Prove that $\sqrt{2}$ is an irrational number.

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Step 1: Assume $\sqrt{2} = \frac{p}{q}$, where $q 
eq 0$ and $p$, $q$ are co-primes.
Step 2: $2q^2 = p^2 \Rightarrow p^2$ is divisible by 2 $\Rightarrow p$ is divisible by 2. Let $p = 2a$.
Step 3: $4a^2 = 2q^2 \Rightarrow q^2 = 2a^2 \Rightarrow q$ is divisible by 2.
Step 4: This contradicts that $p$ and $q$ are co-primes. $	herefore \sqrt{2}$ is irrational.

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