Gujarat BoardEnglish MediumSTD 9MathsNumbers System2 Marks
Question
Examine whether the following numbers are rational or irrational.
$3+\sqrt{3}$
✓
Answer
Let us assume, to the contrary, that $3+\sqrt{3}$ is rational.
Then, $3+\sqrt{3}=\frac{\text{p}}{\text{q}},$ where $p$ and $q$ are coprime and $\text{q}\neq0.$
$\Rightarrow\sqrt{3}=\frac{\text{p}}{\text{q}}-3$
$\Rightarrow\sqrt{3}=\frac{\text{p}-3\text{q}}{\text{q}}$
Since, $p$ and $q$ are are integers.
$\Rightarrow\frac{\text{p}-3\text{q}}{\text{q}}$ is rational.
So, $\sqrt{3}$ is also rational.
But this contradicts the fact that $\sqrt{3}$ is irrational.
This contradiction has arisen because of our incorrect assumption that $3+\sqrt{3}$ is rational.
Hence, $3+\sqrt{3}$ is irrational.
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