document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1107353>Question 1107353</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Explain why 2 + &#8730;3 is an irrational number. </SPAN>\n" );
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document.write( "Proof By Contradiction:\r<BR>\n" );
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document.write( "Let p and q be two integers where q is nonzero. Any rational number is of the form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%2Fq ALIGN=MIDDLE  ALT=\"p%2Fq\"   DISABLED_style= \"max-width:90%; height:auto;\" > (ratio of two integers)\r<BR>\n" );
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document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Bsqrt%283%29 ALIGN=MIDDLE  ALT=\"2%2Bsqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > was rational, then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Bsqrt%283%29=p%2Fq ALIGN=MIDDLE  ALT=\"2%2Bsqrt%283%29=p%2Fq\"   DISABLED_style= \"max-width:90%; height:auto;\" > for a certain (p,q) value. \r<BR>\n" );
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document.write( "Multiplying both sides by q leads us to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=q%282%2Bsqrt%283%29%29+=+p ALIGN=MIDDLE  ALT=\"q%282%2Bsqrt%283%29%29+=+p\"   DISABLED_style= \"max-width:90%; height:auto;\" > which becomes <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2q%2Bq%2Asqrt%283%29+=+p ALIGN=MIDDLE  ALT=\"2q%2Bq%2Asqrt%283%29+=+p\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The right side of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2q%2Bq%2Asqrt%283%29+=+p ALIGN=MIDDLE  ALT=\"2q%2Bq%2Asqrt%283%29+=+p\"   DISABLED_style= \"max-width:90%; height:auto;\" > is an integer (p is an integer)\r<BR>\n" );
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document.write( "The left side of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2q%2Bq%2Asqrt%283%29+=+p ALIGN=MIDDLE  ALT=\"2q%2Bq%2Asqrt%283%29+=+p\"   DISABLED_style= \"max-width:90%; height:auto;\" > should be an integer as well; however, it is not an integer. The expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2q ALIGN=MIDDLE  ALT=\"2q\"   DISABLED_style= \"max-width:90%; height:auto;\" > is surely an integer because 2 times any integer is an integer, but the expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=q%2Asqrt%283%29 ALIGN=MIDDLE  ALT=\"q%2Asqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is NOT an integer. It is only an integer if q was of the form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k%2Asqrt%283%29 ALIGN=MIDDLE  ALT=\"k%2Asqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, but it is not. Again, q is an integer with no irrational parts. \r<BR>\n" );
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document.write( "In short, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2q%2Bq%2Asqrt%283%29 ALIGN=MIDDLE  ALT=\"2q%2Bq%2Asqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > on the left side is NOT an integer while <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p ALIGN=MIDDLE  ALT=\"p\"   DISABLED_style= \"max-width:90%; height:auto;\" > on the right side is an integer.\r<BR>\n" );
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document.write( "So this is where the contradiction lies. This contradiction makes the claim <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Bsqrt%283%29=p%2Fq ALIGN=MIDDLE  ALT=\"2%2Bsqrt%283%29=p%2Fq\"   DISABLED_style= \"max-width:90%; height:auto;\" > to be false, therefore <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Bsqrt%283%29 ALIGN=MIDDLE  ALT=\"2%2Bsqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is not rational. The only thing left is that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Bsqrt%283%29 ALIGN=MIDDLE  ALT=\"2%2Bsqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > must be irrational. <BR>\n" );
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