document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1164515>Question 1164515</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Jacinta says the product of a rational number and an irrational number is always irrational. is that true, does it work for 0?\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #788915 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=788915\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "No, her statement is not true when the rational number is 0.\r<BR>\n" );
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document.write( "This is because 0*x = x*0 = 0. <BR>\n" );
document.write( "Multiply 0 with any number (x) and you'll get 0 as a result.\r<BR>\n" );
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document.write( "So multiplying 0 (which is rational) with any irrational number you want, and you'll get 0 \r<BR>\n" );
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document.write( "If Jacinta said that the rational number was not zero, then her claim would be correct.\r<BR>\n" );
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document.write( "The phrasing \"not zero\" is the same as \"nonzero\".\r<BR>\n" );
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document.write( "Extra Info:\r<BR>\n" );
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document.write( "Let's prove the claim \"the product of a nonzero rational number and an irrational number is always irrational\". We'll use a proof by contradiction.\r<BR>\n" );
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document.write( "Let x be rational and y be irrational.<BR>\n" );
document.write( "Since x is rational, we can use integers p,q to say x = p/q where q is nonzero. <BR>\n" );
document.write( "We'll make p nonzero as well to avoid x = p/q being zero.<BR>\n" );
document.write( "We cannot write y as a fraction of integers since it is irrational. This is the definition of what it means to be irrational. It means \"not rational\".<BR>\n" );
document.write( "Keep this in mind for later. \r<BR>\n" );
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document.write( "Next, let's assume the original claim is flipped to say that multiplying any nonzero rational number with an irrational number is rational.<BR>\n" );
document.write( "In short, let's assume x*y = r, where r is a rational number. <BR>\n" );
document.write( "If x*y was rational, then <BR>\n" );
document.write( "x*y = m/n<BR>\n" );
document.write( "for some integers m,n and n is nonzero.\r<BR>\n" );
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document.write( "Because x = p/q, we can say the following<BR>\n" );
document.write( "x*y = m/n<BR>\n" );
document.write( "(p/q)*y = m/n<BR>\n" );
document.write( "y = (m/n)*(q/p)<BR>\n" );
document.write( "y = (m*q)/(n*p)<BR>\n" );
document.write( "y = (some integer)/(some other integer)<BR>\n" );
document.write( "Side note: multiplying any two integers results in some other integer. <BR>\n" );
document.write( "This shows y is rational, but this contradicts the earlier definition where we made y irrational.\r<BR>\n" );
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document.write( "This contradiction makes the claim \"the product of a nonzero rational number and an irrational number is rational\" to be false. <BR>\n" );
document.write( "Therefore, the opposite claim \"the product of a nonzero rational number and an irrational number is irrational\" must be true. <BR>\n" );
document.write( "This concludes the proof.<BR>\n" );
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