document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=342086>Question 342086</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What is the reasoning behind the fact that it is not possible for the solution of an irrational number multiplied by a whole number (other than 0) to be a rational number? I know that it is not possible, which is the first step of the question, I just don't know the exact reasoning behind it. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #244850 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=244850\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let w = any whole number. We can write 'w' as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=w%2F1 ALIGN=MIDDLE  ALT=\"w%2F1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Also, let i = any irrational number\r<BR>\n" );
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document.write( "Now assume that multiplying 'w' and 'i' will give you a rational number 'r'. We know it's false, but let's just hypothetically say that this is the case. We're basically looking for a contradiction which arises because of this assumption. \r<BR>\n" );
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document.write( "We can represent 'r' by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=a%2Fb ALIGN=MIDDLE  ALT=\"r=a%2Fb\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=wi=r ALIGN=MIDDLE  ALT=\"wi=r\"   DISABLED_style= \"max-width:90%; height:auto;\" > and that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=wi=a%2Fb ALIGN=MIDDLE  ALT=\"wi=a%2Fb\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "Now solve for 'i' to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=i=a%2F%28bw%29 ALIGN=MIDDLE  ALT=\"i=a%2F%28bw%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2F%28bw%29 ALIGN=MIDDLE  ALT=\"a%2F%28bw%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is clearly a rational number, this means that 'i' is a rational number. But wait, we clearly stated that 'i' is an irrational number and it cannot be both. So we have a contradiction.\r<BR>\n" );
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document.write( "So this means that the product of 'w' and 'i' is NOT rational. So it must be irrational (as it's the only other option)\r<BR>\n" );
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document.write( "Therefore, the product of a whole number and an irrational number is an irrational number.\r<BR>\n" );
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document.write( "If you need more help, email me at <a  rel=nofollow HREF=\"mailto:jim_thompson5910@hotmail.com?Subject=Algebra%20Help\">jim_thompson5910@hotmail.com</a>\r<BR>\n" );
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document.write( "Also, feel free to check out my <a  rel=nofollow HREF=\"http://www.freewebs.com/jimthompson5910/home.html\">tutoring website</a>\r<BR>\n" );
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document.write( "Jim </SPAN>\n" );
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