1. **Problem Statement:** Show that the sum of a rational number and an irrational number is irrational. 2. **Definitions:** - A rational number can be expressed as $\frac{p}{q}$ where $p, q$ are integers and $q \neq 0$. - An irrational number cannot be expressed as a ratio of two integers. 3. **Assume the contrary:** Suppose the sum of a rational number $r$ and an irrational number $i$ is rational. Let $r$ be rational and $i$ be irrational. 4. Let the sum be $s = r + i$, and assume $s$ is rational. 5. Rearranging, we get $i = s - r$. 6. Since $s$ and $r$ are rational, their difference $s - r$ is also rational. 7. This implies $i$ is rational, which contradicts the assumption that $i$ is irrational. 8. Therefore, the sum of a rational number and an irrational number must be irrational. **Final answer:** The sum of a rational number and an irrational number is irrational.