document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=454435>Question 454435</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I am trying to find a way to graphically prove dividing two complex numbers.  When I add to complex numbers I can prove it by using the parallelogram method.  How can I prove dividing complex numbers?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #312040 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=312040\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Perhaps use Euler's formula, which says that\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\alpha} = \cos(\alpha) + i\sin(\alpha)\">\r<BR>\n" );
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document.write( "Suppose we want to divide two arbitrary complex numbers (a+bi)/(c+di). If we suppose\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE a+bi = \sqrt{a^2 + b^2}e^{i\alpha}\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE c+di = \sqrt{c^2 + d^2}e^{i\beta}\"> (alpha, beta are the arguments of the complex numbers)\r<BR>\n" );
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document.write( "Then,\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{a+bi}{c+di} = \frac{\sqrt{a^2+b^2}}{\sqrt{c^2+d^2}} \frac{e^{i\alpha}}{e^{i\beta}} = \frac{\sqrt{a^2+b^2}}{\sqrt{c^2+d^2}}e^{i(\alpha - \beta)} = \frac{\sqrt{a^2+b^2}}{\sqrt{c^2+d^2}} (\cos(\alpha - \beta) + i\sin(\alpha - \beta))\">\r<BR>\n" );
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document.write( "Graphing this number isn't too difficult. Here, we locate the complex number with argument alpha - beta and magnitude 1, then scale it by that coefficient on the left. That is the quotient of the two complex numbers. </SPAN>\n" );
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