Convert `(0.875)_(10)` to binary.

Answer

To convert the decimal number \(0.875\) from base \(10\) to binary, we can follow these steps:

### Step 1: Multiply by 2
Take the decimal number \(0.875\) and multiply it by \(2\):
\[ 0.875 \times 2 = 1.75 \]
**Hint:** When multiplying by \(2\), the integer part of the result will be the next binary digit.

### Step 2: Record the Integer Part
The integer part of \(1.75\) is \(1\). This will be the first digit after the binary point:
\[ \text{Binary digit: } 1 \]
Now, we take the fractional part \(0.75\) for the next step.

### Step 3: Multiply the Fractional Part by 2
Now, take the fractional part \(0.75\) and multiply it by \(2\):
\[ 0.75 \times 2 = 1.5 \]
**Hint:** Again, record the integer part of the result.

### Step 4: Record the Integer Part
The integer part of \(1.5\) is \(1\). This will be the second digit after the binary point:
\[ \text{Binary digit: } 1 \]
Now, take the fractional part \(0.5\) for the next step.

### Step 5: Multiply the Fractional Part by 2
Now, take the fractional part \(0.5\) and multiply it by \(2\):
\[ 0.5 \times 2 = 1.0 \]
**Hint:** Record the integer part again.

### Step 6: Record the Integer Part
The integer part of \(1.0\) is \(1\). This will be the third digit after the binary point:
\[ \text{Binary digit: } 1 \]
Now, we have reached a fractional part of \(0\), so we can stop here.

### Final Result
Combining all the binary digits we recorded, we have:
\[ 0.875_{10} = 0.111_2 \]

### Summary of Steps
1. Multiply \(0.875\) by \(2\) → \(1.75\) → Record \(1\)
2. Multiply \(0.75\) by \(2\) → \(1.5\) → Record \(1\)
3. Multiply \(0.5\) by \(2\) → \(1.0\) → Record \(1\)

Thus, the final binary representation of \(0.875\) in base \(2\) is \(0.111\).