document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=113395>Question 113395</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hello out there,I need some help with some really tough problems,I only have two of them,the first one is,Determine whether the number is rational or irrational.If it is irrational,find two consecutive integers between which its graph lies on the number line,-sqrt81.And the last one is,Determine whether sqrt 67 is rational or irrational.If it is irrational,find two consecutive integers between which its graph lies on the number line.Thanks to whoever can help. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #82505 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=82505\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-sqrt%2881%29=-9 ALIGN=MIDDLE  ALT=\"-sqrt%2881%29=-9\"   DISABLED_style= \"max-width:90%; height:auto;\" > because <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+9%2A9=81 ALIGN=MIDDLE  ALT=\"+9%2A9=81\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  9 is clearly a rational number because it can be expressed as the ratio of two integers: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9%2F1 ALIGN=MIDDLE  ALT=\"9%2F1\"   DISABLED_style= \"max-width:90%; height:auto;\" > for example.\r<BR>\n" );
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document.write( "67 is not a perfect square, because <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8%5E2=64 ALIGN=MIDDLE  ALT=\"8%5E2=64\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9%5E2=81 ALIGN=MIDDLE  ALT=\"9%5E2=81\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  So, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2867%29 ALIGN=MIDDLE  ALT=\"sqrt%2867%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is some number between 8 and 9.  But, is it rational or irrational?\r<BR>\n" );
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document.write( "Let's suppose the square root of 67 is a rational number. Then it must be <BR>\n" );
document.write( "of the form a/b, where a and b are whole numbers. Furthermore, we can <BR>\n" );
document.write( "assume that a and b don't have any common factor (otherwise we could <BR>\n" );
document.write( "just divide the numerator and denominator by the common factor).\r<BR>\n" );
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document.write( "So we have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2867%29=a%2Fb ALIGN=MIDDLE  ALT=\"sqrt%2867%29=a%2Fb\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so \r<BR>\n" );
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document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=67=a%5E2%2Fb%5E2 ALIGN=MIDDLE  ALT=\"67=a%5E2%2Fb%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2=67b%5E2 ALIGN=MIDDLE  ALT=\"a%5E2=67b%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "This means that a squared is a multiple of 67. Now you can easily see <BR>\n" );
document.write( "that to get a number that is a multiple of 67 by squaring a, a itself <BR>\n" );
document.write( "must be a multiple of 67, so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+67c ALIGN=MIDDLE  ALT=\"a+=+67c\"   DISABLED_style= \"max-width:90%; height:auto;\" > for some whole number c. Putting <BR>\n" );
document.write( "this back into the equation, we get\r<BR>\n" );
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document.write( "   <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=67%5E2c%5E2+=+67b%5E2 ALIGN=MIDDLE  ALT=\"67%5E2c%5E2+=+67b%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now we divide both sides of the equation by 67, getting\r<BR>\n" );
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document.write( "   <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=67c%5E2=+b%5E2 ALIGN=MIDDLE  ALT=\"67c%5E2=+b%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2 ALIGN=MIDDLE  ALT=\"b%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a multiple of 67, and b must be a multiple of 67.\r<BR>\n" );
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document.write( "But now we are in trouble, because we have shown that both a and b are<BR>\n" );
document.write( "multiples of 67, but we know that a and b have no common factor, since we chose <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Fb ALIGN=MIDDLE  ALT=\"a%2Fb\"   DISABLED_style= \"max-width:90%; height:auto;\" > at the start to be reduced to lowest terms. So <BR>\n" );
document.write( "our original assumption that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2867%29 ALIGN=MIDDLE  ALT=\"sqrt%2867%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > was rational must <BR>\n" );
document.write( "have been wrong.  Therefore, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2867%29 ALIGN=MIDDLE  ALT=\"sqrt%2867%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is irrational.\r<BR>\n" );
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document.write( "Hope this helps,<BR>\n" );
document.write( "John </SPAN>\n" );
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