document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=189409>Question 189409</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Show that 7 + &#8730;2 is not a rational number </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #142196 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=142196\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Garamond\" size=\"+2\">\r<BR>\n" );
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document.write( "I'll do you one better.  Let's prove that the sum of <i><b>any</b></i> rational number and <i><b>any</b></i> irrational number is irrational.\r<BR>\n" );
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document.write( "Assume that there exists an irrational number <i>x</i> and a rational number <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{a}{b}\">, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a, b\ \in\ \I\"> such that the sum:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  x + \frac{a}{b}\"> is rational.\r<BR>\n" );
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document.write( "Then there exists integers <i>c</i> and <i>d</i> such that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{c}{d}\"> is a rational number and,\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  x + \frac{a}{b} = \frac{c}{d}\">\r<BR>\n" );
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document.write( "But then\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  x = \frac{c}{d} - \frac{a}{b} = \frac{bc - ad}{bd}\">\r<BR>\n" );
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document.write( "and since <i>a</i>, <i>b</i>, <i>c</i>, and <i>d</i> are integers, <i>bd</i>, <i>bc</i>, and <i>ad</i> are integers, and further <i>bc</i> - <i>ad</i> is an integer, hence <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{bc - ad}{bd}\"> is rational by definition.\r<BR>\n" );
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document.write( "But that leads to a contradiction because <i>x</i> was originally assumed to be irrational.\r<BR>\n" );
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document.write( "Therefore, reductio ad absurdum, \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  x + \frac{a}{b}\"> is irrational.\r<BR>\n" );
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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" );
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