document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=656388>Question 656388</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please I am very confused with this question:\r<BR>\n" );
document.write( "\n" );
document.write( "What type of real number is the product of an irrational number times a rational numbers?\r<BR>\n" );
document.write( "\n" );
document.write( "My guesses are that the product is an irrational number because for example the\r<BR>\n" );
document.write( "\n" );
document.write( "{{sqrt(16) * sqrt(8) = 6.72...}} which can't be express in form of a fraction which wouldn't make it a rational number\r<BR>\n" );
document.write( "\n" );
document.write( "Thanks </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #409497 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=409497\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Seems reasonable.  Let's assume the opposite is possible; that is, assume that there is a rational number (other than zero) and an irrational number such that the product of the two is rational.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE S\"> be the set of irrational numbers and let <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE s\"> be a number such that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE s\ \in\ S\">.  Let <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE r\"> be a number such that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE r\ \in\ \mathbb{Q}\">.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE r\ \in\ \mathbb{Q}\"> means that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \exists p,\,q\ \in\ \mathbb{Z}\ :\ r\ =\ \frac{p}{q}\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "If the product <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE s\,\cdot\,r\ \in\ \mathbb{Q}\">, then there must be integers <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \exists u,\,v\ \in\ \mathbb{Z}\ :\ s\,\cdot\,r\ =\ \frac{u}{v}\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Substituting:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ s\,\cdot\,\frac{p}{q}\ =\ \frac{u}{v}\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Which is to say:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ s\ =\ \frac{uq}{vp}\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "But since the integers are closed under multiplication, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE s\"> must be the quotient of integers and therefore rational.  This contradicts the initial assumption that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE s\"> is irrational.  Therefore, reductio ad absurdum, the product of a rational number and an irrational number cannot be rational.<BR>\n" );
document.write( " \r<BR>\n" );
document.write( "\n" );
document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi}\ +\ 1\ =\ 0\"><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
document.write( "<div  DISABLED_style= \"text-align:center\"><a  rel=nofollow HREF=\"http://outcampaign.org/\" target=\"_blank\"><IMG STYLE=\"max-width:80%;\" src=\"http://cdn.cloudfiles.mosso.com/c116811/scarlet_A.png\" border=\"0\" alt=\"The Out Campaign: Scarlet Letter of Atheism\" width=\"143\" height=\"122\" /></a></div><BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );