Convert `(0.825)_(10)` to octal.

Answer

To convert the decimal number \(0.825_{10}\) to octal, we will follow a systematic approach by multiplying the decimal fraction by 8 and taking note of the integer parts at each step. Here’s a step-by-step breakdown of the conversion process:

### Step 1: Multiply by 8
Start with the decimal fraction \(0.825\) and multiply it by \(8\).

\[ 0.825 \times 8 = 6.6 \]

### Step 2: Record the Integer Part
The integer part of the result from Step 1 is \(6\). This will be the first digit after the octal point.

### Step 3: Take the Fractional Part
Now, take the fractional part \(0.6\) from the result and multiply it by \(8\).

\[ 0.6 \times 8 = 4.8 \]

### Step 4: Record the Integer Part
The integer part of this result is \(4\). This will be the second digit after the octal point.

### Step 5: Take the Fractional Part
Next, take the fractional part \(0.8\) and multiply it by \(8\).

\[ 0.8 \times 8 = 6.4 \]

### Step 6: Record the Integer Part
The integer part here is \(6\). This will be the third digit after the octal point.

### Step 7: Take the Fractional Part
Now, take the fractional part \(0.4\) and multiply it by \(8\).

\[ 0.4 \times 8 = 3.2 \]

### Step 8: Record the Integer Part
The integer part is \(3\). This will be the fourth digit after the octal point.

### Step 9: Take the Fractional Part
Next, take the fractional part \(0.2\) and multiply it by \(8\).

\[ 0.2 \times 8 = 1.6 \]

### Step 10: Record the Integer Part
The integer part is \(1\). This will be the fifth digit after the octal point.

### Step 11: Take the Fractional Part
Now, take the fractional part \(0.6\) and multiply it by \(8\) again.

\[ 0.6 \times 8 = 4.8 \]

### Step 12: Record the Integer Part
The integer part is \(4\). This will be the sixth digit after the octal point.

### Step 13: Continue the Process
You can see that the fractional part \(0.8\) will repeat the previous steps. Therefore, we can stop here and summarize the results.

### Final Result
Combining all the integer parts we recorded, we get:

\[ (0.646314)_{8} \]

Thus, the octal representation of \(0.825_{10}\) is approximately \(0.646314_{8}\).