document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=320383>Question 320383</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1.If (a+ib)/(c+id)=A+iB ,then prove that (a-ib)/(c-id)=A-iB </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #229488 by </B><A HREF=/tutors/aboutme.mpl?userid=CharlesG2><B>CharlesG2(834)</B></A><img src=\"/images/stars/stars-11.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CharlesG2><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=229488\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> If (a+ib)/(c+id)=A+iB ,then prove that (a-ib)/(c-id)=A-iB:\r<BR>\n" );
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document.write( "i = sqrt(-1), i^2 = -1, a and b and c and d and A and B are all real numbers<BR>\n" );
document.write( "----------<BR>\n" );
document.write( "(a + ib)/(c + id) (when you divide by a complex number you need to multiply top and bottom by its conjugate, in this case c - id)<BR>\n" );
document.write( "(a + ib)/(c + id) * (c - id)/(c - id)<BR>\n" );
document.write( "((a + ib)(c - id))/((c + id)(c - id)) (use FOIL on top and bottom)<BR>\n" );
document.write( "(ac - iad + ibc - (i^2)bd)/(c^2 - icd + icd - (i^2)d^2)<BR>\n" );
document.write( "(ac - iad + ibc - (-1)bd)/(c^2 - icd + icd - (-1)d^2)<BR>\n" );
document.write( "(ac + bd - iad + ibc)/(c^2 + d^2)<BR>\n" );
document.write( "(ac + bd + i(bc - ad))/(c^2 + d^2)<BR>\n" );
document.write( "ac/(c^2 + d^2) + bd/(c^2 + d^2) + i((bc - ad)/(c^2 + d^2))<BR>\n" );
document.write( "A = (ac + bd)/(c^2 + d^2)<BR>\n" );
document.write( "B = (bc - ad)/(c^2 + d^2)<BR>\n" );
document.write( "----------<BR>\n" );
document.write( "(a - ib)/(c - id)  (conjugate of c - id is c + id)<BR>\n" );
document.write( "(a - ib)/(c - id) * (c + id)/(c + id)<BR>\n" );
document.write( "((a - ib)(c + id))/((c - id)(c + id)) (use FOIL on top and bottom)<BR>\n" );
document.write( "(ac + iad - ibc - (i^2)bd)/(c^2 + icd - icd - (i^2)d^2)<BR>\n" );
document.write( "(ac + iad - ibc - (-1)bd)/(c^2 - (-1)d^2)<BR>\n" );
document.write( "(ac + bd - i(bc - ad))/(c^2 + d^2)<BR>\n" );
document.write( "ac/(c^2 + d^2) + bd/(c^2 + d^2) - i((bc - ad)/(c^2 + d^2))<BR>\n" );
document.write( "A = (ac + bd)/(c^2 + d^2)<BR>\n" );
document.write( "B = (bc - ad)/(c^2 + d^2)<BR>\n" );
document.write( "----------<BR>\n" );
document.write( "1. (a+ib)/(c+id) = ac/(c^2 + d^2) + bd/(c^2 + d^2) + i((bc - ad)/(c^2 + d^2))<BR>\n" );
document.write( "(a+ib)/(c+id)=A+iB<BR>\n" );
document.write( "A = (ac + bd)/(c^2 + d^2)<BR>\n" );
document.write( "B = (bc - ad)/(c^2 + d^2)<BR>\n" );
document.write( "2. (a-ib)/(c-id) = ac/(c^2 + d^2) + bd/(c^2 + d^2) - i((bc - ad)/(c^2 + d^2))<BR>\n" );
document.write( "(a-ib)/(c-id)=A-iB<BR>\n" );
document.write( "A = (ac + bd)/(c^2 + d^2)<BR>\n" );
document.write( "B = (bc - ad)/(c^2 + d^2)<BR>\n" );
document.write( "----------<BR>\n" );
document.write( "proven\r<BR>\n" );
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document.write( " </SPAN>\n" );
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