Question

Which number can be added to a rational number a to explain that the some of rational number in the in rational number is inrational

Answer 1

You may add, for example, $\sqrt{2}$, or $\pi$, to any fraction, and the result will be irrational.

In fact, we know that $\sqrt{2}$ or $\pi$ can't be written as fraction. So, let's pretend that

$\dfrac{a}{b} + \sqrt{2} = \dfrac{c}{d}$

which means, the sum of a fraction and an irrational is rational. This can't be true, because it would imply

$\sqrt{2} = \dfrac{c}{d}-\dfrac{a}{b} = \dfrac{bc-ad}{bd}$

And so we have written $\sqrt{2}$ as a fraction, but this is impossible!

So, this proves that the sum of a rational and an irrational is irrational.