document.write( "Question 1181165: Use De Moivre's Theorem to show that integral powers of (-1 + i)/(√2) are real, and which are imaginary \n" ); document.write( "
Algebra.Com's Answer #811005 by ikleyn(52778)\"\" \"About 
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\n" ); document.write( "Use De Moivre's Theorem to show that integral powers of (-1 + i)/(√2) are real, and which are imaginary
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document.write( "The complex number  z = \"%28-1%2Bi%29%2Fsqrt%282%29\"  has the modulus 1  and the argument  \"3pi%2F4\".\r\n" );
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document.write( "So, in cis-form,  z = \"1%2Acis%283pi%2F4%29\" = \"cis%283pi%2F4%29\".\r\n" );
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document.write( "It means that z itself and all integer degrees of z have the modulus 1, i.e. lie on a unit circle \r\n" );
document.write( "in complex plane.\r\n" );
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document.write( "According to De Moivre's theorem, the degrees of z are\r\n" );
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document.write( "    \"z%5E1\" = \"cis%283pi%2F4%29\"\r\n" );
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document.write( "    \"z%5E2\" = \"cis%286pi%2F4%29\" = \"cis%283pi%2F2%29\" = -i   (pure imaginary)\r\n" );
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document.write( "    \"z%5E3\" = \"cis%289pi%2F4%29\"\r\n" );
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document.write( "    \"z%5E4\" = \"cis%2812pi%2F4%29\" = \"cis%283pi%29\" = -1    (real number)\r\n" );
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document.write( "    \"z%5E5\" = \"cis%2815pi%2F4%29\"\r\n" );
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document.write( "    \"z%5E6\" = \"cis%2818pi%2F4%29\" = \"cis%289pi%2F2%29\" = i   (pure imaginary)\r\n" );
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document.write( "    \"z%5E7\" = \"cis%2821pi%2F4%29\"\r\n" );
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document.write( "    \"z%5E8\" = \"cis%2824pi%2F4%29\" = \"cis%286pi%29\" = 1     (real number)\r\n" );
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document.write( "The degrees of z that follow after  \"z%5E8\",  repeat these numbers cyclically\r\n" );
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document.write( "    \"z%5E9\" = \"z%5E1\"  \r\n" );
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document.write( "    \"z%5E10\" = \"z%5E2\" = -i  (imaginary)\r\n" );
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document.write( "    \"z%5E11\" = \"z%5E3\"\r\n" );
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document.write( "    \"z%5E12\" = \"z%5E4\" = -1  (real number)\r\n" );
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document.write( "    \"z%5E13\" = \"z%5E5\"  \r\n" );
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document.write( "    \"z%5E14\" = \"z%5E6\" =  i  (imaginary)\r\n" );
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document.write( "    \"z%5E15\" = \"z%5E7\"\r\n" );
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document.write( "    \"z%5E16\" = \"z%5E8\" =  1  (real number)\r\n" );
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document.write( "So, the pattern is this:  \"z%5En\"  is real     if and only n is of the form  n = 4k  (i.e. n is a multiple of 4), and\r\n" );
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document.write( "                          \"z%5En\"  is pure imaginary if and only n is of the form  n = 4k+2  (i.e. n gives the remainder of 2 when is divided by 4).\r\n" );
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document.write( "ANSWER.  \"z%5En\"  is real if and only if  n == 0  mod 4;\r\n" );
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document.write( "         \"z%5En\"  is pure imaginary if and only if  n == 2  mod 4.\r\n" );
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