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This page contains syllabus and problems of the workshop WSE187 (Spring 2006) in mathematics organized
by Dusa McDuff and Valentina Kiritchenko. In this workshop, we study how to find good approximations for
important real life constants. A good approximation must be both
accurate and easy to use. For instance, our current calendar was designed to approximate the
solar year by means of simple rules for omitting some leap years. However, one can find other simple calendars that give
a better approximation. This will be one of the projects in this workshop. To find good approximations
we will introduce continued fractions - fascinating objects related to number theory.
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| Our course notes | An online book on continued fractions: Fascinating fractions. In particular, the calendar problem is discussed there in great detail. | |
| More on applications of continued fractions to calendars and to well-tempered scales in pianos. | The online encyclopedia Wikipedia, where you can find all sorts of information needed for your projects (like the duration of astronomic year etc.) |
Week by week details:
January 26
We recall some basic properties of integers and prove that the square root of 2 can not be a rational number.
We also discuss Dirichlet's argument explaining why any number must have a good rational approximation.
Using continued fractions, we find two good approximations to pi, namely 22/7 (due to Archimedes)
and 355/133.
January 31 - February 2
We learn more about continued fractions. In particular, we notice that the square root of two has a periodic continued
fraction, namely, [1;2,2,2,2,...]. What about the square roots of 3, 5, 6, 7? We also devised a convenient iterative formula for the convergents of a continued fractions. For the golden ratio, this is the famous formula defining the Fibonacci numbers.
February 7-9
We explore geometry of continued fractions and discuss Pell's equation.
February 14-21
Work on projects.