document.write( "Question 1196157: Find the modulus and argument of √(1+2i)/(1-2i) \n" ); document.write( "
Algebra.Com's Answer #828899 by ikleyn(52781)\"\" \"About 
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\n" ); document.write( "Find the modulus and argument of √(1+2i)/(1-2i)
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\n" ); document.write( "\n" ); document.write( "            The solution by  Edwin is fine.\r
\n" ); document.write( "\n" ); document.write( "            Another solution and another analysis is possible,  which is in my post below.\r
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\n" ); document.write( "\n" ); document.write( "            It allows to get the answer with significantly less computational efforts.\r
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document.write( "So, we consider  \"sqrt%28%281%2B2i%29%2F%281-2i%29%29\".    (1)\r\n" );
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document.write( "First consider the ratio under the square root  \"%281%2B2i%29%2F%281-2i%29\".    (2)\r\n" );
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document.write( "The numerator has the modulus  |1+2i| = \"sqrt%281%5E2+%2B+2%5E2%29\" = \"sqrt%285%29\".\r\n" );
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document.write( "The denominator has the modulus  |1-2i| = \"sqrt%281%5E2+%2B+%28-2%29%5E2%29\" = \"sqrt%285%29\".\r\n" );
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document.write( "Therefore, the ratio  \"%281%2B2i%29%2F%281-2i%29\"  has the modulus of  \"sqrt%285%29%2Fsqrt%285%29\" = 1.\r\n" );
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document.write( "Hence, the original complex number \"sqrt%28%281%2B2i%29%2F%281-2i%29%29\"  has the modulus  \"sqrt%281%29\" = 1.\r\n" );
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document.write( "    +-----------------------------------------------+\r\n" );
document.write( "    |    Thus we completed with the modulus.        |\r\n" );
document.write( "    |    Now, let's determine the argument of (1).  |\r\n" );
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document.write( "Consider the ratio under the square root \"highlight%28highlight%28again%29%29\"  \"%281%2B2i%29%2F%281-2i%29\".    (3)\r\n" );
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document.write( "The numerator has the argument \"a\" in QI such that  tan(a) = 2/1 = 2.\r\n" );
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document.write( "The denominator has the argument \"b\" in QIV such that tan(b) = (-2)/1 = -2.\r\n" );
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document.write( "Therefore, b = -a  (which is obvious).\r\n" );
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document.write( "Hence, the ratio  \"%281%2B2i%29%2F%281-2i%29\"  has the argument  a - b = a - (-a) = 2a.\r\n" );
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document.write( "When we take square root of (3), the argument of the resulting complex number will be \"%282a%29%2F2\" = a.\r\n" );
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document.write( "Thus we get the ANSWER: \r\n" );
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document.write( "    |    the modulus of the sough complex number is 1;                           |\r\n" );
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document.write( "    |    the argument of the sought complex number is \"a\" such that tan(a) = 2,  |\r\n" );
document.write( "    |        i.e. a = arctan(2) = 1.10715 radians = 63.435 degrees (rounded).    |     \r\n" );
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document.write( "For completeness, when we take square root of (3), we obtain, actually, two complex numbers.\r\n" );
document.write( "First one has the argument \"a\", as we found it above; the second complex number has the argument a+\"pi\".\r\n" );
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document.write( "So, the second value of the square root has the modulus of 1 and the argument arctan(2)+\"pi\" = 243.435 degrees.\r\n" );
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\n" ); document.write( "\n" ); document.write( "The answer is the same as in the post by Edwin.\r
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\n" ); document.write( "\n" ); document.write( "On complex numbers,  see relevant lessons\r
\n" ); document.write( "\n" ); document.write( "    - Complex numbers and arithmetical operations on them\r
\n" ); document.write( "\n" ); document.write( "    - Complex plane\r
\n" ); document.write( "\n" ); document.write( "    - Addition and subtraction of complex numbers in complex plane\r
\n" ); document.write( "\n" ); document.write( "    - Multiplication and division of complex numbers in complex plane\r
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\n" ); document.write( "\n" ); document.write( "    - Solved problems on taking roots of complex numbers\r
\n" ); document.write( "\n" ); document.write( "    - Solved problems on arithmetic operations on complex numbers\r
\n" ); document.write( "\n" ); document.write( "    - Solved problem on taking square root of complex number\r
\n" ); document.write( "\n" ); document.write( "in this site.\r
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\n" ); document.write( "\n" ); document.write( "Also,  you have this free of charge online textbook on  ALGEBRA-II  in this site\r
\n" ); document.write( "\n" ); document.write( "    - ALGEBRA-II - YOUR ONLINE TEXTBOOK.\r
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\n" ); document.write( "\n" ); document.write( "The referred lessons are the part of this online textbook under the topic  \"Complex numbers\".\r
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\n" ); document.write( "\n" ); document.write( "Save the link to this textbook together with its description\r
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\n" ); document.write( "\n" ); document.write( "Free of charge online textbook in ALGEBRA-II
\n" ); document.write( "https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson\r
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\n" ); document.write( "\n" ); document.write( "into your archive and use when it is needed.\r
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