Prove that


Prove that $(2+\sqrt{3})$ is an irrational number, given that $\sqrt{3}$ is an irrational number.



Let us assume that $(2+\sqrt{3})$ is a rational number.

Thus, $(2+\sqrt{3})$ can be represented in the form of $\frac{p}{q}$, where $p$ and $q$ are integers, $q \neq 0, p$ and $q$ are co-prime numbers.


$\Rightarrow \sqrt{3}=\frac{p}{q}-2$

$\Rightarrow \sqrt{3}=\frac{p-2 q}{q}$

Since, $\frac{p-2 q}{q}$ is rational $\Rightarrow \sqrt{3}$ is rational

But, it is given that $\sqrt{3}$ is an irrational number.

Therefore, our assumption is wrong.

Hence, $2+\sqrt{3}$ is an irrational number.


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