To express the fraction \(\frac{7}{13}\) as a repeating decimal, we can perform long division of 7 by 13.
- Divide 7 by 13. Since 7 is less than 13, we start by adding a decimal point and appending a zero to the 7, making it 70.
- \( 13 \) goes into \( 70 \) five times because \( 13 \times 5 = 65 \).
- Subtract \( 65 \) from \( 70 \) which gives \( 5 \). Bring down another zero, making it \( 50 \).
- \( 13 \) goes into \( 50 \) three times since \( 13 \times 3 = 39 \).
- Subtract \( 39 \) from \( 50 \) which gives \( 11 \). Bring down another zero, making it \( 110 \).
- \( 13 \) goes into \( 110 \) eight times since \( 13 \times 8 = 104 \).
- Subtract \( 104 \) from \( 110 \) which gives \( 6 \). Bring down another zero, making it \( 60 \).
- \( 13 \) goes into \( 60 \) four times since \( 13 \times 4 = 52 \).
- Subtract \( 52 \) from \( 60 \) which gives \( 8 \). Bring down another zero, making it \( 80 \).
- \( 13 \) goes into \( 80 \) six times since \( 13 \times 6 = 78 \).
- Subtract \( 78 \) from \( 80 \) which gives \( 2 \). Bring down another zero, making it \( 20 \).
- \( 13 \) goes into \( 20 \) once since \( 13 \times 1 = 13 \).
- Subtract \( 13 \) from \( 20 \) which gives \( 7 \). Bring down another zero, making it \( 70 \).
At this point, we can see we are back to \( 70 \), which we started with. This means the decimal will repeat the sequence we have found.
The decimal representation of \(\frac{7}{13}\) is hence:
\[ 0.\overline{538461} \]
So, expressed as a repeating decimal, \( \frac{7}{13} = 0.\overline{538461} \). The digits \( 538461 \) are the repeating digits, marked with a bar.