By actual division, express of the following rational numbers as a repeating decimal.
`2/3`

Answer

To express the rational number \( \frac{2}{3} \) as a repeating decimal through actual division, follow these steps:

### Step-by-Step Solution:

1. **Set Up the Division**: We want to divide 2 by 3. Since 2 is less than 3, we cannot divide directly. We write it as \( 2.000 \) (adding decimal points and zeros).

2. **Perform the Division**:
- Divide 20 by 3.
- 3 goes into 20 six times (since \( 3 \times 6 = 18 \)).
- Write down 6 after the decimal point.

3. **Subtract**:
- Subtract 18 from 20, which gives us a remainder of 2 (i.e., \( 20 - 18 = 2 \)).

4. **Bring Down the Next Zero**:
- Bring down another 0, making it 20 again.

5. **Repeat the Division**:
- Again, divide 20 by 3.
- 3 goes into 20 six times (again, \( 3 \times 6 = 18 \)).
- Write down another 6.

6. **Subtract Again**:
- Subtract 18 from 20, which gives us a remainder of 2 once more.

7. **Continue the Process**:
- Each time you bring down a 0, the process repeats: you will always get 20, divide by 3 to get 6, and subtract to get a remainder of 2.

8. **Conclusion**:
- Since the remainder keeps repeating, we can conclude that \( \frac{2}{3} \) as a decimal is \( 0.666...\), where 6 is the repeating digit.

### Final Answer:
Thus, \( \frac{2}{3} = 0.\overline{6} \) (where the bar indicates that 6 repeats indefinitely).

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