Assume that it is possible for the sum of a rational and an irrational number to be rational. In other words:
$$\frac{a}{b} + k = \frac{c}{d}$$
Rearranging:
$$\begin{align}k &= \frac{c}{d} - \frac{a}{b} \\ &= \frac{bc}{bd} - \frac{ad}{bd} \\ &= \frac{bc - ad}{bd}\end{align}$$
This implies that \(k\) is rational, which is a contradiction. Therefore, the sum of a rational and an irrational is always an irrational.