The number associated with point $A$ on the number line appears to be 0.078. The question asks to round this number to the nearest thousandth.

Since the number is already given to the thousandth place, there is no need to round further. Therefore, the number $A$ rounded to the nearest thousandth is:

$\boxed{0.078}$

Would you like further clarification on rounding or have any additional questions?

Here are 5 related questions to consider:

1. What is the general rule for rounding numbers to the nearest thousandth?
2. How do you round a number like 0.07845 to the nearest thousandth?
3. If $A$ was placed at 0.0787, what would the rounded value be?
4. Can you explain how the number line helps in visualizing rounding?
5. What happens when the digit after the thousandth place is exactly 5?

Tip: When rounding, always look at the digit immediately after the place you're rounding to. If it's 5 or greater, round up; if it's less than 5, round down.