document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=190594>Question 190594</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/0321233832/algebrahomeworkh></A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> Select any imaginary number (of the form \"a + bi,\" where a and b are non-zero real numbers), and another number such that the sum, difference, product, or quotient of the two numbers is a real number.\r<BR>\n" );
document.write( "\n" );
document.write( "Am very lost here. Thanks! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #143086 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=143086\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> It turns out that ANY number of the form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Bbi ALIGN=MIDDLE  ALT=\"a%2Bbi\"   DISABLED_style= \"max-width:90%; height:auto;\" > added to it's complex conjugate of the form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a-bi ALIGN=MIDDLE  ALT=\"a-bi\"   DISABLED_style= \"max-width:90%; height:auto;\" > is \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a%2Bbi%29%2B%28a-bi%29=%28a%2Ba%29%2B%28bi-bi%29=2a%2B0i=2a ALIGN=MIDDLE  ALT=\"%28a%2Bbi%29%2B%28a-bi%29=%28a%2Ba%29%2B%28bi-bi%29=2a%2B0i=2a\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So adding ANY complex number to it's complex conjugate results in a real number.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Example:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let's pick the number <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2B3i ALIGN=MIDDLE  ALT=\"2%2B3i\"   DISABLED_style= \"max-width:90%; height:auto;\" > (where a = 2 and b = 3) and add it to it's complex conjugate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2-3i ALIGN=MIDDLE  ALT=\"2-3i\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%2B3i%29%2B%282-3i%29=%282%2B2%29%2B%283i-3i%29=4%2B0i=4 ALIGN=MIDDLE  ALT=\"%282%2B3i%29%2B%282-3i%29=%282%2B2%29%2B%283i-3i%29=4%2B0i=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "In short <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%2B3i%29%2B%282-3i%29=4 ALIGN=MIDDLE  ALT=\"%282%2B3i%29%2B%282-3i%29=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Note: it turns out that multiplying a complex number by it's complex conjugate also results in a real number (division is a different story however).\r<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );