document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1026729>Question 1026729</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Prove the proposition |z|=1 if {(1+z)/(1-z)} is purely imaginary. Please help. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #642026 by </B><A HREF=/tutors/aboutme.mpl?userid=robertb><B>robertb(5830)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=robertb><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=robertb><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=642026\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2Bz%29%2F%281-z%29 ALIGN=MIDDLE  ALT=\"%281%2Bz%29%2F%281-z%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >  purely imaginary implies that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2Bz%29%2F%281-z%29+=+i%2Aalpha ALIGN=MIDDLE  ALT=\"%281%2Bz%29%2F%281-z%29+=+i%2Aalpha\"   DISABLED_style= \"max-width:90%; height:auto;\" > for some real number <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=alpha ALIGN=MIDDLE  ALT=\"alpha\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "Now let z = a+ib for real numbers a, b.\r<BR>\n" );
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document.write( "==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2B+a+%2B+ib%29%2F%281-a+-+ib%29+=+i%2Aalpha ALIGN=MIDDLE  ALT=\"%281%2B+a+%2B+ib%29%2F%281-a+-+ib%29+=+i%2Aalpha\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2B+a+%2B+ib=%281-a+-+ib%29%2Ai%2Aalpha ALIGN=MIDDLE  ALT=\"1%2B+a+%2B+ib=%281-a+-+ib%29%2Ai%2Aalpha\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2B+a+%2B+ib=i%2Aalpha-a%2Ai%2Aalpha+-+ib%2Ai%2Aalpha ALIGN=MIDDLE  ALT=\"1%2B+a+%2B+ib=i%2Aalpha-a%2Ai%2Aalpha+-+ib%2Ai%2Aalpha\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2B+a+%2B+ib=i%2Aalpha-a%2Ai%2Aalpha+%2B+b%2Aalpha ALIGN=MIDDLE  ALT=\"1%2B+a+%2B+ib=i%2Aalpha-a%2Ai%2Aalpha+%2B+b%2Aalpha\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2Ba%29%2Bib+=+b%2Aalpha%2B+alpha%2A%281-a%29i ALIGN=MIDDLE  ALT=\"%281%2Ba%29%2Bib+=+b%2Aalpha%2B+alpha%2A%281-a%29i\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Ba=b%2Aalpha ALIGN=MIDDLE  ALT=\"1%2Ba=b%2Aalpha\"   DISABLED_style= \"max-width:90%; height:auto;\" >  and  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=+alpha%2A%281-a%29 ALIGN=MIDDLE  ALT=\"b+=+alpha%2A%281-a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "These equations become a system of linear equations in a and b.<BR>\n" );
document.write( "Solving for a and b gives <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+%28-1%2Balpha%5E2%29%2F%281%2Balpha%5E2%29 ALIGN=MIDDLE  ALT=\"a+=+%28-1%2Balpha%5E2%29%2F%281%2Balpha%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=+%282alpha%29%2F%281%2Balpha%5E2%29 ALIGN=MIDDLE  ALT=\"b+=+%282alpha%29%2F%281%2Balpha%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Now <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28z%29+=+sqrt%28a%5E2%2Bb%5E2%29+=+sqrt%28+%28-1%2Balpha%5E2%29%5E2%2F%281%2Balpha%5E2%29%5E2%0D%0A%2B%284alpha%5E2%29%2F%281%2Balpha%5E2%29%5E2%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28+%281-2alpha%5E2%2Balpha%5E4%29%2F%281%2Balpha%5E2%29%5E2%0D%0A%2B%284alpha%5E2%29%2F%281%2Balpha%5E2%29%5E2%29+=+sqrt%28+%281%2B2alpha%5E2%2Balpha%5E4%29%2F%281%2Balpha%5E2%29%5E2%29+=+sqrt%28+%281%2Balpha%5E2%29%5E2%2F%281%2Balpha%5E2%29%5E2%29+=+1 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >, and the statement is proved.\r<BR>\n" );
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