document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=688873>Question 688873</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let z = a + bi represent a general complex number. As noted in the lesson, the conjugate of z, abbreviated conj(z) or conj(a + bi) is the complex number a-bi. Also, the modulus of z, modulus(z) is the \"size\" of z, or SQRT(a2 + b2). Which of the following is true for all complex numbers? \r<BR>\n" );
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document.write( "A. All of the following <BR>\n" );
document.write( "B. z*conj(z) = [modulus(z)]2 <BR>\n" );
document.write( "C. z + conj(z) = 2a <BR>\n" );
document.write( "D. z - conj(z) = 2bi <BR>\n" );
document.write( "E. None of the above \r<BR>\n" );
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document.write( "show your work please (: </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #425706 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=425706\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> z*conj(z) = [modulus(z)]^2 \r<BR>\n" );
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document.write( "(a+bi)(a-bi) = [modulus(z)]^2 \r<BR>\n" );
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document.write( "a^2 - (bi)^2 = [modulus(z)]^2 \r<BR>\n" );
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document.write( "a^2 - b^2i^2 = [modulus(z)]^2 \r<BR>\n" );
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document.write( "a^2 - b^2(-1) = [modulus(z)]^2 \r<BR>\n" );
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document.write( "a^2 + b^2 = [modulus(z)]^2 \r<BR>\n" );
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document.write( "[sqrt(a^2 + b^2)]^2 = [modulus(z)]^2 \r<BR>\n" );
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document.write( "[modulus(z)]^2 = [modulus(z)]^2 \r<BR>\n" );
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document.write( "So z*conj(z) = [modulus(z)]^2  is true.\r<BR>\n" );
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document.write( "z + conj(z) = 2a \r<BR>\n" );
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document.write( "a+bi + a-bi = 2a \r<BR>\n" );
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document.write( "(a+a)+(bi-bi) = 2a \r<BR>\n" );
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document.write( "2a+0bi = 2a \r<BR>\n" );
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document.write( "2a = 2a \r<BR>\n" );
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document.write( "So z + conj(z) = 2a  is true.\r<BR>\n" );
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document.write( "z - conj(z) = 2bi \r<BR>\n" );
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document.write( "a+bi - (a-bi) = 2bi \r<BR>\n" );
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document.write( "a+bi - a+bi = 2bi\r<BR>\n" );
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document.write( "(a-a)+(bi+bi) = 2bi\r<BR>\n" );
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document.write( "0a+2bi = 2bi\r<BR>\n" );
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document.write( "2bi = 2bi\r<BR>\n" );
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document.write( "So z - conj(z) = 2bi  is true.\r<BR>\n" );
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document.write( "Therefore, the answer is choice A. All of the following. </SPAN>\n" );
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