document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=235115>Question 235115</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I understand that the irrationals are not closed under addition, mult, div, subtr, etc.  However,\r<BR>\n" );
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document.write( "Is it possible to construct a line segment on a cartesian coordinate system such that all points on the line have irrational values for the y coordinate when x is rational (i'm using the relationship y = mx + b)?  It's okay if there are restrictions, like, the segment length must be transcendental or something.\r<BR>\n" );
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document.write( "I'm looking for non-zero-length line-segments that avoid all points with rational coordinate pairs.\r<BR>\n" );
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document.write( "I also don't expect this to be true for ALL line segments with, say, irrational coordinates for the endpoints.  I'm just interested in constructing a line segment of non-zero length that has only irrational values for y when x is rational.  If x is irrational i won't care about y - as long as x and y are never rational together.\r<BR>\n" );
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document.write( "I don't know how constraints are explored in mathematics, nor how one looks for solutions to a problem like this, nor how one goes about testing for the existence of such a construction, let alone constructing one.  if this problem is in a well-known class of problems i can research that myself if somebody can help me identify the class. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #173296 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=173296\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> If 'x' is rational, then we can say that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=p%2Fq ALIGN=MIDDLE  ALT=\"x=p%2Fq\"   DISABLED_style= \"max-width:90%; height:auto;\" > where 'p' and 'q' are integers or whole numbers. This is simply the definition of rational numbers. Note: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=q%3C%3E0 ALIGN=MIDDLE  ALT=\"q%3C%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "If we multiply 'x' by 'm', then we get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=mx=m%28p%2Fq%29=%28mp%29%2Fq ALIGN=MIDDLE  ALT=\"mx=m%28p%2Fq%29=%28mp%29%2Fq\"   DISABLED_style= \"max-width:90%; height:auto;\" > which is still rational since 'mp' is an integer (integer multiplication is closed) and 'q' is an integer.\r<BR>\n" );
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document.write( "If we then add on 'b', we get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=mx%2Bb=%28mp%29%2Fq%2Bb=%28mp%29%2Fq%2B%28bq%29%2Fq=%28mp%2Bbq%29%2Fq ALIGN=MIDDLE  ALT=\"mx%2Bb=%28mp%29%2Fq%2Bb=%28mp%29%2Fq%2B%28bq%29%2Fq=%28mp%2Bbq%29%2Fq\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Because <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=mp ALIGN=MIDDLE  ALT=\"mp\"   DISABLED_style= \"max-width:90%; height:auto;\" > is an integer (using the reasoning above) and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=bq ALIGN=MIDDLE  ALT=\"bq\"   DISABLED_style= \"max-width:90%; height:auto;\" > is an integer (same reasoning), <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=mp%2Bbq ALIGN=MIDDLE  ALT=\"mp%2Bbq\"   DISABLED_style= \"max-width:90%; height:auto;\" > is an integer since integer addition is closed. Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=mp%2Bbq ALIGN=MIDDLE  ALT=\"mp%2Bbq\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=q ALIGN=MIDDLE  ALT=\"q\"   DISABLED_style= \"max-width:90%; height:auto;\" > are integers, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28mp%2Bbq%29%2Fq ALIGN=MIDDLE  ALT=\"%28mp%2Bbq%29%2Fq\"   DISABLED_style= \"max-width:90%; height:auto;\" > is rational. \r<BR>\n" );
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document.write( "This means that if 'x' is rational, then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=mx%2Bb ALIGN=MIDDLE  ALT=\"y=mx%2Bb\"   DISABLED_style= \"max-width:90%; height:auto;\" > is rational. So it is never possible to find an irrational 'y' value given a rational 'x' value.\r<BR>\n" );
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document.write( "I'm not sure what you're asking about in terms of the restrictions, but you'll still find that plugging in rational x values will get you rational y values.  </SPAN>\n" );
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