Real Numbers through the Lens of Continued Fractions
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Authors
Varma, Umang
Issue Date
2013
Type
Thesis
Language
en_US
Keywords
Alternative Title
Abstract
Decimal representation of real numbers (or more generally, b-ary representation) give
us a convenient way of expressing real numbers. Although it takes a lot of computation to
find digits in the decimal expansion of irrational numbers (e.g., the decimal expansions
of and e are known only up to a finite number of digits), these expansions help us
find increasingly precise approximations to irrational numbers.
In this paper, we study rational approximations of irrational numbers and their
relationship to continued fractions. We discuss Liouville’s theorem and bounds on how
well irrational numbers may be approximated. We then motivate the study of continued
fractions, and prove their fundamental properties. These properties allow us to prove
an important result about the relationship between rational approximations to irrational
numbers and their continued fraction expansions. Finally, we look at the continued
fraction expansions of famous irrational numbers.
Description
v, 33 p.
Citation
Publisher
Kalamazoo College
License
U.S. copyright laws protect this material. Commercial use or distribution of this material is not permitted without prior written permission of the copyright holder.