document.write( "Question 1196050: Which subsets of the real number system are closed under division by a nonzero number? \n" ); document.write( "
Algebra.Com's Answer #828741 by ikleyn(52750)\"\" \"About 
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\n" ); document.write( "Which subsets of the real number system are closed under division by a nonzero number?
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\n" ); document.write( "\n" ); document.write( "The meaning of your request is really  UNCLEAR.\r
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\n" ); document.write( "\n" ); document.write( "It is written in so mathematically illiterate way,  that I should spend half an hour  (or more)  to explain  WHY  it is so,\r
\n" ); document.write( "\n" ); document.write( "and explain  HOW  it  SHOULD  BE,  which  (time)  I definitely do not have to spend it for nothing.\r
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\n" ); document.write( "\n" ); document.write( "When a person comes with such or similar question,  he/she should have some preliminary pre-requisites \r
\n" ); document.write( "\n" ); document.write( "and formulate his/her question correctly in the frame of these pre-requisites,  in mathematically correct form.\r
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\n" ); document.write( "\n" ); document.write( "Actually,  the field of real numbers has  INFINITELY  MANY  sub-fields.\r
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document.write( "They are sub-fields \r\n" );
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document.write( "    - Q (rational numbers), \r\n" );
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document.write( "    - \"Q%28sqrt%282%29%29\"  (an extension of Q with irrationality \"sqrt%282%29\"),\r\n" );
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document.write( "    - \"Q%28sqrt%283%29%29\"  (an extension of Q with irrationality \"sqrt%283%29\"),\r\n" );
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document.write( "    - \"Q%28sqrt%285%29%29\"  (an extension of Q with irrationality \"sqrt%285%29\"),\r\n" );
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document.write( "       . . . and infinitely many other sub-fields of the form \"Q%28sqrt%28n%29%29\", where positive integer number n is not a perfect square, . . . \r\n" );
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document.write( "    - \"Q%28sqrt%282%29%2C+sqrt%283%29%29\"  (an extension of Q with irrationalities \"sqrt%282%29\" and \"sqrt%283%29\"),\r\n" );
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document.write( "    - \"Q%28sqrt%282%29%2C+sqrt%285%29%29\"  (an extension of Q with irrationalities \"sqrt%282%29\" and \"sqrt%285%29\"),\r\n" );
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document.write( "    - \"Q%28sqrt%283%29%2C+sqrt%285%29%29\"  (an extension of Q with irrationalities \"sqrt%283%29\" and \"sqrt%285%29\"),\r\n" );
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document.write( "    . . . and infinitely many other sub-fields of the form \"Q%28sqrt%28m%29%2Csqrt%28n%29%29\", where m and n are different non-square positive integer numbers, . . . \r\n" );
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document.write( "    - \"Q%28root%283%2C2%29%29\"  (an extension of Q with irrationality \"root%283%2C2%29\"),\r\n" );
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document.write( "    - \"Q%28root%283%2C3%29%29\"  (an extension of Q with irrationality \"root%283%2C3%29\"),\r\n" );
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document.write( "    - \"Q%28root%283%2C5%29%29\"  (an extension of Q with irrationality \"root%283%2C5%29\"),\r\n" );
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document.write( "    . . . and infinitely many other sub-fields of the form \"Q%28root%283%2Cn%29%29\", where integer number n is not a perfect cube, . . . \r\n" );
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document.write( "and so on and so on . . . - - - infinitely many others.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Each of these sub-fields is a sub-set of real numbers,  closed relative addition,  subtraction,
\n" ); document.write( "multiplication and division by a non-zero elements of these sub-fields.\r
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\n" ); document.write( "\n" ); document.write( "By the way,  when in the school,  you learn about getting rid of irrationality  \"m%2Bsqrt%28n%29\"  in denominators,\r
\n" ); document.write( "\n" ); document.write( "you actually learn that the extension  \"Q%28sqrt%28n%29%29\"  is a field over  Q.\r
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