Maharashtra BoardEnglish MediumSTD 9MathsNumber System2 Marks
Question
Examine whether the following numbers are rational or irrational.$\sqrt{7}-2$
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Answer
Let us assume, to the contrary, that $\sqrt{7}-2$ is rational. Then, $\sqrt{7}-2=\frac{\text{p}}{\text{q}},$ where p and q are coprime and $\text{q}\neq0.$$\Rightarrow\sqrt{7}=\frac{\text{p}}{\text{q}}+2$
$\Rightarrow\sqrt{7}=\frac{\text{p}+2\text{q}}{\text{q}}$
Since, p and q are are integers.$\Rightarrow\frac{\text{p}+2\text{q}}{\text{q}}$ is rational.
So, $\sqrt{7}$ is also rational. But this contradicts the fact that $\sqrt{7}$ is irrational. This contradiction has arisen because of our incorrect assumption that $\sqrt{7}-2$ is rational. Hence, $\sqrt{7}-2$ is irrational.
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