document.write( "Question 1207779: Let z and w be complex numbers such that |z| = |w| = 1 and zw is not equal to -1.\r
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Algebra.Com's Answer #845866 by Edwin McCravy(20056) $\"\"$ $\"About$

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document.write( "Here's another way to do the (a) part:\r\n" );
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document.write( "Since  |z| = |w| = 1\r\n" );
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document.write( "This problem is about points on the unit circle, where the horizontal axis\r\n" );
document.write( "is the real axis and the vertical axis is the imaginary axis.\r\n" );
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document.write( "  \r\n" );
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document.write( "All complex numbers on the unit circle have modulus 1., the radius of the\r\n" );
document.write( "unit circle.\r\n" );
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document.write( "The conjugate of any complex number on the unit circle is obviously its\r\n" );
document.write( "reflection in the horizontal (real) axis.  It is also obvious that the \r\n" );
document.write( "argument (angle) of the conjugate is the negative of the argument (angle).\r\n" );
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document.write( "The reciprocal of any complex number is that complex number when multiplied by\r\n" );
document.write( "the complex number, gives the result 1, which is the complex number 1+0i,\r\n" );
document.write( "thought of as the point (1,0).\r\n" );
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document.write( "DeMoivre's theorem tell us that to multiply two complex numbers, we\r\n" );
document.write( "multiply their moduli and add their arguments.  \r\n" );
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document.write( "So when we multiply the arguments of any two complex numbers on the unit circle,\r\n" );
document.write( "we'll get 1.\r\n" );
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document.write( "It's also obvious to see that when you add the arguments, the\r\n" );
document.write( "red and green arcs above, you will get 0 for the argument.   \r\n" );
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document.write( "So it's obvious that the reciprocal and conjugate of any complex number of the\r\n" );
document.write( "unit circle are the same. \r\n" );
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document.write( "This is what the (a) part asks you to prove:   \r\n" );
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document.write( "We can write z and w in trigonometric form:\r\n" );
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document.write( "We will show that the reciprocal equals the conjugate:\r\n" );
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document.write( "Since the denominator equals 1, \r\n" );
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document.write( "So the reciprocal of a complex number on the unit circle is also its\r\n" );
document.write( "conjugate.\r\n" );
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document.write( "Edwin

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