Quick answer
Any irrational number. I can't think of any exception to this statement.
sqrt(5) is irrational. 1/2 * sqrt(5) is still irrational.
-sqrt(5) is irrational. -1/2 * sqrt(5) is still irrational
2 + sqrt(5) is irrational. Multiplying this by 1/2 is still irrational.
1/2(2 + sqrt(5) = 1 + 1/2 * sqrt(5) ) is irrational.
Conclusion
There are no exceptions to this. All irrational numbers produce irrationals when multiplied by 1/2
