document.write( "Question 52818: this question has been driving me nuts for days!!
\n" ); document.write( "it says to prove that:
\n" ); document.write( "(1+z)/(1+conjugate z) = z\r
\n" ); document.write( "\n" ); document.write( "so i have tried all sorts of things, i think that i am meant to change it into polar form to get the answer from that but all i do is end up going round in big circles or else end up with huge long lines of numbers and letters that are impossible to simplify!
\n" ); document.write( "help please! \n" ); document.write( "
Algebra.Com's Answer #35221 by AnlytcPhil(1806)\"\" \"About 
You can put this solution on YOUR website!
: this question has been driving me nuts for days!!\r\n" );
document.write( "it says to prove that:\r\n" );
document.write( "(1+z)/(1+conjugate z) = z \r\n" );
document.write( "                     \r\n" );
document.write( "This is true if and only if |z| = 1.  Are you sure you\r\n" );
document.write( "weren't given that |z| = 1?\r\n" );
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document.write( "You can't prove it in general because it isn't true in\r\n" );
document.write( "general! You can easily DISprove that it is true in \r\n" );
document.write( "general with a counterexample.\r\n" );
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document.write( "Suppose z = 1+i\r\n" );
document.write( "     _\r\n" );
document.write( "Then z = 1-i\r\n" );
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document.write( "Then \r\n" );
document.write( "         _  \r\n" );
document.write( "(1+z)/(1+z) =\r\n" );
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document.write( " 1 + (1+i)     2 + i   2 + i     4 + 4i + i²\r\n" );
document.write( "----------- = -------·------- = ------------- =\r\n" );
document.write( " 1 + (1-i)     2 - i   2 + i        4 - i²\r\n" );
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document.write( "4 + 4i + (-1)     3 + 4i\r\n" );
document.write( "------------- =  -------- = (3/5) + (4/5)i\r\n" );
document.write( "   4 - (-1)         5\r\n" );
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document.write( "which does not equal z, since z = 1+i\r\n" );
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document.write( "=================================================\r\n" );
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document.write( "However if you were given that |z| = 1, and I think\r\n" );
document.write( "you were, then it is true.\r\n" );
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document.write( "Proof:\r\n" );
document.write( "                                       _\r\n" );
document.write( "First we must prove the lemma: |z|² = zz\r\n" );
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document.write( "Suppose z = x+yi where x and y are real numbers\r\n" );
document.write( "       _____\r\n" );
document.write( "|z| = Öx²+y², so |z|² = x²+y²\r\n" );
document.write( " _\r\n" );
document.write( "zz = (x+yi)(x-iy) = x²-y²i² = x²-y²(-1) = x²+y²\r\n" );
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document.write( "Therefore the lemma is true.\r\n" );
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document.write( "     |z| = 1 is given\r\n" );
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document.write( "Square both sides\r\n" );
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document.write( "    |z|² = 1\r\n" );
document.write( "                 _\r\n" );
document.write( "Replace |z|² by zz, using the lemma \r\n" );
document.write( "       _\r\n" );
document.write( "      zz = 1\r\n" );
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document.write( "Add z to both sides\r\n" );
document.write( "       _\r\n" );
document.write( "  z + zz = 1 + z\r\n" );
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document.write( "Factor out z on the left\r\n" );
document.write( "      _\r\n" );
document.write( "z(1 + z) = 1 + z\r\n" );
document.write( "                          _\r\n" );
document.write( "Divide both sides by (1 + z)\r\n" );
document.write( "                        _ \r\n" );
document.write( "       z = (1 + z)/(1 + z)\r\n" );
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document.write( "QED\r\n" );
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document.write( "Edwin
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