Question

which of the following correctly describes the quotient of a nonzero rational number and an irrational number? a. always irrational b. always rational c. can be either rational or irrational depending on the rationals used d. a combination of rational and irrational

Explanation:

We use proof by contradiction. Assume the quotient of a nonzero rational number $\frac{a}{b}$ (where $a,b$ are integers, $a
eq0,b
eq0$) and an irrational number $x$ is rational, so $\frac{\frac{a}{b}}{x}=\frac{c}{d}$ (where $c,d$ are integers, $c
eq0,d
eq0$). Rearranging gives $x=\frac{ad}{bc}$. Since $a,b,c,d$ are integers, $\frac{ad}{bc}$ is rational, which contradicts $x$ being irrational. Thus the quotient must be irrational.

Answer:

A. always irrational