document.write( "Question 1181500: Prove that (z^n) + (z*)^n is always real for integer n \n" ); document.write( "
Algebra.Com's Answer #811401 by ikleyn(52790)\"\" \"About 
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\n" ); document.write( "Prove that (z^n) + (z*)^n is always real for integer n
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\n" ); document.write( "\n" ); document.write( "            In this problem,  it is assumed  (hidden assumption)  that  z  is a complex number   z = a + bi.\r
\n" ); document.write( "\n" ); document.write( "            Another hidden assumption is that  z*  is the complex conjugate to  z.\r
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document.write( "In the binomial expansion of  z^n = \"%28a%2Bbi%29%5En\",  the imaginary parts (addends) are those and only those \r\n" );
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document.write( "that contain the factors  (ib)  in ODD  degrees.\r\n" );
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document.write( "Similar is for the binomial expansion of  (z*)^n = \"%28a-bi%29%5En\" :  the imaginary parts (addends) are those and only those \r\n" );
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document.write( "that contain the factors  (-ib)  in ODD  degrees.\r\n" );
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document.write( "But the corresponding imaginary addends in binomial expansions  z^n  and  (z*)^n  go with opposite signs, and, THEREFORE,\r\n" );
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document.write( "they cancel each other in the sum  z^n + (z*)^n.\r\n" );
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document.write( "Therefore, the sum  z^n + (z*)^n  is a real number.\r\n" );
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