To find where \( \sqrt{10} \) lies on the number line, we can first estimate its value.
Calculating:
\[ \sqrt{9} = 3 \quad \text{and} \quad \sqrt{16} = 4 \]
Since \( 9 < 10 < 16 \), we know:
\[ 3 < \sqrt{10} < 4 \]
To get a better estimate, we can test the value \( 3.1 \):
\[ (3.1)^2 = 9.61 \quad \text{(too low)} \]
Now let’s try \( 3.2 \):
\[ (3.2)^2 = 10.24 \quad \text{(too high)} \]
So, \( \sqrt{10} \) is between \( 3.1 \) and \( 3.2 \).
Next, we can get a more precise estimate. Trying \( 3.15 \):
\[ (3.15)^2 = 9.9225 \quad \text{(too low)} \]
Now try \( 3.16 \):
\[ (3.16)^2 = 10.0356 \quad \text{(too high)} \]
So, we conclude that:
\[ 3.15 < \sqrt{10} < 3.16 \]
Approximately, we can round \( \sqrt{10} \) to \( 3.16 \), which is slightly above \( 3.1 \) and below \( 3.2 \).
Now, if we want to express \( \sqrt{10} \) to the nearest half, we find that it is closest to \( 3.5 \).
However, strictly on whole numbers and the rational number, to the nearest half:
The whole number range for \( \sqrt{10} \) is between \( 3 \) and \( 4 \), and the nearest half for rational approximation is \( 3.5 \).
Thus, we can say:
\(\sqrt{10}\) is between 3 and 3.5.