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document.write( "Let z and w be complex numbers such that |z| = |w| = 1 and zw is not equal to -1.
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document.write( "(a) Prove that conjugate {z} = 1/z and conjugate{w} = 1/w
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document.write( "(b) Prove that ={z + w}/{zw + 1} is a real number.
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document.write( " Proof for part (a)\r
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document.write( "It is well known fact that the product 
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= |z|^2.\r\n" );
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document.write( "Indeed, if z = a + bi, then = a - bi, and, therefore,\r\n" );
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document.write( " . = (a+bi)*(a-bi) = 
= 
= 
= 
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document.write( "But we are given that |z| = 1; hence, = 1.\r\n" );
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document.write( "Thus . = 1.\r\n" );
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document.write( "Hence, = 
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document.write( "It is what was required to prove.\r\n" );
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document.write( "For 
= 
the proof is the same.\r\n" );
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document.write( "It is, actually, the same statement as = , but expressed using letter w instead of z.\r\n" );
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document.write( "At this point, proof for part (a) is complete.\r
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document.write( " Proof for part (b)\r
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document.write( "z and w are unit complex numbers, in the sense that their modules are equal to 1 (given).\r\n" );
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document.write( "Hence, in geometric presentation in complex plane, z and w are adjacent side of a rhombus.\r\n" );
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document.write( "Then, according to the parallelogram rule of adding complex numbers, the sum z+w\r\n" );
document.write( "is the diagonal of the rhombus on side z and w.\r\n" );
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document.write( "According to properties of a rhombus, its diagonal is a bisector of the angle \r\n" );
document.write( "between vectors z and w.\r\n" );
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document.write( "Therefore, for the argument of the complex number z+w we can write\r\n" );
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document.write( " arg(z+w) = 
mod 
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document.write( "Now consider complex number zw. As the product of two complex unit numbers, it is a unit\r\n" );
document.write( "complex number, too, i.e. has the modulus 1, and its argument is the sum of arguments z and w\r\n" );
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document.write( " arg(zw) = arg(z) + arg(w) mod . (2)\r\n" );
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document.write( "Also, notice that \"1\" is a complex unit number too, with the argument arg(1) = 0.\r\n" );
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document.write( "Applying the same proof as in (1), we get\r\n" );
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document.write( " arg(zw+1) = 
= 
= 
mod . (3)\r\n" );
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document.write( "Using (2), we can continue line (3) this way\r\n" );
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document.write( " arg(zw+1) = 
mod . (4)\r\n" );
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document.write( "Comparing (1) with (4), we see that\r\n" );
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document.write( " arg(z+w) = arg(zw+1) mod 
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document.write( "It means that complex numbers z+w and (zw+1) are either co-directed vectors or oppositely directed vectors.\r\n" );
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document.write( "In any case, it implies that the number 
has the argument 0 or mod ,\r\n" );
document.write( "i.e. is a real number.\r\n" );
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document.write( "At this point, the proof (b) is complete.\r\n" );
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document.write( "Solved, proved and completed, in full.\r
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document.write( "![\"\"](\"/images/stars/stars-13.gif\")
