To determine the number of irrational numbers between \(\sqrt{11}\) and \(\sqrt{17}\), we start by calculating the approximate values of these square roots:
- \(\sqrt{11} \approx 3.3166\)
- \(\sqrt{17} \approx 4.1231\)
The interval between \(\sqrt{11}\) and \(\sqrt{17}\) is approximately \( (3.3166, 4.1231) \).
Now, within any interval of real numbers, there are infinitely many irrational numbers. This is because the set of real numbers is dense, meaning between any two distinct real numbers (whether rational or irrational), there exists another real number, and specifically, both rational and irrational numbers can be found.
Thus, since there is a non-empty interval \( (3.3166, 4.1231) \) that contains both rational and irrational numbers, we conclude that there are infinitely many irrational numbers in this interval.
Therefore, the answer is:
\[ \text{There are infinitely many irrational numbers between } \sqrt{11} \text{ and } \sqrt{17}. \]
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