Convert `(38.21)_(10)` to its binary equivalent.

Answer

To convert the decimal number \( (38.21)_{10} \) to its binary equivalent, we need to handle the integer part (38) and the fractional part (0.21) separately.

### Step 1: Convert the Integer Part (38) to Binary

1. **Divide the integer by 2 and record the quotient and remainder**:
- \( 38 \div 2 = 19 \) remainder \( 0 \)
- \( 19 \div 2 = 9 \) remainder \( 1 \)
- \( 9 \div 2 = 4 \) remainder \( 1 \)
- \( 4 \div 2 = 2 \) remainder \( 0 \)
- \( 2 \div 2 = 1 \) remainder \( 0 \)
- \( 1 \div 2 = 0 \) remainder \( 1 \)

2. **Write the remainders in reverse order**:
- Reading the remainders from bottom to top gives us \( 100110 \).

Thus, the binary equivalent of \( 38 \) is \( (100110)_2 \).

### Step 2: Convert the Fractional Part (0.21) to Binary

1. **Multiply the fractional part by 2 and record the integer part**:
- \( 0.21 \times 2 = 0.42 \) → integer part \( 0 \)
- \( 0.42 \times 2 = 0.84 \) → integer part \( 0 \)
- \( 0.84 \times 2 = 1.68 \) → integer part \( 1 \)
- \( 0.68 \times 2 = 1.36 \) → integer part \( 1 \)
- \( 0.36 \times 2 = 0.72 \) → integer part \( 0 \)
- \( 0.72 \times 2 = 1.44 \) → integer part \( 1 \)
- \( 0.44 \times 2 = 0.88 \) → integer part \( 0 \)
- \( 0.88 \times 2 = 1.76 \) → integer part \( 1 \)
- \( 0.76 \times 2 = 1.52 \) → integer part \( 1 \)
- \( 0.52 \times 2 = 1.04 \) → integer part \( 1 \)
- \( 0.04 \times 2 = 0.08 \) → integer part \( 0 \)
- \( 0.08 \times 2 = 0.16 \) → integer part \( 0 \)
- \( 0.16 \times 2 = 0.32 \) → integer part \( 0 \)
- \( 0.32 \times 2 = 0.64 \) → integer part \( 0 \)
- \( 0.64 \times 2 = 1.28 \) → integer part \( 1 \)

2. **Stop when you reach a satisfactory precision or a repeating pattern**:
- The integer parts collected are \( 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1 \).

Thus, the binary equivalent of \( 0.21 \) is approximately \( (0.0011010111100001)_2 \).

### Final Step: Combine Both Parts

Combining both parts, we get:
\[ (38.21)_{10} = (100110.0011010111100001)_2 \]