Give an irrational numbers between `0.505005000 . . . and 0.525225222 . . . .

Explanation

To find an irrational number between the two given numbers \(0.505005000...\) and \(0.525225222...\), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Numbers**:
Let \(a = 0.505005000...\) and \(b = 0.525225222...\).

2. **Compare the Numbers**:
We need to confirm that \(a < b\).
- The first digit after the decimal for both numbers is 5.
- The second digit for \(a\) is 0, while for \(b\) it is 2.
- Since \(0 < 2\), we can conclude that \(a < b\).

3. **Find a Midpoint**:
To find a number between \(a\) and \(b\), we can calculate the average of \(a\) and \(b\):
\[ x = \frac{a + b}{2} = \frac{0.505005000 + 0.525225222}{2} \]

4. **Calculate the Average**:
Performing the addition:
\[ 0.505005000 + 0.525225222 = 1.030230222 \]
Now divide by 2:
\[ x = \frac{1.030230222}{2} = 0.515115111 \]

5. **Check if \(x\) is Irrational**:
The number \(0.515115111...\) is a repeating decimal, which can be expressed as a fraction. Therefore, we need to find a different irrational number.

6. **Construct an Irrational Number**:
We can create an irrational number by adding a small irrational component to our midpoint.
For example, we can take:
\[ x = 0.515115111 + \sqrt{2}/1000 \]
This will ensure that \(x\) remains between \(a\) and \(b\) and is irrational.

7. **Final Answer**:
Thus, one possible irrational number between \(0.505005000...\) and \(0.525225222...\) is:
\[ x = 0.515115111 + \frac{\sqrt{2}}{1000} \]