How many irrational numbers are there between 11−−√ and 17−−√?

Answer 1

First, we need to find the ceiling of $\sqrt{11}$ and the floor of $\sqrt{17}$, since the irrational numbers between $\sqrt{11}$ and $\sqrt{17}$ will correspond to the irrational numbers between $\sqrt{11}$ and $\lceil\sqrt{11}\rceil$, and between $\lfloor\sqrt{17}\rfloor$ and $\sqrt{17}$. Note that $\sqrt{11}$ is approximately $3.317$, while $\sqrt{17}$ is approximately $4.123$. Thus, the irrational numbers between $\sqrt{11}$ and $\sqrt{17}$ will correspond to the irrational numbers between $3$ and $4$. Since there are no integers between $3$ and $4$, there are $\boxed{0}$ irrational numbers between $\sqrt{11}$ and $\sqrt{17}$. (There are still infinitely many, but none of them are rational.)