document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1208936>Question 1208936</A>: <I><SPAN STYLE=\"word-break: break-all;\"> True or false? The sum of rational and irrational number is rational </SPAN>\n" );
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document.write( "Answer: <font color=red size=4>False</font>\r<BR>\n" );
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document.write( "Explanation<BR>\n" );
document.write( "x = rational number<BR>\n" );
document.write( "y = irrational number<BR>\n" );
document.write( "x = a/b for some integers a,b where b is nonzero<BR>\n" );
document.write( "y cannot be written as a fraction of integers because it is set up to be irrational.\r<BR>\n" );
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document.write( "Assume x+y was rational. <BR>\n" );
document.write( "We'll do a proof by contradiction to show that x+y is instead irrational.<BR>\n" );
document.write( "x+y = rational<BR>\n" );
document.write( "x+y = p/q<BR>\n" );
document.write( "(a/b) + y = p/q<BR>\n" );
document.write( "y = (p/q) - (a/b)<BR>\n" );
document.write( "y = (bp)/(bq) - (aq)/(bq)<BR>\n" );
document.write( "y = (bp-aq)/(bq)<BR>\n" );
document.write( "y = (some integer)/(some integer)<BR>\n" );
document.write( "y = rational number<BR>\n" );
document.write( "But wait, we made y irrational and now it's rational. <BR>\n" );
document.write( "This is clearly a contradiction.\r<BR>\n" );
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document.write( " <BR>\n" );
document.write( "A number cannot be both rational and irrational at the same time. <BR>\n" );
document.write( "The term \"irrational\" literally means \"not rational\". <BR>\n" );
document.write( "This contradiction means the assumption x+y = rational is false, so x+y = irrational must be the case.\r<BR>\n" );
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document.write( "A template to write down in your notes would be <font color=blue>rational + irrational = irrational</font><BR>\n" );
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