You may add, for example, [tex] \sqrt{2} [/tex], or [tex] \pi[/tex], to any fraction, and the result will be irrational.
In fact, we know that [tex] \sqrt{2} [/tex] or [tex] \pi[/tex] can't be written as fraction. So, let's pretend that
[tex] \dfrac{a}{b} + \sqrt{2} = \dfrac{c}{d} [/tex]
which means, the sum of a fraction and an irrational is rational. This can't be true, because it would imply
[tex] \sqrt{2} = \dfrac{c}{d}-\dfrac{a}{b} = \dfrac{bc-ad}{bd} [/tex]
And so we have written [tex] \sqrt{2} [/tex] as a fraction, but this is impossible!
So, this proves that the sum of a rational and an irrational is irrational.
