document.write( "Question 998517: prove that rational+irrational=irrational \n" ); document.write( "

Algebra.Com's Answer #616276 by Edwin McCravy(20055) $\"\"$ $\"About$

You can put this solution on YOUR website!
prove that rational+irrational=irrational
\n" ); document.write( "

\r\n" );
document.write( "Let x be given to be a rational number and y be given to\r\n" );
document.write( "be an irrational number.\r\n" );
document.write( "\r\n" );
document.write( "We want to show that x+y is irrational.\r\n" );
document.write( "\r\n" );
document.write( "There exist integers p and q such that x = p/q\r\n" );
document.write( "\r\n" );
document.write( "For contradiction, assume x+y is rational\r\n" );
document.write( "\r\n" );
document.write( "Then there exist integers r and s such that x+y = r/s.\r\n" );
document.write( "\r\n" );
document.write( "Then p/q + y = r/s \r\n" );
document.write( "           y = r/s - p/q\r\n" );
document.write( "           y = (rq-ps)/(sq)\r\n" );
document.write( "\r\n" );
document.write( "But that says that y is rational.  But it was give to\r\n" );
document.write( "be irrational.  So we have reached a contradiction, \r\n" );
document.write( "and therefore the assumption that x+y is rational is\r\n" );
document.write( "false.  Therefore x+y is irrational.\r\n" );
document.write( "\r\n" );
document.write( "Edwin

\n" ); document.write( "

\n" );