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.
$\Rightarrow\sqrt{3}=\frac{\text{p}-3\text{q}}{\text{q}}$
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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