Question

What irrational number is between square root of 14 and square root of 18

Answer 1

To find an irrational number between \(\sqrt{14}\) and \(\sqrt{18}\), we first need to approximate these square roots.

Calculating the approximate values:

\[
\sqrt{14} \approx 3.74
\]
\[
\sqrt{18} \approx 4.24
\]

Now we see that the number \(4\) lies between these two square roots. The number \(4\) itself is not irrational, but we can find an irrational number between \(\sqrt{14}\) and \(\sqrt{18}\).

One way to generate an irrational number in this range is to take the average of \(\sqrt{14}\) and \(\sqrt{18}\). The average is given by:

\[
x = \frac{\sqrt{14} + \sqrt{18}}{2}
\]

Calculating this gives:

\[
\sqrt{14} \approx 3.74, \quad \sqrt{18} \approx 4.24
\]
\[
x \approx \frac{3.74 + 4.24}{2} \approx \frac{8}{2} = 4
\]

But note that while \(x\) could be \(4\), to find an irrational number closer to the square roots, we can choose a number like:

\[
\frac{\sqrt{14} + \sqrt{18}}{2} + \frac{1}{10} \text{ or } \frac{\sqrt{14} + \sqrt{18}}{2} - \frac{1}{10}
\]

For instance, we can consider:

\[
\sqrt{15} \text{ or } \sqrt{16} \text{ (which is 4 and isn't irrational)}
\]

Another valid choice is:

\[
\sqrt{15} \approx 3.87
\]

This is an irrational number that lies between \(\sqrt{14}\) and \(\sqrt{18}\).

Thus, a reasonable irrational number between \(\sqrt{14}\) and \(\sqrt{18}\) is:

\[
\sqrt{15}
\]