To convert the repeating decimal \( 0.888... \) into a fraction, we can use the following method:
Let \( x = 0.888... \).
Next, multiply both sides of the equation by 10 to shift the decimal point one place to the right:
\[ 10x = 8.888... \]
Now we have two equations:
- \( x = 0.888... \)
- \( 10x = 8.888... \)
Next, we can subtract the first equation from the second:
\[ 10x - x = 8.888... - 0.888... \]
This simplifies to:
\[ 9x = 8 \]
Now, to solve for \( x \), we divide both sides by 9:
\[ x = \frac{8}{9} \]
Thus, the repeating decimal \( 0.888... \) can be expressed as the rational number \( \frac{8}{9} \).