To find the whole number and the rational number (to the nearest half) that \( \sqrt{20} \) falls between, let's first compute the value of \( \sqrt{20} \).
Step 1: Calculate \( \sqrt{20} \)
We know that:
\[ \sqrt{20} = \sqrt{4 \times 5} = \sqrt{4} \times \sqrt{5} = 2\sqrt{5} \]
Now to find a decimal approximation of \( \sqrt{5} \), we can use a calculator or approximate it. The value of \( \sqrt{5} \) is approximately \( 2.236 \).
Then,
\[ \sqrt{20} \approx 2 \times 2.236 \approx 4.472 \]
Step 2: Determine whole numbers around \( \sqrt{20} \)
We can see that \( 4.472 \) is between the whole numbers \( 4 \) and \( 5 \).
Step 3: Find the rational number to the nearest half
To find the nearest half, we can round \( 4.472 \) to the nearest half unit. The nearest halves to \( 4.472 \) are:
- \( 4.0 \)
- \( 4.5 \)
- \( 5.0 \)
Since \( 4.472 \) is closer to \( 4.5 \) than to \( 4.0 \), we consider \( 4.5 \) as the nearest half.
Conclusion
Thus, \( \sqrt{20} \) is between:
- Whole number: \( 4 \) and \( 5 \)
- Nearest half: \( 4.5 \)
So, \( \sqrt{20} \) is between the whole number \( 4 \) and the rational number \( 4.5 \).
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