To convert the fraction \( \frac{7}{13} \) into a decimal, you can perform long division.
- Divide 7 by 13. Since 7 is less than 13, the quotient starts with 0.
- Add a decimal point and a zero, which gives you 70. Now divide 70 by 13, which goes 5 times (since \( 13 \times 5 = 65 \)).
- Subtract 65 from 70 to get a remainder of 5.
- Bring down another 0, making it 50. Dividing 50 by 13 goes 3 times (since \( 13 \times 3 = 39 \)).
- Subtract 39 from 50 to get a remainder of 11.
- Bring down another 0, making it 110. Dividing 110 by 13 goes 8 times (since \( 13 \times 8 = 104 \)).
- Subtract 104 from 110 to get a remainder of 6.
- Bring down another 0, making it 60. Dividing 60 by 13 goes 4 times (since \( 13 \times 4 = 52 \)).
- Subtract 52 from 60 to get a remainder of 8.
- Bring down another 0, making it 80. Dividing 80 by 13 goes 6 times (since \( 13 \times 6 = 78 \)).
- Subtract 78 from 80 to get a remainder of 2.
- Bring down another 0, making it 20. Dividing 20 by 13 goes 1 time (since \( 13 \times 1 = 13 \)).
- Subtract 13 from 20 to get a remainder of 7.
Notice that at this point, we have returned to a remainder of 7, which is where we started the division. This indicates that the decimal will start repeating.
Thus, the repeating decimal for \( \frac{7}{13} \) is:
\[ 0.\overline{538461} \]
So, \( \frac{7}{13} = 0.\overline{538461} \).