To convert the repeating decimal \( 0.8\overline{8} \) (where the bar indicates that the 8 repeats indefinitely) to a rational number, we can follow these steps:
Let \( x = 0.8888\ldots \).
We can multiply both sides of the equation by 10 to shift the decimal point:
\[ 10x = 8.8888\ldots \]
Now we have two equations:
- \( x = 0.8888\ldots \)
- \( 10x = 8.8888\ldots \)
Next, we subtract the first equation from the second:
\[ 10x - x = 8.8888\ldots - 0.8888\ldots \]
This simplifies to:
\[ 9x = 8 \]
Now, solve for \( x \):
\[ x = \frac{8}{9} \]
Thus, the rational number equivalent of \( 0.8\overline{8} \) is \( \frac{8}{9} \).