document.write( "Question 1189372: The complex numbers z and w satisfy |z| = |w| = 1 and zw is not equal to -1.\r
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Algebra.Com's Answer #820754 by ikleyn(52787) $\"\"$ $\"About$

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\n" ); document.write( "The complex numbers z and w satisfy |z| = |w| = 1 and zw is not equal to -1.
\n" ); document.write( "(a) Prove that overline{z} = 1/z and overline{w} = 1/w.
\n" ); document.write( "(b) Prove that (z + w)/(zw + 1) is a real number.
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\n" ); document.write( "\n" ); document.write( " Part (a)\r
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document.write( "Part (a) is the widely known fact. Students learn it on the beginner steps of studying complex numbers.\r\n" );
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document.write( "The proof is very short and straightforward.\r\n" );
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document.write( "Notice, that in part (a), we should prove only first statement for z, since the statement for w is mathematically THE SAME.\r\n" );
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document.write( "    If z = a + bi, then overline(z) = a - bi.\r\n" );
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document.write( "    If |z| = 1, it means that  = 1, or, which is the same, a^2 + b^2 = 1.\r\n" );
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document.write( "    Now,   =  =  =  = a-bi = overline(z), \r\n" );
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document.write( "    and the statement is proved.\r\n" );
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\n" ); document.write( "\n" ); document.write( " Part (b)\r
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document.write( "Part (b) is not widely known, which makes it interesting.\r\n" );
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document.write( "    Since |z| = 1 and |w| = 1, it means that z and w are the unit vectors of the length 1: their endpoints lie on the unit circle.\r\n" );
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document.write( "    To calculate (z+w), apply the parallelogram's rule.  Since the sides of the parallelogram on vectors z and w are equal,\r\n" );
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document.write( "    the parallelogram is a rhombus.  The sum (z+w) is the diagonal of the parallelogram, and since parallelogram is a rhombus,\r\n" );
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document.write( "    arg(z+w) is  EITHER   OR  .   Here arg() means the argument of complex number.\r\n" );
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document.write( "    The first case   arg(z+w) =  is when the angle between vectors z and w is less than  :  |arg(z)-arg(w)| <= .\r\n" );
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document.write( "    The second case  arg(z+w) =  is when the angle between vectors z and w is greater than  :  |arg(z)-arg(w)| > .\r\n" );
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document.write( "    Notice that by the modulo of ,  arg(z+w) =    always.\r\n" );
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document.write( "    Further, the product zw is the unit vector, again, so the same formulas are applicable to vectors zw and 1 = (1,0).\r\n" );
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document.write( "    Notice that arg(zw) = arg(z) + arg(w), so arg(zw+1) is EITHER ,  or  , depending\r\n" );
document.write( "    on the angle between vectors zw and 1 = (1,0).\r\n" );
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document.write( "    But in any case,  the vectors (z+w) and (zw+1) are EITHER parallel OR anti-parallel (opposite).\r\n" );
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document.write( "    By the modulo of ,  arg(zw+1) =    always.\r\n" );
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document.write( "    By the rule of argument of quotient for complex numbers, it means that the ratio   is real number.\r\n" );
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document.write( "    This real number is EITHER positive (when the vectors (z+w) and (zw+1) are parallel), \r\n" );
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document.write( "                          OR   negative (when the vectors (z+w) and (zw+1) are anti-parallel).                     \r\n" );
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\n" ); document.write( "\n" ); document.write( "At this point, the proof is completed.\r
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