To convert the repeating decimal \( 0.18\overline{18} \) (which means the digits "18" repeat indefinitely) into a fraction, we can follow these steps:
Let: \[ x = 0.181818...\]
Next, we can multiply both sides of the equation by 100 (since the repeating block is two digits long): \[ 100x = 18.181818...\]
Now, we can subtract the first equation from the second: \[ 100x - x = 18.181818... - 0.181818...\] This simplifies to: \[ 99x = 18\]
Now, solve for \( x \): \[ x = \frac{18}{99}\]
Next, we simplify the fraction: \[ \frac{18}{99} = \frac{2}{11} \] (after dividing both the numerator and the denominator by 9).
Thus, \( 0.18\overline{18} \) as a fraction is: \[ \boxed{\frac{2}{11}} \]