By actual division, express of the following rational numbers as a repeating decimal.
`13/11`

Answer

To express the rational number \( \frac{13}{11} \) as a repeating decimal through actual division, follow these steps:

### Step 1: Set up the division
We need to divide 13 by 11. We can write this as \( 13 \div 11 \).

### Step 2: Perform the division
1. **Determine how many times 11 fits into 13**:
- 11 goes into 13 one time (1).
- Write down 1 as the whole number part of the quotient.

2. **Subtract**:
- \( 13 - 11 = 2 \).
- Now we have a remainder of 2.

### Step 3: Continue with decimal division
1. **Add a decimal point and a zero**:
- Since 2 is less than 11, we add a decimal point to the quotient and a zero to the remainder, making it 20.

2. **Divide again**:
- 11 goes into 20 one time (1).
- Write down 1 after the decimal point.

3. **Subtract**:
- \( 20 - 11 = 9 \).
- Now we have a remainder of 9.

### Step 4: Repeat the process
1. **Add another zero**:
- Again, since 9 is less than 11, we add a zero, making it 90.

2. **Divide**:
- 11 goes into 90 eight times (8).
- Write down 8 after the decimal point.

3. **Subtract**:
- \( 90 - 88 = 2 \).
- Now we have a remainder of 2 again.

### Step 5: Identify the repeating pattern
1. **Notice the cycle**:
- We see that we are back to a remainder of 2, which we had earlier. This means the decimal will repeat the sequence we found.
- The decimal representation so far is \( 1.181818... \).

### Final Representation
We can express \( \frac{13}{11} \) as:
\[ 1.181818... = 1.18\overline{18} \]
This indicates that "18" is the repeating part of the decimal.