Question

the product of non zero rational and an irrational is

Answer 1

Question #15073 The product of non zero rational and an irrational is irrational. Please explain.

Solution Let $\alpha$ be irrational number and $q = \frac{m}{n}$ be any non-zero rational number $m \neq 0$ . Assume that $\alpha \cdot q = r$ , where $r$ is some rational number. So, $r = \frac{l}{k}$ , thus $\alpha = \frac{ln}{km}$ , which contradicts the fact that $\alpha$ is irrational. Hence, $\alpha \cdot q$ is irrational, providing that $q$ is non zero rational.