At Dr. Math, I read about irrational number being not a closed system. The writer proved it by showing that multiplied by itself produces a non-irrational number, and that is also a non-irrational number.

Now, if I multiply by 2, I get an irrational number , and when I divide by 2, I also get an irrational number .

The question remains that if I have or , where , will I definitely get an irrational number? Is there an existing theorem in regard to this?