document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=896000>Question 896000</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Are these statements always, sometimes, or never true? If never, what is a counterexample. If sometimes true, give an example AND counterexample.\r<BR>\n" );
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document.write( "1) The difference between any complex number a+bi (b doesn't equal zero) and its conjugate is a real number.(I said ALWAYS because it turned out to always be 0)\r<BR>\n" );
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document.write( "2) The product of any two imaginary numbers bi (b doesn't equal zero) and di (d doesn't equal zero) is a positive real number. (I said NEVER TRUE because i * i is -1)<BR>\n" );
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document.write( "Please Answer! Thanks for the help! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #543166 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=543166\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> when you add a + bi and a - bi, the answer is going to always be 2a + 0i because bi - bi = 0i which is equal to 0.<BR>\n" );
document.write( "since the imaginary component becomes 0, the result will always be a real number.<BR>\n" );
document.write( "this statement is always true.\r<BR>\n" );
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document.write( "the second statement is:\r<BR>\n" );
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document.write( "The product of any two imaginary numbers bi (b doesn't equal zero) and di (d doesn't equal zero) is a positive real number.\r<BR>\n" );
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document.write( "bi * di is equal to -b*d because i^2 is always equal to -1.\r<BR>\n" );
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document.write( "here's where you can get tripped up.\r<BR>\n" );
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document.write( "the only requirement for b and d is that they are not equal to 0.\r<BR>\n" );
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document.write( "they can be positive or negative.\r<BR>\n" );
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document.write( "if they are both positive, then their product is positive and so the result will be negative when you multiply them by -1.\r<BR>\n" );
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document.write( "if they are both negative, then their product is positive and so the result will be negative when you multiply them by -1.\r<BR>\n" );
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document.write( "if one of them is positive and one of them is negative, then their product is negative and the result will be positive when you multiply them by -1.\r<BR>\n" );
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document.write( "so the answer to the second questions is sometimes, since there is no restriction on whether b or d is positive or negative.\r<BR>\n" );
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document.write( "the only restriction is that they do not equal 0.\r<BR>\n" );
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