document.write( "Question 123519This question is from textbook Discrete Mathematics
\n" ); document.write( ": Assuming the truth of the theorem that states that √n is irrational whenever n is a positive integer that is not a perfect square, prove that √2 + √3 is irrational. \n" ); document.write( "

Algebra.Com's Answer #90651 by Edwin McCravy(20054) $\"\"$ $\"About$

You can put this solution on YOUR website!
Assuming the truth of the theorem that states that √n is irrational whenever n is a positive integer that is not a perfect square, prove that √2 + √3 is irrational.
\n" ); document.write( "

\r\n" );
document.write( "\r\n" );
document.write( "For contradiction, assume √2 + √3 is rational\r\n" );
document.write( "\r\n" );
document.write( "Then there exist two positive integers, p and q such that\r\n" );
document.write( "\r\n" );
document.write( "            √2 + √3 = p/q \r\n" );
document.write( "\r\n" );
document.write( "Square both sides:\r\n" );
document.write( "\r\n" );
document.write( "                  (√2 + √3)² = p²/q²\r\n" );
document.write( "\r\n" );
document.write( "Both sides are rational since the square of a rational number is\r\n" );
document.write( "rational.\r\n" );
document.write( "\r\n" );
document.write( "          (√2 + √3)(√2 + √3) = p²/q²\r\n" );
document.write( "\r\n" );
document.write( "Use FOIL\r\n" );
document.write( "\r\n" );
document.write( "   √2√2 + √2√3 + √3√2 + √3√3 = p²/q²        \r\n" );
document.write( "\r\n" );
document.write( "Using properties of radicals:\r\n" );
document.write( "            \r\n" );
document.write( "             2 + √6 + √6 + 3 = p²/q²\r\n" );
document.write( "\r\n" );
document.write( "Combining like terms:\r\n" );
document.write( "\r\n" );
document.write( "                     5 + 2√6 = p²/q²\r\n" );
document.write( "\r\n" );
document.write( "Subtracting 5 from both sides\r\n" );
document.write( "\r\n" );
document.write( "                         2√6 = p²/q² - 5\r\n" );
document.write( "\r\n" );
document.write( "Both sides are rational because the difference of two rational\r\n" );
document.write( "numbers is rational\r\n" );
document.write( "                         \r\n" );
document.write( "Multiply both sides by rational number 1/2\r\n" );
document.write( "\r\n" );
document.write( "                          √6 = (p²/q² - 5)/2\r\n" );
document.write( "\r\n" );
document.write( "Both sides are rational since the product of two rational\r\n" );
document.write( "numbers is rational.\r\n" );
document.write( "\r\n" );
document.write( "                           √6 is rational\r\n" );
document.write( "\r\n" );
document.write( "We assume √n is irrational whenever n is a positive integer \r\n" );
document.write( "that is not a perfect square.\r\n" );
document.write( "\r\n" );
document.write( "6 is not a perfect square, so √6 is irrational.\r\n" );
document.write( "\r\n" );
document.write( "So we have reached a contradiction.  Therefore the assumption\r\n" );
document.write( "\r\n" );
document.write( "                      √2 + √3 is rational\r\n" );
document.write( "\r\n" );
document.write( "is false.  Therefore √2 + √3 is irrational.\r\n" );
document.write( "\r\n" );
document.write( "Edwin

\n" ); document.write( "

\n" );