document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=497707>Question 497707</A>: <I><SPAN STYLE=\"word-break: break-all;\"> can you name the subsets of the real numbers to which each number belongs?<BR>\n" );
document.write( "3/4       -8          <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2pi ALIGN=MIDDLE  ALT=\"2pi\"   DISABLED_style= \"max-width:90%; height:auto;\" >         45,368       <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%287%29 ALIGN=MIDDLE  ALT=\"sqrt%287%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >    -2/3 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #336834 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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/cgi-bin/show-face.mpl?user=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=336834\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\"><BR>\n" );
document.write( "If a number cannot be represented by the quotient of two integers, then the number is Irrational.  The number isn't insane, but it cannot be represented as a ratio.  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \mathbb{I}\ =\ \{x\,|\,x\ \notin\ \mathbb{Q}\}\">\r<BR>\n" );
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document.write( "If a number can be represented by the ratio of two integers, the number is Rational.  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \mathbb{Q}\ =\ \{x\,|\,x\ =\ \frac{p}{q},\text{ where } p,\,q\ \in\ \mathbb{Z}\}\">\r<BR>\n" );
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document.write( "The positive and negative whole numbers are the Integers.  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \mathbb{Z}\ =\ \{...-3, -2, -1, 0, 1, 2, 3, ...\}\">\r<BR>\n" );
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document.write( "Non-negative whole numbers are the Natural numbers.  (Actually, there are varying opinions on the exact definition of the set of Whole Numbers and the set of Natural numbers. Whole Numbers could mean Natural Numbers in the sense of positive integers, i.e. <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \mathbb{N}\ =\ \{1, 2, 3, ...\}\">.  Or Whole Numbers could mean Natural Numbers in the sense of non-negative integers, i.e. <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \mathbb{N}\ =\ \{0, 1, 2, 3, ...\}\">.  Or Whole Numbers could be used to mean the entire set of Integers, i.e. <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \mathbb{Z}\ =\ \{...-3, -2, -1, 0, 1, 2, 3, ...\}\">)\r<BR>\n" );
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document.write( "Note that Natural numbers are a proper subset of the Integers which, in turn, is a proper subset of the Rationals.\r<BR>\n" );
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document.write( "45,368 is Natural, Whole, Integer, AND Rational.\r<BR>\n" );
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document.write( "-8 is Whole (maybe), Integer, and Rational\r<BR>\n" );
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document.write( "All fractions and explicitly stated decimals, even those that repeat forever, are Rational.\r<BR>\n" );
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document.write( "Everything else is Irational.\r<BR>\n" );
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document.write( "Any number that is the root of a non-zero polynomial function with rational coefficients is an Algebraic number.  Any other number is a Trancendental number.  Only one of the numbers in your list is Trancendental.  Can you find it?\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi} + 1 = 0\"><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
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