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Unformatted text preview: Liouvilles Theorem : Calculus Jonathan L.F. King University of Florida, Gainesville FL 32611-2082, USA [email protected] Webpage http://www.math.ufl.edu/ squash/ 2 February, 2011 (at 21:07 ) Abstract: This is Liouvilles proof of Liouvilles thm on rational approximations of numbers. Souvenir. The degree of an algebraic num- ber is the degree of the smallest-degree non-zip intpoly having as a zero. Whenever I write a rational, e.g p q , the denom- inator q is always positive . Warmup. Here is a classic theorem ( due probably to Liouville ). 1 : Theorem. Fix an R . Then - p q 6 1 /q 2 : for a seq. of rationals p q with arbitrarily large q . Proof. WLOG is irrational. Take a large N and look at 0 , , 2 ,..., [ N- 1] , where we in- terprete these in the circle group, mod 1. By PHP ( Pigeon-hole Principle ), for some indices j&lt;k we must have the circle-gp distance J k ,j K 6 1 /N ....
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This note was uploaded on 01/26/2012 for the course MAC 3472 taught by Professor Jury during the Fall '07 term at University of Florida.
- Fall '07