InterviewSolution
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InterviewSolution
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Let x and y be rational and irrational numbers , respectively. Is x+y necessarily an irrational number ? Give an example in suppot of your answer. |
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Answer» Yes ,(x+y) is necessarily an irrational number . e.g… Let `" " x=2, Y=sqrt3` Then, ` " " x+y =2+sqrt3` if possible, let x+y =2 `+ sqrt3` be a rational number. Consider , ` " " a=2,+sqrt3` On Squaring both sides, we get `a^(2)=(2+sqrt3)^(2) " " [ "unsing identity"(a+b)^(2)=a^(2) + b^(2) + 2ab]` ` implies " " a^(2) = 2^(2) + (sqrt3)^(2))+2(2)(sqrt(3))` `a^(2)=4 +3+4sqrt3 implies (a^(2)-7)/4=sqrt(3)` So, a is rational `Rightarrow (a^(2) -7)/4` is rational ` implies sqrt3` is rational. But, this contradicits the fact that `sqrt3` is an irrational number. thus, our assumption is wrong. Hence, x+ y is an irraional number. |
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