document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=539691>Question 539691</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Suppose z=a+bi is a complex number, and w=x+yi is another complex number such that z+w is a real number and zw is also a real number. Show that w must be the conjugate of z. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #353510 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=353510\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> If z+w is real then <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE Im(z+w) = 0 \Rightarrow b+y = 0\"> (here, Im(z) stands for the imaginary part of z), so b = -y. Also, if zw is real, then\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE Im((a+bi)(x+yi)) = 0\">. We can replace y with -b.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE Im((a+bi)(x-bi)) = 0 \Rightarrow -abi + bxi = 0 \Rightarrow a = x\">\r<BR>\n" );
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document.write( "Since a = x and y = -b, z and w are conjugates of each other. </SPAN>\n" );
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