The geometry of continued fractions as analysed by considering Möbius transformations acting on the hyperbolic plane

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dc.contributor.author van Rensburg, Richard
dc.date.accessioned 2012-02-24T06:01:29Z
dc.date.available 2012-02-24T06:01:29Z
dc.date.issued 2012-02-24
dc.identifier.uri http://hdl.handle.net/10539/11345
dc.description M.Sc., Faculty of Science, University of the Witwatersrand, 2011 en_US
dc.description.abstract Continued fractions have been extensively studied in number-theoretic ways. In this text, we will illuminate some of the geometric properties of contin- ued fractions by considering them as compositions of MÄobius transformations which act as isometries of the hyperbolic plane H2. In particular, we examine the geometry of simple continued fractions by considering the action of the extended modular group on H2. Using these geometric techniques, we prove very important and well-known results about the convergence of simple con- tinued fractions. Further, we use the Farey tessellation F and the method of cutting sequences to illustrate the geometry of simple continued fractions as the action of the extended modular group on H2. We also show that F can be interpreted as a graph, and that the simple continued fraction expansion of any real number can be can be found by tracing a unique path on this graph. We also illustrate the relationship between Ford circles and the action of the extended modular group on H2. Finally, our work will culminate in the use of these geometric techniques to prove well-known results about the relationship between periodic simple continued fractions and quadratic irrationals. en_US
dc.language.iso en en_US
dc.subject Transformations (mathematics) en_US
dc.subject Numbers, complex en_US
dc.subject Geometry, hyperbolic en_US
dc.subject Matrix groups en_US
dc.title The geometry of continued fractions as analysed by considering Möbius transformations acting on the hyperbolic plane en_US
dc.type Thesis en_US


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