To convert the fraction \( \frac{4}{11} \) into a repeating decimal, we can perform the long division of 4 by 11.
- Set up the division: 4.000000... divided by 11.
- \( 11 \) goes into \( 40 \) three times since \( 11 \times 3 = 33 \).
- Subtract \( 33 \) from \( 40 \) to get \( 7 \).
- Bring down the next zero, making it \( 70 \).
- \( 11 \) goes into \( 70 \) six times since \( 11 \times 6 = 66 \).
- Subtract \( 66 \) from \( 70 \) to get \( 4 \).
- Bring down another zero, making it \( 40 \) again.
- Repeat the steps: \( 11 \) goes into \( 40 \) three times, leaving \( 7 \).
- It will continue this way, alternating between \( 3 \) and \( 6 \).
Thus, the decimal representation of \( \frac{4}{11} \) is:
\[ 0.\overline{36} \]
This indicates that the digits \( 36 \) repeat indefinitely.