document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=345766>Question 345766</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How are realnumber broken down </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #247282 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=247282\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Garamond\" size=\"+2\">\r<BR>\n" );
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document.write( "I assume you are asking about the subsets of the set of real numbers.\r<BR>\n" );
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document.write( "Real numbers consist of the rational numbers (those numbers that can be expressed as the quotient of two integers) and the irrational numbers (NOT the quotient of two integers)\r<BR>\n" );
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document.write( "Irrational numbers are of two kinds, algebraic and transcendental.  The algebraic irrationals are roots of non-constant polynomial equations with rational coefficients, example: <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \sqrt{2}\">.  The transcendental irrationals are not roots of such equations, example: <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \pi\">\r<BR>\n" );
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document.write( "Rational numbers consist of integers and non-integer quotients of integers.\r<BR>\n" );
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document.write( "Integers are positive and negative whole numbers plus zero.  The positive whole numbers are the Natural Numbers.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 3\"> is Natural, an Integer, Rational, and Real\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{2}{3}\"> is Rational and Real\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large -4\"> is an Integer, Rational, and Real\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \sqrt{3}\"> is Algebraic Irrational (one of the roots of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x^2\ -\3\ =\0\">) and Real\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large e\"> is Transcendental Irrational (base of the natural logs) and Real\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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