To find an irrational number between 4 and 8, we need to consider the properties of irrational numbers. An irrational number cannot be expressed as a simple fraction and has a non-repeating, non-terminating decimal expansion.
A straightforward approach to identify an irrational number within that range is to take the square root of a non-perfect square between the two endpoints.
For example, consider the number 5. Let's calculate the square root of 5:
\[ \sqrt{5} \approx 2.236 \]
Since 2.236 is between the integers 2 and 3, we can add this value to 4:
\[ 4 + \sqrt{5} \approx 4 + 2.236 = 6.236 \]
Now, 6.236 is clearly between 4 and 8, and since \(\sqrt{5}\) is irrational, \(4 + \sqrt{5}\) is also irrational.
Alternatively, we could also look for other irrational numbers like \(\pi\) or \(e\). The value of \(\pi\) is approximately 3.14, and:
\[ 4 + \pi \approx 4 + 3.14 = 7.14 \]
This is again between 4 and 8, and since \(\pi\) is irrational, \(4 + \pi\) is also irrational.
Therefore, one example of an irrational number between 4 and 8 is:
\[ 4 + \sqrt{5} \quad \text{or} \quad 4 + \pi \]
Both of these are valid irrational numbers located in that range.