If x is rational and y is irrational, then x + y is irrational
Assume $x + y$ is rational
\begin{equation} \begin{aligned} \Rightarrow & \frac{n}{m} = x + y && m, n \in \mathbb{N} \\ \Rightarrow & \frac{n}{m} - \frac{n_1}{m_1} = y && m_1, n_1 \in \mathbb{N} \\ \Rightarrow & \frac{n m_1}{m m_1} - \frac{n_1 m}{m m_1} = y \\ \Rightarrow & \frac{n m_1 - n_1 m}{m m_1} = y \\ \Rightarrow & \mbox{y is rational} \\ \Rightarrow & \mbox{this contracts y is irrational} \\ \Rightarrow & \mbox{y is irrational} \\ \end{aligned} \end{equation}