Abstract
The pigeonhole principle is defined and related to cardinality. It is applied to show that decimal expansions of rational numbers always repeat. It is shown that general lossless compression of files is impossible as another application. A third application given is that for any irrational number \(x\), the set of fractional parts of \(nx\) with \(n\) an integer gets arbitrarily close to zero.
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© 2015 Springer International Publishing Switzerland
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Joshi, M. (2015). The Pigeonhole Principle. In: Proof Patterns. Springer, Cham. https://doi.org/10.1007/978-3-319-16250-8_3
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DOI: https://doi.org/10.1007/978-3-319-16250-8_3
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Publisher Name: Springer, Cham
Print ISBN: 978-3-319-16249-2
Online ISBN: 978-3-319-16250-8
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