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InterviewSolution
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Prove that `sqrt(11)` is irrational. |
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Answer» Let, if possible, `sqrt(11)` be rational and its simplest form be`(a)/(b).` Then a and b are integers and having no common factor other than 1 and `b ne 0`. Now, `sqrt(11) = (a)/(b)` `rArr` `11 = (a^(2))/(b^(2))` `rArr` `a^(2) = 11b^(2)" "`...(1) `therefore` As `11b^(2)` is divisible by 11. `rArr` `a^(2)` is divisible by 11. `rArr` a is divisible by 11. Let a = 11c, for some integer c. From equation (1) `(11c)^(2) = 11b^(2)` `rArr` `b^(2) = 11c^(2)` But `11c^(2)` is divisible by 11. `therefore b^(2)` is divisible by 11. `rArr` b is divisible by 11. Let b = 11d, for some integer d. Thus, 11 is a common factor of a and b both. But it contradicts the fact that a and b have no common factor other than 1. So, our supposition is wrong. Hence, `sqrt(11)` is irrational. `" "`Hence Proved. |
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