document.write( "Question 1156628: Find all complex numbers z such that z^2=z with line on top or complex conjugate \n" ); document.write( "

Algebra.Com's Answer #779388 by ikleyn(52788) $\"\"$ $\"About$

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\n" ); document.write( "Find all complex numbers z such that z^2=z with line on top or complex conjugate
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document.write( "It is better and much easier to analyse and to solve this problem in polar trigonometric form.\r\n" );
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document.write( "If z^2 = z complex cojugate,  then, firstly, the modulus of z is equal to 1.\r\n" );
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document.write( "In other words, z lies on the unit circle in a coordinate plane.\r\n" );
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document.write( "Next, if the argument of z is polar angle ,  then the polar angle of z^2 is  ,\r\n" );
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document.write( "while the polar angle of (z conjugate) is    or  .\r\n" );
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document.write( "So, we get the equation for the polar angle\r\n" );
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document.write( "    Case 1.   = -,  which implies   = 0,  or   = 0.   Then the solution is z = 1.\r\n" );
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document.write( "OR\r\n" );
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document.write( "    Case 2.   = ,  which implies   = .    Hence,   = .\r\n" );
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document.write( "ANSWER.  There are TWO solutions.  One solution is z = 1.\r\n" );
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document.write( "          The other solution is  z =  = .\r\n" );
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