document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=550693>Question 550693</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What does it mean if your solution is a complex number? Explain what a complex solution means both algebraically and graphically.Thanks a ton!! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #359631 by </B><A HREF=/tutors/aboutme.mpl?userid=richard1234><B>richard1234(7193)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=richard1234><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=richard1234><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=359631\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> It simply means that replacing x or whatever variable with a complex number satisfies the equation. Graphically, this is more difficult since you would need 3D graphs or graphs with colors to represent values.\r<BR>\n" );
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document.write( "Here is an example of a well-known function, called the Riemann zeta function. It is often defined as\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \zeta(s) = \frac{1}{1^s} + \frac{1}{2^s} + \frac{1}{3^s} + ...\">\r<BR>\n" );
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document.write( "and uses a process called analytic continuation to extend the domain to complex numbers. If you copy/paste the link, you will see a graph of the zeta function to the right side:\r<BR>\n" );
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document.write( "http://en.wikipedia.org/wiki/Riemann_zeta_function\r<BR>\n" );
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document.write( "The Riemann Hypothesis claims that all \"nontrivial\" complex s satisfying <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \zeta(s) = 0\"> have a real part of 1/2. Graphically, this means that all such solutions lie on a line. If you look at the graph, you will see that the black spots indeed appear to lie on that same line. Prove the Riemann Hypothesis, and you will be US$1,000,000 richer (seriously). Keep in mind that the problem was first posed over 150 years ago, and no one has solved it yet.\r<BR>\n" );
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