Algebra.Com's Answer #850849 by ikleyn(52781)![\"\"](\"/images/stars/stars-13.gif\")  
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document.write( "The complex numbers z and w satisfy |z| = |w| = 1 and zw is not equal to -1.
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document.write( "(a) Prove that \overline{z} = {1}/{z} and \overline{w} = {1}/{w}.\r
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document.write( "(b) Prove that {z + w}/{zw + 1} is a real number.\r
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document.write( "Can you please explain in detail? I'm trying to grasp every aspect of the problem. Thanks
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document.write( " Here, I will prove (b).\r
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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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document.write( "At this point, the proof is completed. \r\n" );
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