document.write( "Question 1204483: Prove that the set of all n-tuples of rational numbers {(q1, q2, . . . , qn)|qi ∈ Q} ⊂ R^n is NOT a subspace of R^n. \n" ); document.write( "

Algebra.Com's Answer #840799 by ikleyn(52781) $\"\"$ $\"About$

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document.write( "The simplest way is to start from 1d-case.\r\n" );
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document.write( "1d-case is simply the set of rational numbers as a part of all real numbers on the number line.\r\n" );
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document.write( "If you multiply any rational number by irrational number, you will get irrational number,\r\n" );
document.write( "which is obvious.\r\n" );
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document.write( "Thus, when you multiply rational number by irrational, you leave the set of rational numbers.\r\n" );
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document.write( "Mathematicians say that \"rational numbers cannot withstand multiplication by irrational numbers \r\n" );
document.write( "and become irrationals\".\r\n" );
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document.write( "It explains everything in 1d-case.  In multidimensional space, everything is the same (everything is similar).\r\n" );
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