To determine which irrational number can be added to [tex]\(\pi\)[/tex] to produce a rational result, we should evaluate each option carefully:
1. [tex]\(\frac{1}{\pi}\)[/tex]:
When we add [tex]\(\frac{1}{\pi}\)[/tex] to [tex]\(\pi\)[/tex], we get:
[tex]\[
\pi + \frac{1}{\pi}
\][/tex]
Since [tex]\(\pi\)[/tex] is an irrational number, [tex]\(\frac{1}{\pi}\)[/tex] is also irrational. The sum of an irrational number and an irrational number is generally irrational. Therefore, [tex]\(\pi + \frac{1}{\pi}\)[/tex] is irrational.
2. [tex]\(-\frac{1}{\pi}\)[/tex]:
When we add [tex]\(-\frac{1}{\pi}\)[/tex] to [tex]\(\pi\)[/tex], we get:
[tex]\[
\pi - \frac{1}{\pi}
\][/tex]
Similar to the previous case, since both [tex]\(\pi\)[/tex] and [tex]\(-\frac{1}{\pi}\)[/tex] are irrational numbers, their sum is generally irrational. Thus, [tex]\(\pi - \frac{1}{\pi}\)[/tex] is irrational.
3. [tex]\(-\pi\)[/tex]:
When we add [tex]\(-\pi\)[/tex] to [tex]\(\pi\)[/tex], we get:
[tex]\[
\pi + (-\pi) = 0
\][/tex]
Zero is a rational number. Hence, adding [tex]\(-\pi\)[/tex] to [tex]\(\pi\)[/tex] results in a rational number.
4. [tex]\(\pi\)[/tex]:
When we add [tex]\(\pi\)[/tex] to [tex]\(\pi\)[/tex], we get:
[tex]\[
\pi + \pi = 2\pi
\][/tex]
Since [tex]\(\pi\)[/tex] is irrational, [tex]\(2\pi\)[/tex] is also irrational. Therefore, the sum [tex]\(\pi + \pi\)[/tex] is irrational.
Based on the analysis above, the only option that results in a rational number when added to [tex]\(\pi\)[/tex] is:
[tex]\[
-\pi
\][/tex]
Thus, the correct answer is:
[tex]\[
-\pi
\][/tex]
