Liouville's Approximation Theorem - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-05-21T17:46:11Z http://mathoverflow.net/feeds/question/97426 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/97426/liouvilles-approximation-theorem Liouville's Approximation Theorem Bob Davis 2012-05-19T20:01:52Z 2012-05-19T20:29:06Z <p>I am trying to understand how Liouville's Theorem would apply to complex numbers. </p> <p>alpha an irrational algebraic, then there exist a constant, c, depending on alpha such that for all rationals, the inequality is satisfied. </p> <p>c/q^d &lt;= abs(alpha-(p/q))</p> <p>Assume x is a complex irrational algebraic. In the theorem's inequality, would alpha be seperately the Re(x) and Im(x) of the complex number or would it be the magnitude of x?</p> <p>If x is complex algebraic, is its real and imaginary parts necessarily algebraic? </p> <p>I am asking because I'm trying to prove Liouville's theorem for alpha a complex number. </p>