document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1207460>Question 1207460</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Use z = a + bi and w = c + di to show that (z • w) bar = z bar • w bar.\r<BR>\n" );
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document.write( "Note: one bar line over (z • w).\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #845339 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=845339\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "a,b,c,d are real numbers<BR>\n" );
document.write( "i = sqrt(-1) represents an imaginary number\r<BR>\n" );
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document.write( "z = a+bi<BR>\n" );
document.write( "zBar = horizontal bar over \"a+bi\" = complex conjugate of z<BR>\n" );
document.write( "zBar = a-bi<BR>\n" );
document.write( "w = c+di<BR>\n" );
document.write( "wBar = c-di\r<BR>\n" );
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document.write( "z*w = (a+bi)*(c+di)<BR>\n" );
document.write( "z*w = a*(c+di)+bi*(c+di)<BR>\n" );
document.write( "z*w = ac+adi+bci+bdi^2<BR>\n" );
document.write( "z*w = ac+adi+bci+bd*(-1)<BR>\n" );
document.write( "z*w = (ac-bd) + (ad+bc)i<BR>\n" );
document.write( "(z*w)bar = (ac-bd) - (ad+bc)i<BR>\n" );
document.write( "We'll return to this later.\r<BR>\n" );
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document.write( "zBar*wBar = (a-bi)*(c-di)<BR>\n" );
document.write( "zBar*wBar = a*(c-di)-bi*(c-di)<BR>\n" );
document.write( "zBar*wBar = ac-adi-bci+bdi^2<BR>\n" );
document.write( "zBar*wBar = ac-adi-bci+bd*(-1)<BR>\n" );
document.write( "zBar*wBar = ac-adi-bci-bd<BR>\n" );
document.write( "zBar*wBar = (ac-bd)+(-adi-bci)<BR>\n" );
document.write( "zBar*wBar = (ac-bd)-(ad+bc)i<BR>\n" );
document.write( "This is an identical match with the conclusion of the previous paragraph.\r<BR>\n" );
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document.write( "Therefore we have shown that (z*w)bar = zBar*wBar is indeed the case.<BR>\n" );
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