The sum, difference, product and quotient of two non-zero real numbers, from which one is rational and the other irrational, is irrational.

Proof. Let $a$ be a rational and $\alpha$ irrational number. Here we prove only that $\frac{a}{\alpha}$ is irrational -- the other cases are similar. If $\frac{a}{\alpha}$ were a rational number $r \neq 0$ then also $\alpha = ar^{-1}$ , would be rational as a product of two rationals. This contradiction shows that $\frac{a}{\alpha}$ is irrational.

Note. In the result, the words real, rational and irrational may be replaced resp. by the words complex, algebraic and transcendental or resp. by the words complex, real and imaginary (the last term here meaning, as commonly in Continental Europe, a complex number having non-zero imaginary part).