To express \( \frac{7}{13} \) as a decimal, we can perform long division.
- Divide 7 by 13. Since 7 is less than 13, we start with 0. and add decimal places.
- Multiply by 10 to get 70. Divide 70 by 13, which gives 5 (since \( 13 \times 5 = 65 \)).
- Subtract 65 from 70 to get 5. Bring down another 0 to make it 50.
- Divide 50 by 13, which gives 3 (since \( 13 \times 3 = 39 \)).
- Subtract 39 from 50 to get 11. Bring down another 0 to make it 110.
- Divide 110 by 13, which gives 8 (since \( 13 \times 8 = 104 \)).
- Subtract 104 from 110 to get 6. Bring down another 0 to make it 60.
- Divide 60 by 13, which gives 4 (since \( 13 \times 4 = 52 \)).
- Subtract 52 from 60 to get 8. Bring down another 0 to make it 80.
- Divide 80 by 13, which gives 6 (since \( 13 \times 6 = 78 \)).
- Subtract 78 from 80 to get 2. Bring down another 0 to make it 20.
- Divide 20 by 13, which gives 1 (since \( 13 \times 1 = 13 \)).
- Subtract 13 from 20 to get 7. Bring down another 0 to make it 70.
At this point, we notice that we have returned to a previous remainder (5 from step 3), indicating that the digits will start repeating.
We summarize the long division results: \[ \frac{7}{13} = 0.538461538461... \] Thus, we can denote the repeating decimal as: \[ 0.\overline{538461} \]
In conclusion, \[ \frac{7}{13} = 0.\overline{538461} \] with the digits 538461 repeating indefinitely.