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Prove that root 6 is a irrational number????? |
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Answer» Let √6 be a rational number , then√6 = p÷q , where p,q are integers , q not = 0 and p,q have no common factors ( except 1 )=> 6 = p² ÷ q²=> p² = 2q² ................(i)As 2 divides 6q² , so 2 divides p² but 2 is a prime number=> 2 divides pLet p = 2m , where m is an integer .Substituting this value of p in (i) , we get(2m)² = 6q²=> 2m² = 3q²As 2 divides 2m² , 2 divides 3q²=> 2 divides 3 or 2 divide q²But 2 does not divide 3 , therefore , 2 divides q²=> 2 divides qThus , p and q have a common factor 2 . This contradicts that p and q have no common factors ( except 1 ).Hence , our supposition is wrong . Therefore , √6 is an irrational number. Dgtfgi |
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