document.write( "Question 1207483: Simplify the square root. Write your answer in the form a + bi. \r
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\n" ); document.write( "\n" ); document.write( "Do I FOIL the radicand? \n" ); document.write( "
Algebra.Com's Answer #845381 by ikleyn(52754)\"\" \"About 
You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( "Do I FOIL the radicand?
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document.write( "Working with radicand, you may do FOIL or you may use shortcut  \"%28a%2Bb%29%2A%28a-b%29\" = \"a%5E2\" - \"b%5E2\".\r\n" );
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document.write( "In any case and in any way, you will get the same value of -25 for the radicand.\r\n" );
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document.write( "So, your first simplification step is  \"sqrt%28%284%2B3i%29%2A%283i-4%29%29\" = \"sqrt%28-25%29\".\r\n" );
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document.write( "    |   But the most interesting things start from here.   |\r\n" );
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document.write( "The fact is that square root of complex number in complex domain ALWAYS has TWO VALUES,\r\n" );
document.write( "that have opposite signs. Concretely,\r\n" );
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document.write( "    \"sqrt%28-25%29\" = +/- 5i.\r\n" );
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document.write( "Yes, I did not misspeak: \"sqrt%28-25%29\" has two values. One value is 5i, another value is -5i.\r\n" );
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document.write( "It is because  \"%285i%29%5E2\" = -25  and  \"%28-5i%29%5E2\" = -25.\r\n" );
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document.write( "In real number domain, there is a selected set of positive numbers;\r\n" );
document.write( "therefore, for positive numbers we can get agreement that the square root\r\n" );
document.write( "of positive real number is a positive real number.\r\n" );
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document.write( "But in complex domain, there is no such selected set as positive numbers.\r\n" );
document.write( "Therefore, the square root ALWAYS has TWO different values in complex domain.\r\n" );
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document.write( "There is only one exclusion from this rule: it is the number 0 (zero), which has only one \r\n" );
document.write( "value of the square root.  All other (non-zero) complex numbers have TWO values\r\n" );
document.write( "of square root, and these two values have EQUAL RIGHTS to exist as the values of the square root\r\n" );
document.write( "in complex domain.\r\n" );
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document.write( "Therefore, the right answer to this problem question is that  \"sqrt%28%284+%2B+3i%29%2A%283i+-+4%29%29\" has\r\n" );
document.write( "two different values, namely, 5i and -5i.\r\n" );
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document.write( "You can write them in the form a+bi:  5i = 0 + 5i;\r\n" );
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document.write( "                                     -5i = 0 - 5i.\r\n" );
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document.write( "The answer in WolframAlpha \r\n" );
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document.write( "https://www.wolframalpha.com/input?i=sqrt%28+%284%2B3i%29%283i-4%29+%29+%3D+5i\r\n" );
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document.write( "stating that \"sqrt%28-25%29\" = 5i  is INCORRECT in this case, because it is INCOMPLETE.  \r\n" );
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document.write( "        It gives only one value  5i  for  \"sqrt%28-25%29\"  in complex domain,\r\n" );
document.write( "        while there are TWO different values for it,  5i  and  -5i.\r\n" );
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document.write( "SIMILARLY, \"sqrt%28-1%29\" has two values, i and -i.\r\n" );
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document.write( "    I will shock you even more, when I'll say that in complex domain, \r\n" );
document.write( "    the n-th root of any non-zero complex number has n different values.\r\n" );
document.write( "                            But it is TRUE.\r\n" );
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document.write( "In fact, only those, who professionally know complex numbers and who learned/studied\r\n" );
document.write( "a full course of complex numbers analysis in combination with abstract algebra\r\n" );
document.write( "at universities, do understand this fully.\r\n" );
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document.write( "Only those have really NON-SELF-CONTRADICTORY understanding of this subject in their minds\r\n" );
document.write( "(under the condition that they learned from good professors and from good textbooks).\r\n" );
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\n" ); document.write( "\n" ); document.write( "So,  let me repeat it again,  because it is of great importance.\r
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document.write( "    The answer in WolframAlpha \r\n" );
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document.write( "    https://www.wolframalpha.com/input?i=sqrt%28+%284%2B3i%29%283i-4%29+%29+%3D+5i\r\n" );
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document.write( "    stating that \"sqrt%28-25%29\" = 5i  is INCORRECT in this case, because it is INCOMPLETE.  \r\n" );
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document.write( "            It gives only one value  5i  for  \"sqrt%28-25%29\"  in complex domain,\r\n" );
document.write( "            while there are TWO different values for it,  5i  and  -5i.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Thus the reference to  WolframAlpha  in this case  \" for confirmation \"  does not work.\r
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\n" ); document.write( "\n" ); document.write( "WolframAlpha  is  WRONG  in this case.\r
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\n" ); document.write( "\n" ); document.write( "Dear tutor @math_tutor2020,  read  THIS  from Wikipedia
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\n" ); document.write( "\n" ); document.write( "https://en.wikipedia.org/wiki/Nth_root#:~:text=Every%20non%2Dzero%20number%20x,root%20of%200%20is%200.&text=All%20nth%20roots%20of%20rational,of%20integers%20are%20algebraic%20integers.\r
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\n" ); document.write( "\n" ); document.write( "Read this part,  highlighted in the referred text:\r
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document.write( "    Every non-zero number x, real or complex, \r\n" );
document.write( "    has n different complex number nth roots. \r\n" );
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\n" ); document.write( "\n" ); document.write( "It is well known fact after  deMoivre,  Euler,  Riemann . . . \r
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\n" ); document.write( "\n" ); document.write( "And after reading it,  please stop argue with me.\r
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\n" ); document.write( "\n" ); document.write( "Regarding notices by @math_tutor2020 about values of square roots.\r
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document.write( "On real number domain, \r\n" );
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document.write( "    - square roots ALWAYS have two values for positive numbers: one positive value and one (opposite) negative value;\r\n" );
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document.write( "    - square root         has  one value  for the number of zero;\r\n" );
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document.write( "    - square root is not defined for negative numbers.\r\n" );
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document.write( "Sometime (quite often), we want to consider square root functions over real numbers.\r\n" );
document.write( "Then, to satisfy a vertical line test for functions, we accept an agreement that we consider \r\n" );
document.write( "only positive values of square root function.\r\n" );
document.write( "It is critically important for a student to know and to understand, that it is only an AGREEMENT;\r\n" );
document.write( "that square root ALWAYS has two values due to its nature (when defined).  \r\n" );
document.write( "It is only by agreement we accept that the square root real functions take only positive values.\r\n" );
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document.write( "In complex number domain, square root ALWAYS has two values.\r\n" );
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document.write( "In school course of complex numbers, square root functions are not considered.\r\n" );
document.write( "In university courses of complex analysis, there are two levels of consideration square root functions\r\n" );
document.write( "and, generally, n-root functions.\r\n" );
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document.write( "One level, which is basic for all specialties, except of graduate in Math, considers square functions \r\n" );
document.write( "as two-values and n-root functions as n-values.\r\n" );
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document.write( "Other level, for graduate students in Math, specializing in Topology, Algebraic geometry, number theory,\r\n" );
document.write( "consider square root functions and n-root functions over Riemannian surfaces,\r\n" );
document.write( "which makes them 1-valued. But this level uses totally different conceptions and is far from the school level.\r\n" );
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document.write( "        It is what (ideally) a person who studies \r\n" );
document.write( "        this great piece of Math should know.\r\n" );
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document.write( "In reality, of 70,000 US school Math teachers, probably, only 70 persons do understand it clearly.\r\n" );
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document.write( "The rest of the teachers do not have clear understanding on the subject and are totally \r\n" );
document.write( "intimidated by the vertical line test for square root functions.\r\n" );
document.write( "They themselves are intimidated and intimidate their students, too;\r\n" );
document.write( "so, in this community, there is NO right understanding of the nature of things.\r\n" );
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