You may add, for example,
, or
, to any fraction, and the result will be irrational.
In fact, we know that
or
can't be written as fraction. So, let's pretend that
![\dfrac{a}{b} + \sqrt{2} = \dfrac{c}{d}](https://tex.z-dn.net/?f=%20%5Cdfrac%7Ba%7D%7Bb%7D%20%2B%20%5Csqrt%7B2%7D%20%3D%20%5Cdfrac%7Bc%7D%7Bd%7D%20)
which means, the sum of a fraction and an irrational is rational. This can't be true, because it would imply
![\sqrt{2} = \dfrac{c}{d}-\dfrac{a}{b} = \dfrac{bc-ad}{bd}](https://tex.z-dn.net/?f=%20%5Csqrt%7B2%7D%20%3D%20%5Cdfrac%7Bc%7D%7Bd%7D-%5Cdfrac%7Ba%7D%7Bb%7D%20%3D%20%5Cdfrac%7Bbc-ad%7D%7Bbd%7D%20)
And so we have written
as a fraction, but this is impossible!
So, this proves that the sum of a rational and an irrational is irrational.