By actual division, express of the following rational numbers as a repeating decimal.
`22/7`

Answer

To express the rational number \( \frac{22}{7} \) as a repeating decimal through actual division, follow these steps:

### Step 1: Set up the division
We need to divide 22 by 7. We write it as:
\[ 22 \div 7 \]

### Step 2: Perform the division
1. **First Division**:
- 7 goes into 22 three times (since \( 7 \times 3 = 21 \)).
- Write down 3 and subtract \( 21 \) from \( 22 \):
\[ 22 - 21 = 1 \]
- So, we have \( 3 \) as the whole number part and a remainder of \( 1 \).

2. **Add Decimal Point**:
- Since we have a remainder, we add a decimal point and a zero to the remainder, making it \( 10 \).

3. **Second Division**:
- 7 goes into 10 once (since \( 7 \times 1 = 7 \)).
- Write down \( 1 \) and subtract \( 7 \) from \( 10 \):
\[ 10 - 7 = 3 \]

4. **Add Another Zero**:
- Add a zero to the remainder, making it \( 30 \).

5. **Third Division**:
- 7 goes into 30 four times (since \( 7 \times 4 = 28 \)).
- Write down \( 4 \) and subtract \( 28 \) from \( 30 \):
\[ 30 - 28 = 2 \]

6. **Add Another Zero**:
- Add a zero to the remainder, making it \( 20 \).

7. **Fourth Division**:
- 7 goes into 20 two times (since \( 7 \times 2 = 14 \)).
- Write down \( 2 \) and subtract \( 14 \) from \( 20 \):
\[ 20 - 14 = 6 \]

8. **Add Another Zero**:
- Add a zero to the remainder, making it \( 60 \).

9. **Fifth Division**:
- 7 goes into 60 eight times (since \( 7 \times 8 = 56 \)).
- Write down \( 8 \) and subtract \( 56 \) from \( 60 \):
\[ 60 - 56 = 4 \]

10. **Add Another Zero**:
- Add a zero to the remainder, making it \( 40 \).

11. **Sixth Division**:
- 7 goes into 40 five times (since \( 7 \times 5 = 35 \)).
- Write down \( 5 \) and subtract \( 35 \) from \( 40 \):
\[ 40 - 35 = 5 \]

12. **Add Another Zero**:
- Add a zero to the remainder, making it \( 50 \).

13. **Seventh Division**:
- 7 goes into 50 seven times (since \( 7 \times 7 = 49 \)).
- Write down \( 7 \) and subtract \( 49 \) from \( 50 \):
\[ 50 - 49 = 1 \]

### Step 3: Identify the repeating pattern
At this point, we notice that we are back to a remainder of \( 1 \), which is where we started after the whole number division. This indicates that the decimal will repeat from here.

### Final Result
Thus, the decimal representation of \( \frac{22}{7} \) is:
\[ 3.142857142857\ldots \]
or simply \( 3.\overline{142857} \), indicating that \( 142857 \) is the repeating part.