Question

the difference of a rational number and an irrational number is _blank_ an irrational number.

which word correctly fills in the blank to create a true statement?

sometimes
always
never

Explanation:

Step 1: Assume the difference is rational

Let \(r\) be a rational number and \(i\) be an irrational number. Assume their difference is a rational number \(q\):

$$r - i = q$$

Step 2: Solve for the irrational number

Rearranging the equation to solve for \(i\) gives:

$$i = r - q$$

Step 3: Analyze the contradiction

Since the difference of two rational numbers \(r\) and \(q\) must be rational, \(i\) must be rational, which contradicts the assumption that \(i\) is irrational. Thus, the difference is always irrational.

Answer:

always