Proof. Since $\alpha$ and $\beta$ are rational, it follows that $\alpha=a / b$ and $\beta=c / d$, where $a, b, c$, and $d$ are integers with $b \neq 0$ and $d \neq 0$. Then, each of the numbers
$$\begin{aligned} \alpha+\beta & =a / b+c / d=(a d+b c) / b d \\ \alpha-\beta & =a / b-c / d=(a d-b c) / b d \\ \alpha \beta & =(a / b) \cdot(c / d)=a c / b d \\ \alpha / \beta & =(a / b) /(c / d)=a d / b c \quad(\beta \neq 0) \end{aligned}$$
is rational, since it is the quotient of two integers with denominator different from zero.