$\sqrt{2}$ .xyz $\approx$ 1.414213562.xyz

Assume, for the sake of contradiction, that $\sqrt{2}$ is a rational number.
Then $\sqrt{2}$ may be represented as an irreducible fraction of two coprime integers $a$ and $b$. $$ \begin{align*} \sqrt{2} &= \frac{a}{b} \\ 2b^2 &=a^2 \\ \end{align*} $$ Since $a^2$ is even, $a$ is also even as squares of odd integers are always odd. Let $a=2c$. $$ \begin{align*} 2b^2 &= (2c)^2 \\ b^2 &= 2c^2 \\ \end{align*} $$ Since $b$ is also even, this contradicts the definition of $a$ and $b$ that they must be coprime. Thus $\sqrt{2}$ must not be rational.

$\therefore \; \sqrt{2}$ is irrational.