document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1097476>Question 1097476</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find a rational number and irrational number between the two given numbers:<BR>\n" );
document.write( "6.21744...6.22746 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #711974 by </B><A HREF=/tutors/aboutme.mpl?userid=math_helper><B>math_helper(2461)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_helper><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=math_helper><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=711974\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> To find a rational number, note that\r<BR>\n" );
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document.write( "6.21744 < 6.22000 < 6.22746\r<BR>\n" );
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document.write( "and 6.22 can be written 622/100 or reduced to  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+highlight%28311%2F50%29+ ALIGN=MIDDLE  ALT=\"+highlight%28311%2F50%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "EDIT: I had accidentally typed out an extra zero to make 1000 in the denominator.  Edited to fix it.  Sorry about that.      <BR>\n" );
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document.write( "For the irrational number, let's call it x\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+6.21744+%3C+x+%3C+6.22746 ALIGN=MIDDLE  ALT=\"+6.21744+%3C+x+%3C+6.22746\"   DISABLED_style= \"max-width:90%; height:auto;\" >   and x irrational (so x^2 is not an integer)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+6217.44+%3C+1000x+%3C+6227.46+ ALIGN=MIDDLE  ALT=\"+6217.44+%3C+1000x+%3C+6227.46+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%286217.44%29%5E2+%3C+1000000x%5E2+%3C+%286227.46%29%5E2+ ALIGN=MIDDLE  ALT=\"+%286217.44%29%5E2+%3C+1000000x%5E2+%3C+%286227.46%29%5E2+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now if we restrict our range to:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=++%286218%29%5E2+%3C+1000000x%5E2+%3C+%286219%29%5E2+ ALIGN=MIDDLE  ALT=\"++%286218%29%5E2+%3C+1000000x%5E2+%3C+%286219%29%5E2+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "We can just pick an integer between <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+6218%5E2+ ALIGN=MIDDLE  ALT=\"+6218%5E2+\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+6219%5E2+ ALIGN=MIDDLE  ALT=\"+6219%5E2+\"   DISABLED_style= \"max-width:90%; height:auto;\" > (excluding endpoints) and <BR>\n" );
document.write( "set <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+1000000x%5E2+ ALIGN=MIDDLE  ALT=\"+1000000x%5E2+\"   DISABLED_style= \"max-width:90%; height:auto;\" > to that:\r<BR>\n" );
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document.write( "For example, one solution is:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+1000000x%5E2+=+6218%5E2+%2B+1+=+38663525+ ALIGN=MIDDLE  ALT=\"+1000000x%5E2+=+6218%5E2+%2B+1+=+38663525+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=++x%5E2+=+38663525%2F1000000+ ALIGN=MIDDLE  ALT=\"++x%5E2+=+38663525%2F1000000+\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+highlight%28x+=+sqrt%2838.663525%29%29+ ALIGN=MIDDLE  ALT=\"+highlight%28x+=+sqrt%2838.663525%29%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >   is irrational and is approximately 6.2180001, which falls between the limits given.   This is just one of many, many solutions.<BR>\n" );
document.write( "—<BR>\n" );
document.write( "The reason it is irrational is every integer between <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%286218%29%5E2+=+38663524+ ALIGN=MIDDLE  ALT=\"+%286218%29%5E2+=+38663524+\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%286219%29%5E2+=+38675961+ ALIGN=MIDDLE  ALT=\"+%286219%29%5E2+=+38675961+\"   DISABLED_style= \"max-width:90%; height:auto;\" >,  excluding endpoints, can not be a perfect square, and hence its square root is irrational.\r<BR>\n" );
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