Question

show and example of an irrational number between 9 and 11

Answer 1

An example of an irrational number between 9 and 11 is \( \sqrt{100} + \sqrt{2} \).

Calculating this, we have:

\[
\sqrt{100} = 10
\]

So,

\[
\sqrt{100} + \sqrt{2} = 10 + \sqrt{2}
\]

Since \( \sqrt{2} \) is approximately 1.414, we find:

\[
10 + \sqrt{2} \approx 10 + 1.414 = 11.414
\]

However, to ensure it falls between 9 and 11, we can simply consider:

\(
10 - \sqrt{2}
\)

This gives us an irrational number:

\[
10 - \sqrt{2} \approx 10 - 1.414 = 8.586
\]

To find an example that is definitely between 9 and 11, we can consider other values, such as:

\(\sqrt{97}\) or \(\sqrt{99}\)

For example, calculating \(\sqrt{97}\):

\[
\sqrt{97} \approx 9.848
\]

This confirms that \(\sqrt{97}\) is an irrational number that lies between 9 and 11.