Answer
**Short Answer:** No, the average of two irrational numbers need not be irrational.
Work Step by Step
**Short Answer:** No, the average of two irrational numbers need not be irrational. For example, consider
\[
\sqrt{2}
\quad\text{and}\quad
2 - \sqrt{2}.
\] Both are irrational, yet their average is
\[
\frac{\sqrt{2} + (2 - \sqrt{2})}{2}
\;=\; \frac{2}{2}
\;=\; 1,
\] which is rational. This provides a counterexample.
