textstyle\strut1 \over \textstyle\strut{a_{3} + \ldots}}}}}$$ denoted by $[a_1;a_2,a_3\ldots]$, for simplicity. Every real number has a continued fraction expansion and the ones which terminate are precisely the rational numbers (unlike for decimal expansions where ${1\over 3}=.333 \ldots$). Many irrational numbers have interesting continued fraction representations. For example, $\sqrt2=[1;2,2,2\ldots]$, the golden ratio ${1+\sqrt5\over2} =[1;1,1,1\ldots]$, and $\tan(1)=[1;1,1,3,1,5,1,7\ldots]$

There is also a vast literature on continued fractions which includes many applications. Continued fractions have been used to obtain rational approximations for irrational numbers; to obtain integer solutions to equations of the form $x^{2}-d y^{2}=1$, where $d$ is a positive integer (known as the Fermat-Pell equations); to approximate the eigenvalues and eigenvectors of certain large matrices; and even to find all batting records that round to a particular batting average!

The goal of a thesis in this area would be to study the theory and applications of continued fractions using Mathematica to explore examples that are too intricate to do by hand.

Math Department web pages Created: 15 Apr 2002 Last modified: 25 Apr 2008 15:38:02 Comments to: math@union.edu

Thesis Topics - S. Niefield Thesis Topics for 2008-2009 Thesis Topics - K. Lesh