If 2004.4375 is converted from base 10 to base 2, then find the number of digits in the binary expression including decimal and fractional parts.

Options

A. 14
B. 15
C. 17
D. 18

Answer

To convert the decimal number 2004.4375 from base 10 to base 2 and find the total number of digits in the binary representation, including both the integer and fractional parts, we can follow these steps:

### Step 1: Convert the Integer Part to Binary
1. **Identify the integer part**: The integer part of 2004.4375 is 2004.
2. **Divide by 2**: Start dividing the integer part by 2 and keep track of the quotient and remainder.
- 2004 ÷ 2 = 1002, remainder = 0
- 1002 ÷ 2 = 501, remainder = 0
- 501 ÷ 2 = 250, remainder = 1
- 250 ÷ 2 = 125, remainder = 0
- 125 ÷ 2 = 62, remainder = 1
- 62 ÷ 2 = 31, remainder = 0
- 31 ÷ 2 = 15, remainder = 1
- 15 ÷ 2 = 7, remainder = 1
- 7 ÷ 2 = 3, remainder = 1
- 3 ÷ 2 = 1, remainder = 1
- 1 ÷ 2 = 0, remainder = 1

3. **Write the binary representation**: Collect the remainders from bottom to top.
- The binary representation of 2004 is **11111010004**.

### Step 2: Convert the Fractional Part to Binary
1. **Identify the fractional part**: The fractional part of 2004.4375 is 0.4375.
2. **Multiply by 2**: Multiply the fractional part by 2 and take the integer part as the next binary digit.
- 0.4375 × 2 = 0.875 → integer part = 0
- 0.875 × 2 = 1.75 → integer part = 1
- 0.75 × 2 = 1.5 → integer part = 1
- 0.5 × 2 = 1.0 → integer part = 1

3. **Write the binary representation**: Collect the integer parts.
- The binary representation of 0.4375 is **.0111**.

### Step 3: Combine the Integer and Fractional Parts
- Combine the integer and fractional parts:
- The complete binary representation of 2004.4375 is **1111101000.0111**.

### Step 4: Count the Total Number of Digits
1. **Count digits**: Count the total number of digits in the binary representation including the decimal point.
- Integer part: 11 digits (1111101000)
- Fractional part: 4 digits (0111)
- Decimal point: 1 digit

Total digits = 11 (integer) + 4 (fractional) + 1 (decimal point) = **16 digits**.

### Final Answer
The total number of digits in the binary expression of 2004.4375 is **16**.

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