Question

Using a number line, what whole number and rational number to the nearest half is sqrt 10 between?

Answer 1

To determine where \(\sqrt{10}\) lies on a number line and find the whole number and rational number to the nearest half, we can start by calculating the approximate value of \(\sqrt{10}\).

Since \(3^2 = 9\) and \(4^2 = 16\), we know that:

\[
3 < \sqrt{10} < 4
\]

Now, to find a more precise value, we can do some additional checks:

- \(3.1^2 = 9.61\) which is less than 10
- \(3.2^2 = 10.24\) which is greater than 10

Thus, we can narrow it down further:

\[
3.1 < \sqrt{10} < 3.2
\]

To estimate further, we can check:

- \(3.15^2 = 9.9225\) which is less than 10
- \(3.16^2 = 10.0576\) which is greater than 10

This gives us:

\[
3.15 < \sqrt{10} < 3.16
\]

To round \(\sqrt{10}\) to the nearest half, we can look at the two nearest halves around \(3.15\) and \(3.16\):

- The nearest halves are \(3.0\) and \(3.5\).
- Since \(3.15\) is closer to \(3.0\) (specifically, \(3.15 - 3.0 = 0.15\) and \(3.5 - 3.15 = 0.35\)), we round \(\sqrt{10}\) down to \(3.0\).

Thus, to the nearest half, we conclude that:

1. The whole number \(\sqrt{10}\) is between is \(3\).
2. The rational number to the nearest half is \(3.0\).

Therefore, \(\sqrt{10}\) is between \(3\) and \(3.0\) on the number line.