Algebra.Com's Answer #846675 by ikleyn(52790)![\"\"](\"/images/stars/stars-13.gif\")  
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document.write( "True or False: If S, P, and A are the cube roots of a complex number, then Arg(S)+Arg(P)=2×Arg(A).
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document.write( "Mathematical symbol Arg (written with capital \"A\") is the principal value of the argument \r\n" );
document.write( "of a complex number. In distinction from \"arg\" (written with lowercase \"a\"), Arg is always\r\n" );
document.write( "in the interval [ , ), while arg can differ from Arg by any integer multiple of .\r\n" );
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document.write( "Let's consider first a simplest case of cube roots S, P and A of the number 1 (real one).\r\n" );
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document.write( "They have principal arguments 0,  and  .\r\n" );
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document.write( "If to take these cube roots and their principal arguments in this order \r\n" );
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document.write( " s = 0, p = , a = ,\r\n" );
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document.write( "where s = Arg(S), p = Arg(P), a = arg(A), then the equality s + p = 2a becomes 0 + =  ,\r\n" );
document.write( "which does not work (is FALSE). It works ONLY if we consider the equality by the modulo .\r\n" );
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document.write( "If we take the roots and the sum of their principal arguments in other order, like s + a = 2p, we see that it works.\r\n" );
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document.write( "So, in order for the equality s + p = 2a be universal and work for any order of addends,\r\n" );
document.write( "the formulation of the problem must be changed. The equality s + p = 2a should be considered by modulo .\r\n" );
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document.write( "Then this modified statement will be true for any permutations (and for any order) of cube roots.\r\n" );
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document.write( "So, the correct formulation in this problem MUST BE\r\n" );
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document.write( " Arg(S) + Arg(P) = 2*Arg(A) (mod ).\r\n" );
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document.write( "In this formulation, the statement of the problem is true independently of the order of addends.\r\n" );
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document.write( "For the proof in the general case, let S be the cube root of the complex number with the smallest principal argument s;\r\n" );
document.write( "let P be the cube root of the complex number with the intermediate principal argument p;\r\n" );
document.write( "and let A be the cube root of the complex number with the greatest principal argument \"a\".\r\n" );
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document.write( "Then, according to the de Moivre formula\r\n" );
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document.write( " p = s + , a = s + .\r\n" );
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document.write( "So , from one side, Arg(S) + Arg(P) = s + p = s + (s +  = 2s + ;\r\n" );
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document.write( " from the other side, 2*Arg(A) = 2*(s +  = 2s +  .\r\n" );
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document.write( "But is the same as modulo . Thus, we proved that\r\n" );
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document.write( " Arg(S) + Arg(P) = 2*Arg(A) (mod ) (1)\r\n" );
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document.write( "in this order.\r\n" );
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document.write( "Furthermore, it is obvious, that equality (1) is valid for any permutation of the roots;\r\n" );
document.write( "so, it is valid UNIVERSALLY for any order of the principal arguments.\r\n" );
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document.write( "At this point the proof and the solution are completed.\r\n" );
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document.write( "Solved.\r
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document.write( "But please note that it is valid universally ONLY for the modified formulation,
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document.write( "where the equality is by modulo .\r
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document.write( "Otherwise, it is EITHER invalid, OR is invalid as a universal equality for any/arbitrary order of the cube roots
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document.write( "(and, respectively, for any/arbitrary order of their principal arguments).\r
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