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

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Step 1: Let us assume that $2 + \sqrt{3}$ is rational. Let $2 + \sqrt{3} = \frac{p}{q}$; $q 
eq 0$ and $p$, $q$ are integers. $\Rightarrow \sqrt{3} = \frac{p - 2q}{q}$
Step 2: $p$ and $q$ are integers, $	herefore p - 2q$ is an integer. $\Rightarrow \frac{p - 2q}{q}$ is a rational number.
Step 3: This contradicts our assumption that $\sqrt{3}$ is an irrational number. $	herefore 2 + \sqrt{3}$ is an irrational number.

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