Math Department Colloquium

The next Math Colloquium will be held Tuesday, September 20 at 3:30 in VH1224. David Garth will be speaking on Fractions, Pi, and Quasicrystals. Refreshments will be provided.

ABSTRACT

A rational number is one that can be expressed as a ratio of two integers. A real number that is not rational is said to be irrational. In the field of rational approximation one is interested in approximating an irrational number, like Pi, by a rational number whose denominator is as small as possible. For example, 22/7 is a good approximation to Pi, having a single digit denominator that approximates Pi to two decimal digits. In this talk we will show how to find other such fractions of small denominator that serve to approximate well a given irrational number. In fact, our method will provide a means for finding all such fractions. This method has a nice geometric interpretation and will provide an introduction to the theory of continued fractions. It will also illustrate a surprising application to the study of quasicrystals in solid state physics.