Originally Posted by

**novice** At Dr. Math, I read about irrational number being not a closed system. The writer proved it by showing that $\displaystyle \sqrt{2}$ multiplied by itself produces a non-irrational number, and that $\displaystyle (\pi-1)+(6-\pi)=5$ is also a non-irrational number.

Now, if I multiply $\displaystyle \sqrt{2}$ by 2, I get an irrational number $\displaystyle 2\sqrt{2}$, and when I divide $\displaystyle \sqrt{2} $ by 2, I also get an irrational number $\displaystyle \frac{\sqrt{2}}{2}$.

The question remains that if I have $\displaystyle b\sqrt{a}$ or $\displaystyle \frac{\sqrt{a}}{b}$, where $\displaystyle a,b \in \mathbb{Z}$, will I definitely get an irrational number? Is there an existing theorem in regard to this?