The presentation focused on the material presented by Ian Short in his article "Ford Circles, Continued Fractions, and Rational Approximations," which was published in
The American Mathematical Monthly in February 2011. Ford Circles are a geometric representation of the relationship between continued fractions and approximation of real numbers by rational numbers. An introduction to some of these essential concepts were presented before delving into the material Short presented which further illuminated the mathematics behind rational approximation of real numbers. Short proves two major theorems along with a series of lemmas and corollaries. By proving these statements,
Charles Michel ultimately showed that several of the key properties of continued fraction expansion and Diophantine approximations can be proven using Ford Circles and rewritten in terms of relationships between tangential Ford Circles, at the Math 450 Senior Seminar that was held on Tuesday, March 10.
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