Question

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

Explanation:

By proof by contradiction: Assume a rational number $r = \frac{p}{q}$ (where $p,q$ are integers, $q
eq0$) and an irrational number $x$. Suppose $r - x$ is rational, so $r - x = \frac{m}{n}$ (integers $m,n$, $n
eq0$). Rearranging gives $x = r - \frac{m}{n} = \frac{p}{q} - \frac{m}{n} = \frac{pn - qm}{qn}$, which would make $x$ rational, contradicting that $x$ is irrational. Thus, the difference must be irrational.

Answer:

A. always irrational