Question

Show an example of a rational number that is between √10 and √11.

Answer 1

To find a rational number between \( \sqrt{10} \) and \( \sqrt{11} \), we first need to estimate the values of these square roots.

Calculating \( \sqrt{10} \):
\[
\sqrt{10} \approx 3.162
\]

Calculating \( \sqrt{11} \):
\[
\sqrt{11} \approx 3.317
\]

Now that we have approximate values for \( \sqrt{10} \) and \( \sqrt{11} \), we can find a rational number between these two values.

One simple rational number we can choose is \( 3.2 \).

Now we can verify that \( 3.2 \) is indeed between \( \sqrt{10} \) and \( \sqrt{11} \):
- \( 3.162 < 3.2 < 3.317 \)

Thus, \( 3.2 \) is a rational number that lies between \( \sqrt{10} \) and \( \sqrt{11} \). Another example could be \( \frac{32}{10} \) or \( \frac{16}{5} \), which is equivalent to \( 3.2 \).

Therefore, you can use \( 3.2 \) or \( \frac{16}{5} \) as a rational number between \( \sqrt{10} \) and \( \sqrt{11} \).