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A Geometric Representation of Continued Fractions

Alan F. Beardon and Ian Short
The American Mathematical Monthly
Vol. 121, No. 5 (May 2014), pp. 391-402
DOI: 10.4169/amer.math.monthly.121.05.391
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Page Count: 12
Topics: Fractions, Integers
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A Geometric Representation of Continued Fractions
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Abstract Inspired by work of Ford, we describe a geometric representation of real and complex continued fractions by chains of horocycles and horospheres in hyperbolic space. We explore this representation using the isometric action of the group of Möbius transformations on hyperbolic space, and prove a classical theorem on continued fractions.

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