Express each of the following as a recurring decimal :
`2/13`

Answer

To express the fraction \( \frac{2}{13} \) as a recurring decimal, we will perform long division. Here’s a step-by-step solution:

### Step 1: Set up the division
We need to divide 2 by 13. Since 2 is less than 13, we will start by adding a decimal point and a zero to make it 20.

### Step 2: Divide 20 by 13
- 13 goes into 20 **1** time.
- Multiply: \( 1 \times 13 = 13 \).
- Subtract: \( 20 - 13 = 7 \).

### Step 3: Bring down the next zero
Now, we bring down another zero, making it 70.

### Step 4: Divide 70 by 13
- 13 goes into 70 **5** times.
- Multiply: \( 5 \times 13 = 65 \).
- Subtract: \( 70 - 65 = 5 \).

### Step 5: Bring down the next zero
Bring down another zero, making it 50.

### Step 6: Divide 50 by 13
- 13 goes into 50 **3** times.
- Multiply: \( 3 \times 13 = 39 \).
- Subtract: \( 50 - 39 = 11 \).

### Step 7: Bring down the next zero
Bring down another zero, making it 110.

### Step 8: Divide 110 by 13
- 13 goes into 110 **8** times.
- Multiply: \( 8 \times 13 = 104 \).
- Subtract: \( 110 - 104 = 6 \).

### Step 9: Bring down the next zero
Bring down another zero, making it 60.

### Step 10: Divide 60 by 13
- 13 goes into 60 **4** times.
- Multiply: \( 4 \times 13 = 52 \).
- Subtract: \( 60 - 52 = 8 \).

### Step 11: Bring down the next zero
Bring down another zero, making it 80.

### Step 12: Divide 80 by 13
- 13 goes into 80 **6** times.
- Multiply: \( 6 \times 13 = 78 \).
- Subtract: \( 80 - 78 = 2 \).

### Step 13: Bring down the next zero
Bring down another zero, making it 20 again.

### Step 14: Recognize the repeating pattern
At this point, we notice that we are back to dividing 20 by 13, which we already did in Step 2. This indicates that the decimal will start repeating.

### Final Result
Putting it all together, we find:
\[ \frac{2}{13} = 0.153846 \overline{153846} \]
The bar indicates that the digits 153846 repeat indefinitely.