Write the decimal representations of the following rational numbers:
`(3)/(7)`

Answer

To find the decimal representation of the rational number \( \frac{3}{7} \), we can follow these steps:

### Step 1: Set up the division
We need to divide 3 by 7. Since 3 is smaller than 7, we will add a decimal point and a zero to make it 30.

### Step 2: Perform the division
Now, we divide 30 by 7.
- 7 goes into 30 four times (since \( 7 \times 4 = 28 \)).
- Write down 4 after the decimal point.

### Step 3: Subtract and bring down the next zero
- Subtract 28 from 30, which gives us 2.
- Bring down another 0 to make it 20.

### Step 4: Continue the division
Now, divide 20 by 7.
- 7 goes into 20 two times (since \( 7 \times 2 = 14 \)).
- Write down 2 in the decimal representation.

### Step 5: Subtract and bring down the next zero
- Subtract 14 from 20, which gives us 6.
- Bring down another 0 to make it 60.

### Step 6: Repeat the division
Now, divide 60 by 7.
- 7 goes into 60 eight times (since \( 7 \times 8 = 56 \)).
- Write down 8 in the decimal representation.

### Step 7: Subtract and bring down the next zero
- Subtract 56 from 60, which gives us 4.
- Bring down another 0 to make it 40.

### Step 8: Continue the division
Now, divide 40 by 7.
- 7 goes into 40 five times (since \( 7 \times 5 = 35 \)).
- Write down 5 in the decimal representation.

### Step 9: Subtract and bring down the next zero
- Subtract 35 from 40, which gives us 5.
- Bring down another 0 to make it 50.

### Step 10: Continue the division
Now, divide 50 by 7.
- 7 goes into 50 seven times (since \( 7 \times 7 = 49 \)).
- Write down 7 in the decimal representation.

### Step 11: Subtract and bring down the next zero
- Subtract 49 from 50, which gives us 1.
- Bring down another 0 to make it 10.

### Step 12: Continue the division
Now, divide 10 by 7.
- 7 goes into 10 one time (since \( 7 \times 1 = 7 \)).
- Write down 1 in the decimal representation.

### Step 13: Subtract and bring down the next zero
- Subtract 7 from 10, which gives us 3.
- Bring down another 0 to make it 30.

### Step 14: Identify the repeating pattern
Notice that we are back to dividing 30 by 7 again, which means the digits will start repeating.

### Final Result
Thus, the decimal representation of \( \frac{3}{7} \) is:
\[ 0.428571428571... \]
This can be written as \( 0.\overline{428571} \), indicating that "428571" repeats indefinitely.

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