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Prove that $2 + \sqrt{3}$ is an irrational number, given that $\sqrt{3}$ is an irrational number.
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Let us assume that $2 + \sqrt{3}$ is rational
Let $2 + \sqrt{3} = \frac{P}{q}$ ; $q \neq 0$ and p, q are integers
$\Rightarrow \sqrt{3} = \frac{p-2q}{q}$
p and q are integers, $\therefore p - 2q$ is an integer
$\Rightarrow \frac{p - 2q}{q}$ is a rational number
$\Rightarrow \sqrt{3}$ is a rational number which contradicts our assumption that $\sqrt{3}$ is an irrational number.
$\Rightarrow 2 + \sqrt{3}$ is an irrational number
Let $2 + \sqrt{3} = \frac{P}{q}$ ; $q \neq 0$ and p, q are integers
$\Rightarrow \sqrt{3} = \frac{p-2q}{q}$
p and q are integers, $\therefore p - 2q$ is an integer
$\Rightarrow \frac{p - 2q}{q}$ is a rational number
$\Rightarrow \sqrt{3}$ is a rational number which contradicts our assumption that $\sqrt{3}$ is an irrational number.
$\Rightarrow 2 + \sqrt{3}$ is an irrational number